Highly rotating viscous compressible fluids in presence of capillarity effects

Abstract

We study here a singular limit problem for a Navier–Stokes–Korteweg system with Coriolis force, in the domain $\mathbb {R}^2\times \,]0,1[\,$ and for general ill-prepared initial data. Taking the Mach and the Rossby numbers proportional to a small parameter $\varepsilon \rightarrow 0$, we perform the incompressible and high rotation limits simultaneously; moreover, we consider both the constant and vanishing capillarity regimes. In this last case, the limit problem is identified as a 2-D incompressible Navier–Stokes equation in the variables orthogonal to the rotation axis; if the capillarity is constant, instead, the limit equation slightly changes, keeping however a similar structure, due to the presence of an additional surface tension term. In the vanishing capillarity regime, various rates at which the capillarity coefficient goes to 0 are considered: in general, this produces an anisotropic scaling in the system. The proof of the results is based on suitable applications of the RAGE theorem, combined with microlocal symmetrization arguments.

1 Introduction

Let us consider, in space dimension $d=3$, the Navier–Stokes–Korteweg system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\mathrm{div}\,\left( \rho \,u\right) =0 \\ \partial _t\left( \rho \,u\right) +\mathrm{div}\,(\rho \,u\otimes u)+\nabla P(\rho )+ e^3\times \rho u\,-\,\mathrm{div}\,(\mu (\rho )\,Du)\\ \quad -\kappa \,\rho \,\nabla (\sigma '(\rho )\,\Delta \sigma (\rho ))=0, \end{array}\right. } \end{aligned}$$

which describes the evolution of a compressible viscous fluid under the action both of the surface tension and of the Coriolis force.

In the previous system, the scalar function $\rho =\rho (t,x)\ge 0$ represents the density of the fluid, while $u=u(t,x)\in \mathbb {R}^3$ is its velocity field. The function $P(\rho )$ is the pressure of the fluid, and throughout this paper we will suppose it to be given by the Boyle law $P(\rho )=\rho ^\gamma /\gamma$, for some $1<\gamma \le 2$ (these conditions will be justified in Sect. 2). The positive function $\mu (\rho )$ represents the viscosity coefficient, the parameter $\kappa >0$ is the capillarity coefficient and the term $\sigma (\rho )\ge 0$ takes into account the surface tension. Here, we will always assume (we will motivate this choice below)

$$\begin{aligned} \mu (\rho )=\nu \,\rho \quad \text{ and } \quad \sigma (\rho )=\rho , \end{aligned}$$

for some fixed number $\nu >0$. Finally, the term

$$\begin{aligned} e^3\,\times \,\rho u\,:=\,(-\rho \,u^2,\,\rho \,u^1,\,0) \end{aligned}$$

represents the Coriolis force, which acts on the system due to the rotation of the Earth. Here we have supposed that the rotation axis is parallel to the $x^3$-axis and constant. Notice that this approximation is valid in regions which are very far from the equatorial zone and from the poles, and which are not too extended: in general, the dependence of the Coriolis force on the latitude should be taken into account (see e.g. works by Gallagher and Saint-Raymond [14–16]). On the other hand, for simplicity we are neglecting the effects of the centrifugal force.

Now, taking a small parameter $\varepsilon \in \,]0,1]$, we perform the scaling $t\mapsto \varepsilon t$, $u\mapsto \varepsilon u$, $\mu (\rho )\mapsto \varepsilon \mu (\rho )$ and we set $\kappa =\varepsilon ^{2\alpha }$, for some . Then, keeping in mind the assumptions we fixed above, we end up with the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho _\varepsilon +\mathrm{div}\,\left( \rho _\varepsilon u_\varepsilon \right) =0 \\ \partial _t\left( \rho _\varepsilon u_\varepsilon \right) +\mathrm{div}\,(\rho _\varepsilon u_\varepsilon \otimes u_\varepsilon ) \\ \quad +\dfrac{1}{\varepsilon ^2}\,\nabla P(\rho _\varepsilon )+ \dfrac{1}{\varepsilon }\,e^3\times \rho _\varepsilon u_\varepsilon -\nu \mathrm{div}\,(\rho _\varepsilon Du_\varepsilon )- \dfrac{1}{\varepsilon ^{2(1-\alpha )}}\,\rho _\varepsilon \nabla \Delta \rho _\varepsilon =0. \end{array}\right. } \end{aligned}$$

(1)

Notice that the previous scaling corresponds to supposing both the Mach number and the Rossby number to be proportional to $\varepsilon$ (see e.g. paper [19], or Chapter 4 of book [12]). We are interested in studying the asymptotic behavior of weak solutions to the previous system, in the regime of small $\varepsilon$, namely for $\varepsilon \rightarrow 0$. In particular, this means that we are performing the incompressible limit, the high rotation limit and, when $\alpha >0$, the vanishing capillarity limit simultaneously.

Many are the mathematical contributions to the study of the effects of fast rotation on fluid dynamics, under different assumptions (about e.g. incompressibility of the fluid, about the domain and the boundary conditions...). We refer e.g. to book [8] and the references therein for an extensive analysis of this problem for incompressible viscous fluids. In the context of compressible fluids there are, to our knowledge, few works, dealing with different models: among others that we are going to present more in detail below, we quote here [4, 16] as important contributions.

In the recent paper [11], Feireisl, Gallagher and Novotný studied the incompressible and high rotation limits together, for the 3-D compressible barotropic Navier–Stokes system with Coriolis force, in the general instance of ill-prepared initial data. Their asymptotic result relies on the spectral analysis of the singular perturbation operator: by use of the celebrated RAGE theorem (see e.g. books [9, 28]), they were able to prove some dispersion properties due to fast rotation, from which they deduced strong convergence of the velocity fields, and this allowed them to pass to the limit in the weak formulation of the system. In paper by Feireisl et al. [10], the effect of the centrifugal force was added to the previous system. Notice that this term scales as $1/\varepsilon ^2$; hence, they got interested in both the isotropic and multi-scale limit: namely, the Mach number was supposed to be proportional to $\varepsilon ^m$, for $m=1$ in the former instance (as in [11]), $m>1$ in the latter. Let us just point out that, in the analysis of the isotropic scaling (i.e. $m=1$), they had to resort to compensated compactness arguments (used for the first time in [15] in the context of rotating fluids) in order to pass to the limit in the weak formulation: as a matter of fact, the singular perturbation operator had variable coefficients, and spectral analysis tools were no more available. In both previous works [10, 11], it is proved that the limit system is a 2-D viscous quasi-geostrophic equation for the limit density (or, better, for the limit r of the quantities $r_\varepsilon =\varepsilon ^{-1}\left( \rho _\varepsilon -1\right)$), which can be interpreted as a sort of stream-function for the limit divergence-free velocity field. We remark also that the authors were able to establish stability (and then uniqueness) for the limit equation under an additional regularity hypothesis on the limit-point of the initial velocity fields.

The fact that the limit equation is two-dimensional is a common feature in the context of fast rotating fluids (see for instance book [8]): indeed, it is the expression of a well-known physical phenomenon, the Taylor–Proudman theorem (see e.g. [27]). Namely, in the asymptotic regime, the high rotation tends to stabilize the motion, which becomes constant in the direction parallel to the rotation axis: the fluid moves along vertical coloumns (the so called “Taylor–Proudman coloumns”), and the flow is purely horizontal.

Let us come back now to our problem for the Navier–Stokes–Korteweg system (1).

We point out that the general Navier–Stokes–Korteweg system, that we introduced at the beginning, has been widely studied (mostly with no Coriolis force), under various choices of the functions $\mu (\rho )$ and $\sigma (\rho )$: one can refer e.g. to papers [7, 18, 21]. In fact, this gives rise to many different models, which are relevant, for instance, in the context of quantum hydrodynamics.

For the previous system supplemented with our special assumptions and with no rotation term, in [5], Bresch, Desjardins and Lin proved the existence of global in time “weak” solutions. Actually, they had to resort to a modified notion of weak solution: indeed, any information on the velocity field u and its gradient is lost when the density vanishes; on the other hand, lower bounds for $\rho$ seem not to be available in the context of weak solutions. This makes it impossible to pass to the limit in the non-linear terms when constructing a solution to the system. The authors overcame such an obstruction choosing test functions whose support is concentrated on the set of positive density: namely, one has to evaluate the momentum equation not on a classical test function $\varphi$, but rather on $\rho \,\varphi$, and this leads to a slightly different weak formulation of the system (see also Definition 2.2 below). This modified formulation is made possible exploiting additional regularity for $\rho$, which is provided by the capillarity term: in fact, a fundamental issue in the analysis of [5] was the proof of the conservation of a second energy (besides the classical one) for this system, the so called BD entropy, which allows to control higher order space derivatives of the density term.

Still using this special energy conservation, in [2], Bresch and Desjardins were able to prove existence of global in time weak solutions (in the classical sense) for a 2-D viscous shallow water model in a periodic box (we refer also to [3] for the explicit construction of the sequence of approximate solutions). The system they considered there is very similar to the previous one, but it presents two additional friction terms: a laminar friction and a turbolent friction. The latter plays a similar role to the capillarity (one does not need both to prove compactness properties for the sequence of smooth solutions), while the former gives integrability properties on the velocity field u: this is why they did not need to deal with the modified notion of weak solutions. In the same work [2], the authors were able to consider also the fast rotation limit in the instance of well-prepared initial data, and to prove the convergence to the viscous quasi-geostrophic equations (as mentioned above for papers [10, 11]). Let us point out that their argument in passing to the limit relies on the use of the modulated energy method.

The recent paper [19], by Jüngel, Lin and Wu, deals with a very similar problem: namely, incompressible and high rotation limit in the two dimensional torus $\mathbb {T}^2$ for well-prepared initial data, but combined also with a vanishing capillarity limit (more specifically, like in system (1), with ). On the one hand, the authors were able to treat more general forms of the Navier–Stokes–Korteweg system, with different functions $\mu (\rho )$ and $\sigma (\rho )$; on the other hand, for doing this they had to work in the framework of (local in time) strong solutions. Again by use of the modulated energy method, they proved the convergence of the previous system to the viscous quasi-geostrophic equation: as a matter of fact, due to the vanishing capillarity regime, no surface tension terms enter into the singular perturbation operator, and the limit system is the same as in works [2, 10, 11].

In the present paper, we consider system (1) in the infinite slab $\Omega =\mathbb {R}^2\times \,]0,1[\,$, supplemented with complete slip boundary conditions, and with ill-prepared initial data. Our goal is to study the asymptotic behavior of weak solutions in the regime of low Mach number and low Rossby number, possibly combining these effects with the vanishing capillarity limit.

We stress here the following facts. First of all, we do not deal with general viscosity and surface tension functions: we fix both $\mu (\rho )$ and $\sigma (\rho )$ as specified above. Second point: we consider a 3-D domain, but we impose complete slip boundary conditions, in order to avoid boundary layers effects, which we will not treat in this paper. Finally, our analysis relies on the techniques used in [11], and this allows us to deal with general ill-prepared initial data, i.e. initial densities $(\rho _{0,\varepsilon })_\varepsilon$ (of the form $\rho _{0,\varepsilon }=1+\varepsilon r_{0,\varepsilon }$) and initial velocities $(u_{0,\varepsilon })_\varepsilon$, both bounded in suitable spaces, which do not necessarily belong to the kernel of the singular perturbation operator.

For any fixed $\varepsilon \in \,]0,1]$, the existence of global in time “weak” solutions to our system can be proved in the same way as in [5] (see the discussion in Sect. 2.2 for more details). As a matter of fact, energy methods still work, due to the skew-symmetry of the Coriolis operator; moreover, a control of the rotation term (but not uniformly in $\varepsilon$) is easy to get in the BD entropy estimates: this guarantees additional regularity for the density, and the possibility to prove convergence of the sequence of smooth solutions to a weak one. However, a uniform control on the higher order derivatives of the densities is fundamental in our study, in order to pass to the limit for $\varepsilon \rightarrow 0$. It can be obtained (see Sect. 3.2) arguing like in the proof of the BD entropy estimates, but showing a uniform bound for the rotation term. This is the first delicate point of our analysis: the problem comes from the fact that we have no control on the velocity fields $u_\varepsilon$. For the same reason, we resort to the notion of “weak” solution developed by Bresch, Desjardins and Lin (see Definition 2.2); notice that this leads to handle new non-linear terms arising in the modified weak formulation of the momentum equation.

Let us spend now a few words on the limit system (see the explicit expression in Theorems 2.4 and 2.6), which will be studied in Sect. 3.3. Both for vanishing and constant capillarity regimes, we still find that the limit velocity field u is divergence-free and horizontal and depends just on the horizontal variables (in accordance with the Taylor–Proudman theorem); moreover, a relation links u to the limit density r, which can still be seen as a sort of stream-function for u. In the instance of constant capillarity, this relation slightly changes, giving rise to a more complicated equation for r (compare Eqs. (11) and (12) below): indeed, a surface tension term enters into play in the singular perturbation operator, and hence in the limit equation.

We first focus on the vanishing capillarity case, and more precisely on the choice $\alpha =1$ (treated in Sect. 4). Formally, the situation is analogous to the one of paper [11], and the analysis of Feireisl et al. still applies, after handling some technical points in order to adapt their arguments to the modified weak formulation. The main issue is the analysis of acoustic waves: also in this case, we can apply the RAGE theorem to prove dispersion of the components which are in the subspace orthogonal to the kernel of the singular perturbation operator. This is the key point in order to pass to the limit in the non-linear terms, and to get convergence. In the end we find (as in [11]) that the weak solutions of our system tend to a weak solution of a 2-D quasi-geostrophic equation (see (11) below).

Let us now turn our attention to the case of constant capillarity, i.e. $\alpha =0$ (see Sect. 5). Here, the capillarity term scales as $1/\varepsilon ^2$, so it is of the same order as the pressure and the rotation operator. As a consequence, the singular perturbation operator (say) $\mathcal {A}_0$ presents an additional term, and it is no more skew-adjoint with respect to the usual $L^2$ scalar product. However, on the one hand $\mathcal {A}_0$ has still constant coefficients, so that spectral analysis is well-adapted: direct computations show that the point spectrum still coincides with the kernel of the wave propagator, as in the previous case $\alpha =1$. On the other hand, passing in Fourier variables it is easy to find a microlocal symmetrizer (in the sense of Métivier, see [24]) for our system, i.e. a scalar product with respect to which $\mathcal {A}_0$ is still skew-adjoint: this allows us to apply again the RAGE theorem with respect to the new scalar product, and to recover the convergence result by the same techniques as above. We remark that we use here the additional regularity for the density (the new inner product involves two space derivatives for r). We also point out that, for $\alpha =0$, the limit equation becomes (12), which presents a similar structure to the one of (11), but where new terms appear.

The case (see Sect. 6) is technically more complicated, because this choice introduces an anisotropy of scaling in the system for acoustic waves. We will need to treat this anisotropy as a perturbation term in the acoustic propagator: also in this case we can resort to spectral analysis methods, and we can symmetrize our system, but now both the acoustic propagator and the microlocal symmetrizer depend on $\varepsilon$, via the perturbation term. So, we first need to prove a RAGE-type theorem (see Theorem 6.8) for families of operators and symmetrizers: the main efforts in the analysis are devoted to this. Then, the proof of the convergence can be performed exactly as in the previous cases: again, the limit system is identified as the 2-D quasi-geostrophic equation (11).

Before going on, let us present the organization of the paper.

In the next section we collect our assumptions, and we state our main results. Besides, we recall the definition of weak solution we adopt in the sequel and we spend a few words about the existence theory. Some requirements imposed in this definition are justified by a priori bounds, which we show in Sect. 3; there, we also identify the weak limits u and r and establish some of their properties. Section 4 is devoted to the proof of the result for $\alpha =1$, while in Sect. 5 we deal with the case $\alpha =0$. The anisotropic scaling is treated in Sect. 6. Finally, we collect in the appendix some auxiliary results from Littlewood–Paley theory.

2 Main hypotheses and results

This section is devoted to present our main results. First of all, we collect our working assumptions. Then, we give the definition of weak solutions we will adopt throughout all the paper, and which is based on the one of [5]. Finally, we state our main theorems about the asymptotic limit.

2.1 Working setting and main assumptions

We fix the infinite slab

$$\begin{aligned} \Omega =\mathbb {R}^2\,\times \,\,]0,1[ \end{aligned}$$

and we consider in $\mathbb {R}_+\times \Omega$ the scaled Navier–Stokes–Korteweg system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\mathrm{div}\,\left( \rho u\right) =0 \\ \partial _t\left( \rho u\right) +\mathrm{div}\,(\rho u\otimes u)+\dfrac{1}{\varepsilon ^2}\,\nabla P(\rho )+ \dfrac{1}{\varepsilon }\,e^3\times \rho u-\nu \mathrm{div}\,(\rho Du)\\ \quad -\dfrac{1}{\varepsilon ^{2(1-\alpha )}}\,\rho \nabla \Delta \rho =0, \end{array}\right. } \end{aligned}$$

(2)

where $\nu >0$ denotes the viscosity of the fluid, D is the viscous stress tensor defined by

$$\begin{aligned} Du\,:=\,\frac{1}{2}\,(\nabla u+^t\nabla u), \end{aligned}$$

$e^3=(0,0,1)$ is the unit vector directed along the $x^3$-coordinate, and is a fixed parameter. Taking different values of $\alpha$, we are interested in performing a low capillarity limit (for ), with capillarity coefficient proportional to $\varepsilon ^{2\alpha }$, or in leaving the capillarity constant (i.e. choosing $\alpha =0$).

We supplement system (2) by complete slip boundary conditions for u and Neumann boundary conditions for $\rho$: this allows us to avoid boundary layers effects, which go beyond the scopes of the present paper and will not be discussed here. If we denote by n the unitary outward normal to the boundary $\partial \Omega$ of the domain (simply, $\partial \Omega =\{x^3=0\}\cup \{x^3=1\}$), we impose

$$\begin{aligned} \left( u\cdot n\right) _{|\partial \Omega }=u^3_{|\partial \Omega }=0,\quad \left( \nabla \rho \cdot n\right) _{|\partial \Omega }=\partial _3\rho _{|\partial \Omega }=0,\quad ((Du)n\times n)_{|\partial \Omega }=0. \end{aligned}$$

(3)

In the previous system (2), the scalar function $\rho \ge 0$ represents the density of the fluid, $u\in \mathbb {R}^3$ its velocity field, and $P(\rho )$ its pressure, given by the $\gamma$-law

$$\begin{aligned} P(\rho )\,:=\,\frac{1}{\gamma }\,\rho ^\gamma ,\quad \text{ for } \text{ some } \quad 1\,<\,\gamma \,\le \,2. \end{aligned}$$

(4)

The requirement $\gamma \le 2$ is motivated by the fact that we need, roughly speaking, $\rho -1$ of finite energy, both for the existence theory and the asymptotic limit. In fact, by point (i) of Lemma 7.3 in the Appendix, when $\gamma >2$ we miss the property $\rho -1\,\in \,L^\infty _T(L^2)$ (recall that the pressure term gives informations on the low frequencies of the density). About the existence theory, notice that no problems of this type appeared in [5], since the considered domain was bounded; concerning the singular limit analysis, we refer to estimate (15) and Remark 3.4.

Remark 2.1

Let us point out here that Eqs. (2), supplemented by boundary conditions (3), can be recast as a periodic problem with respect to the vertical variable, in the new domain

$$\begin{aligned} \Omega =\mathbb {R}^2\,\times \,\mathbb {T}^1,\quad \text{ with } \quad \mathbb {T}^1\,:=\,[-1,1]/\sim , \end{aligned}$$

where $\sim$ denotes the equivalence relation which identifies $-1$ and 1. As a matter of fact, it is enough to extend $\rho$ and $u^h$ as even functions with respect to $x^3$, and $u^3$ as an odd function: the equations are invariant under such a transformation.

In what follows, we will always assume that such modifications have been performed on the initial data, and that the respective solutions keep the same symmetry properties.

Here we will consider the general instance of ill-prepared initial data $(\rho ,u)_{|t=0}=(\rho _{0,\varepsilon },u_{0,\varepsilon })$. Namely, we will suppose the following on the family $(\rho _{0,\varepsilon },\,u_{0,\varepsilon })_{\varepsilon >0}$:

• (i) $\rho _{0,\varepsilon }=1+\varepsilon \,r_{0,\varepsilon }$, with $(r_{0,\varepsilon })_\varepsilon \,\subset \,H^1(\Omega )\cap L^\infty (\Omega )$ bounded;

• (ii) $(u_{0,\varepsilon })_\varepsilon \,\subset \,L^2(\Omega )$ bounded.

Up to extraction of a subsequence, we can suppose that

$$\begin{aligned} r_{0,\varepsilon }\,\rightharpoonup \,r_0\quad \text{ in } \;H^1(\Omega )\quad \text{ and } \quad u_{0,\varepsilon }\,\rightharpoonup \,u_0\quad \text{ in } \;L^2(\Omega ), \end{aligned}$$

(5)

where we denoted by $\rightharpoonup$ the weak convergence in the respective spaces.

2.2 Weak solutions

In the present paragraph, we define the notion of weak solution for our system: it is based on the one given in [5]. Essentially, we need to localize the equations on sets where $\rho$ is positive (see also the discussion in the Sect. 1): this is achieved, formally, by testing the momentum equation on functions of the form $\rho \psi$, where $\psi \in \mathcal {D}$ is a classical test function.

First of all, we introduce the internal energy, i.e. the scalar function $h=h(\rho )$ such that

$$\begin{aligned} h''(\rho )=\frac{P'(\rho )}{\rho }=\rho ^{\gamma -2}\quad \text{ and } \quad h(1)=h'(1)=0\,; \end{aligned}$$

let us define then the energies

$$\begin{aligned} E_\varepsilon [\rho ,u](t):= & {} \int _\Omega \left( \dfrac{1}{\varepsilon ^2}\,h(\rho )+\dfrac{1}{2}\,\rho \,|u|^2+ \dfrac{1}{2\,\varepsilon ^{2(1-\alpha )}}\,|\nabla \rho |^2\right) dx \end{aligned}$$

(6)

$$\begin{aligned} F_\varepsilon [\rho ](t):= & {} \frac{\nu ^2}{2}\int _\Omega \rho \,|\nabla \log \rho |^2\,dx\;=\; 2\,\nu ^2\int _\Omega \left| \nabla \sqrt{\rho }\right| ^2\,dx. \end{aligned}$$

(7)

We will denote by $E_\varepsilon [\rho _0,u_0]\,\equiv \,E_\varepsilon [\rho ,u](0)$ and by $F_\varepsilon [\rho _0]\,\equiv \,F_\varepsilon [\rho ](0)$ the same quantities, when computed on the initial data $(\rho _0,u_0)$.

Here we present the definition. The integrability properties required in points (i)–(ii)–(v) will be justified by the computations of Sects. 3.1 and 3.2.

Definition 2.2

Fix $(\rho _0,u_0)$ such that $E_\varepsilon [\rho _0,u_0]+F_\varepsilon [\rho _{0}]\,<\,+\infty$.

We say that $(\rho ,u)$ is a weak solution to system (2)–(3) in $[0,T[\,\times \Omega$ (for some $T>0$) with initial data $(\rho _0,u_0)$ if the following conditions are fulfilled:

• (i) $\rho \ge 0$ almost everywhere, $\rho -1\in L^\infty ([0,T[\,;L^\gamma (\Omega ))$, $\nabla \rho$ and $\nabla \sqrt{\rho }\;\in L^\infty ([0,T[\,;L^2(\Omega ))$ and $\nabla ^2\rho \in L^2([0,T[\,;L^2(\Omega ))$;

• (ii) $\sqrt{\rho }\,u\,\in L^\infty ([0,T[\,;L^2(\Omega ))$ and $\sqrt{\rho }\,Du\,\in L^2([0,T[\,;L^2(\Omega ))$;

• (iii) the mass equation is satisfied in the weak sense: for any $\phi \in \mathcal {D}([0,T[\,\times \Omega )$ one has

  $$\begin{aligned} -\int ^T_0\int _\Omega (\rho \,\partial _t\phi +\rho \,u\,\cdot \,\nabla \phi )\,dx\,dt=\int _\Omega \rho _0\,\phi (0)\,dx; \end{aligned}$$

  (8)

• (iv) the momentum equation is verified in the following sense: for any $\psi \in \mathcal {D}([0,T[\times \Omega )$ one has

  $$\begin{aligned}&\int _\Omega \rho _0^2u_0\cdot \psi (0)\,dx\!=\!\int ^T_0\!\!\int _\Omega \bigg (\!-\!\rho ^2u\cdot \partial _t\psi \,-\,\rho u\otimes \rho u:\nabla \psi \!+\! \rho ^2\left( u\cdot \psi \right) \mathrm{div}\,u \nonumber \\&\quad -\,\frac{\gamma }{\varepsilon ^2(\gamma +1)}\,P(\rho )\rho \,\mathrm{div}\,\psi +\frac{1}{\varepsilon }\,e^3\times \rho ^2u\cdot \psi + \nu \rho Du:\rho \nabla \psi \,+ \nonumber \\&\quad +\nu \rho Du:\left( \psi \otimes \nabla \rho \right) \!+\!\frac{1}{\varepsilon ^{2(1-\alpha )}}\,\rho ^2\Delta \rho \,\mathrm{div}\,\psi \!+\! \frac{2}{\varepsilon ^{2(1-\alpha )}}\,\rho \Delta \rho \nabla \rho \cdot \psi \bigg )\,dx\,dt;\nonumber \\ \end{aligned}$$

  (9)

• (v) for almost every $t\in \,]0,T[\,$, the following energy inequalities hold true:

  $$\begin{aligned}&E_\varepsilon [\rho ,u](t)+\nu \int ^t_0\int _\Omega \rho \,\left| Du\right| ^2\,dx\,d\tau \le E_\varepsilon [\rho _0,u_0] \\&F_\varepsilon [\rho ](t)+\dfrac{\nu }{\varepsilon ^2}\int ^t_0\int _\Omega P'(\rho )\,|\nabla \sqrt{\rho }|^2\,dx\,d\tau \\&\quad \qquad \qquad +\dfrac{\nu }{\varepsilon ^{2(1-\alpha )}}\int ^t_0\int _\Omega \left| \nabla ^2\rho \right| ^2\,dx\,d\tau \le C\,(1+T), \end{aligned}$$

  for some constant C depending just on $(E_\varepsilon [\rho _{0},u_0],F_\varepsilon [\rho _{0}],\nu )$.

Remark 2.3

Notice that, under our hypotheses (recall points (i)–(ii) in Sect. 2.1), the energies of the initial data are uniformly bounded with respect to $\varepsilon$:

$$\begin{aligned} E_\varepsilon [\rho _{0,\varepsilon },u_{0,\varepsilon }]+F_\varepsilon [\rho _{0,\varepsilon }]\,\le \,K_0, \end{aligned}$$

where the constant $K_0>0$ is independent of $\varepsilon$.

Let us spend a few words on existence of weak solutions to our equations.

Existence of weak solutions. We sketch here the proof of existence of global in time weak solutions to our system, in the sense specified by Definition 2.2. The main point is the construction of a sequence of smooth approximate solutions, which respect the energy and BD entropy estimates we will establish in Propositions 3.1, 3.2 and 3.3. This can be done arguing as in paper [3] by Bresch and Desjardins (see also [25, 26]): namely we have to introduce regularizing terms, both for u and $\rho$, in the momentum equations, depending on small parameters that we will let vanish in a second moment.

Before presenting the details, let us remark that, in [3], the construction is performed for a shallow water system with two additional drag terms: a laminar friction and a turbulent friction. As remarked there, the former item is needed to have integrability properties for the velocity field, while the latter plays a role analogous to capillarity in stability analysis, and one do not need both of them to recover existence of weak solutions. In any case, precisely due to the laminar friction (which could be also replaced by a so-called “cold pressure” term), one can get back to the classical notion of weak solutions; on the contrary, in our case, since we miss integrability properties for u, we have to resort to the modified notion of Definition 2.2, the problem relying in proving compactness of the sequence of approximate solutions.

We also point out that this construction works in the simplest case when the density dependent viscosity coefficient and the surface tension term are taken linear in $\rho$. For the general equations written at the beginning of the Sect. 1, the explicit construction of smooth approximate solutions, which respect the BD entropy structure of the system, is much more complicated, and deep advances have been obtained just recently. In the matter of this, we refer e.g. to works [6, 22, 29, 30].

Let us come back to the explicit construction. The parameter $\varepsilon >0$ is kept fixed at this level: for any $\varepsilon$, we want to construct a weak solution $(\rho _\varepsilon ,u_\varepsilon )$ to system (2). Let us take two additional small parameters $\delta >0$ and $\eta >0$, and let us consider the system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\rho +\mathrm{div}\,\left( \rho u\right) =0 \\ \partial _t\left( \rho u\right) +\mathrm{div}\,(\rho u\otimes u)+\dfrac{1}{\varepsilon ^2}\,\nabla P(\rho )+ \dfrac{1}{\varepsilon }\,e^3\times \rho u- \\ \quad -\nu \mathrm{div}\,(\rho Du)-\dfrac{1}{\varepsilon ^{2(1-\alpha )}}\,\rho \nabla \Delta \rho = \delta \rho \nabla \Delta ^{2s+1}\rho -\delta \nabla \pi (\rho )-\eta \Delta ^2u. \end{array}\right. } \end{aligned}$$

(10)

The hypercapillarity $\rho \nabla \Delta ^{2s+1}\rho$ (for some $s>2$ large enough) is needed to smooth out the density function; the artificial “cold pressure” $\pi (\rho )=-\rho ^{-3}$ serves to keep it bounded away from 0. Finally, the hyperviscosity term $\Delta ^2u$ helps to regularize the velocity field.

Therefore, for any fixed positive $\delta$ and $\eta$, system (10) becomes parabolic in u, and then it admits a unique global solution $(\rho _{\varepsilon ,\delta ,\eta },\,u_{\varepsilon ,\delta ,\eta })$. Furthermore, as observed in [3], the new terms are compatible with the mathematical structure of the original equations, so that one can find uniform (with respect to $(\delta ,\eta )$ parameters) energy and BD entropy estimates. For the explicit estimates and a proof of them, we refer to Sects. 3.1 and 3.2.

Now, the first step is to pass to the limit for $\eta \rightarrow 0$: this can be done as in Subsection 2.3 of [3]. Notice that we can still recover compactness of the velocity fields thanks to the uniform (in $\eta$ but not in $\delta$) upper and lower bounds for $\rho$. In this way, we get a solution $(\rho _{\varepsilon ,\delta },\,u_{\varepsilon ,\delta })$ to Eq. (10) supplemented with the choice $\eta =0$. Then, the final step consists in passing to the limit also for $\delta \rightarrow 0$: as remarked above, we have to work with the modified weak formulation of Definition 2.2, for which the stability analysis performed in [5] applies. We refer to [3, 25] for more details.

In the end we have proved that, under the hypotheses fixed in Sect. 2.1, for any initial datum $(\rho _{0,\varepsilon },u_{0,\varepsilon })$ there exists a global weak solution $(\rho _\varepsilon ,u_\varepsilon )$ to problem (2)–(3) in $\mathbb {R}_+\times \Omega$, in the sense specified by Definition 2.2 above.

2.3 Main results

We are interested in studying the asymptotic behaviour of a family of weak solutions $(\rho _\varepsilon ,\,u_\varepsilon )_\varepsilon$ to system (2) for the parameter $\varepsilon \rightarrow 0$. As we will see in a while (see Theorems 2.4 and 2.6), one of the main features is that the limit-flow will be two-dimensional and horizontal along the plane orthogonal to the rotation axis.

Then, let us introduce some notations to describe better this phenomenon. We will always decompose $x\in \Omega$ into $x=(x^h,x^3)$, with $x^h\in \mathbb {R}^2$ denoting its horizontal component. Analogously, for a vector-field $v=(v^1,v^2,v^3)\in \mathbb {R}^3$ we set $v^h=(v^1,v^2)$, and we define the differential operators $\nabla _h$ and $\mathrm{div}\,_{\!h}$ as the usual operators, but acting just with respect to $x^h$. Finally, we define the operator $\nabla ^\perp _h\,:=\,(-\partial _2,\,\partial _1)$.

We can now state our main result in the vanishing capillarity case. The particular choice of the pressure law, i.e. $\gamma =2$, will be commented in Remark 3.4.

Theorem 2.4

Let us take in (2) and $\gamma =2$ in (4).

Let $(\rho _\varepsilon ,\,u_\varepsilon )_\varepsilon$ be a family of weak solutions to system (2)–(3) in $[0,T]\times \Omega$, in the sense of Definition 2.2, related to initial data $(\rho _{0,\varepsilon },u_{0,\varepsilon })_\varepsilon$ satisfying the hypotheses (i)–(ii) and (5), and the symmetry assumptions of Remark 2.1. Let us define the scalar quantity $r_\varepsilon \,:=\,\varepsilon ^{-1}(\rho _\varepsilon -1)$.

Then, up to the extraction of a subsequence, one has the following properties:

• (a) $r_\varepsilon \,\rightharpoonup \,r$ in $L^\infty ([0,T];L^2(\Omega ))\,\cap \,L^2([0,T];H^1(\Omega ))$;

• (b) $\sqrt{\rho _\varepsilon }\,u_\varepsilon \,\rightharpoonup \,u$ in $L^\infty ([0,T];L^2(\Omega ))$ and $\sqrt{\rho _\varepsilon }\,Du_\varepsilon \rightharpoonup Du$ in $L^2([0,T];L^2(\Omega ))$;

• (c) $r_\varepsilon \,\rightarrow \,r$ and $\rho _\varepsilon ^{3/2}\,u_\varepsilon \,\rightarrow \,u$ (strong convergence) in $L^2([0,T];L^2_{loc}(\Omega ))$,

where $r=r(x^h)$ and $u=(u^h(x^h),0)$ are linked by the relation $u^h=\nabla ^\perp _hr$. Moreover, r satisfies (in the weak sense) the quasi-geostrophic type equation

$$\begin{aligned} \partial _t(r\,-\,\Delta _hr)+\nabla ^\perp _hr\,\cdot \,\nabla _h\Delta _hr+\frac{\nu }{2}\,\Delta _h^2r=0 \end{aligned}$$

(11)

supplemented with the initial condition $r_{|t=0}=\overline{r}_0$, where $\overline{r}_0\,\in \,H^1(\mathbb {R}^2)$ is the unique solution of

$$\begin{aligned} (\mathrm{Id}\,-\Delta _h)\,\overline{r}_0=\int _0^1(\omega ^3_0+r_0)\,dx^3, \end{aligned}$$

with $r_0$ and $u_0$ which are defined in (5), $\omega _0=\nabla \times u_0$ which is the vorticity of $u_0$ and $\omega _0^3$ its third component.

Remark 2.5

Let us point out that, the limit equation being the same, the uniqueness criterion given in Theorem 1.3 of [11] still holds true under our hypothesis.

Then, if $\omega ^3_0\in L^2(\Omega )$, the solution r to Eq. (11) is uniquely determined by the initial condition in the subspace of distributions such that $\nabla _hr\in L^\infty (\mathbb {R}_+;H^1(\mathbb {R}^2))\,\cap \,L^2(\mathbb {R}_+;\dot{H}^2(\mathbb {R}^2))$, and the whole sequence of weak solutions converges to it.

Let us now turn our attention to the case $\alpha =0$, i.e. when the capillarity coefficient is taken to be constant.

Theorem 2.6

Let us take $\alpha =0$ in (2) and $1<\gamma \le 2$ in (4).

Let $(\rho _\varepsilon ,\,u_\varepsilon )_\varepsilon$ be a family of weak solutions to system (2)–(3) in $[0,T]\times \Omega$, in the sense of Definition 2.2, related to initial data $(\rho _{0,\varepsilon },u_{0,\varepsilon })_\varepsilon$ satisfying the hypotheses $\mathrm{(i)-(ii)}$ and (5), and the symmetry assumptions of Remark 2.1. We define $r_\varepsilon \,:=\,\varepsilon ^{-1}\left( \rho _\varepsilon -1\right)$, as before.

Then, up to the extraction of a subsequence, one has the convergence properties

• (a*) $r_\varepsilon \,\rightharpoonup \,r$ in $L^\infty ([0,T];H^1(\Omega ))\,\cap \,L^2([0,T];H^2(\Omega ))$

and the same (b) and (c) stated in Theorem 2.4, where, this time, $r=r(x^h)$ and $u=(u^h(x^h),0)$ are linked by the relation $u^h=\nabla ^\perp _h\left( \mathrm{Id}\,-\Delta _h\right) r$. Moreover, r solves(in the weak sense) the modified quasi-geostrophic equation

$$\begin{aligned} \partial _t(r\,-\,\Delta _hr+\Delta _h^2r)+\nabla ^\perp _h(\mathrm{Id}\,-\Delta _h)r\,\cdot \,\nabla _h\Delta _h^2r+ \frac{\nu }{2}\,\Delta _h^2(\mathrm{Id}\,-\Delta _h)r=0 \end{aligned}$$

(12)

supplemented with the initial condition $r_{|t=0}=\widetilde{r}_0$, where $\widetilde{r}_0\,\in \,H^3(\mathbb {R}^2)$ is the unique solution of

$$\begin{aligned} (\mathrm{Id}\,-\Delta _h+\Delta _h^2)\,\widetilde{r}_0=\int _0^1(\omega ^3_0+r_0)\,dx^3. \end{aligned}$$

3 Preliminaries and uniform bounds

The present section is devoted to stating the main properties of the family of weak solutions of Theorems 2.4 and 2.6.

First of all, we prove energy and BD entropy estimates for our system, uniformly with respect to the parameter $\varepsilon$. This will justify the properties required in Definition 2.2. From them, we will infer additional uniform bounds and further properties the family $(\rho _\varepsilon ,u_\varepsilon )_\varepsilon$ enjoys. Finally, we will derive some constraints on its weak limit.

3.1 Energy and BD entropy estimates

Suppose that $(\rho ,u)$ is a smooth solution to system (2) in $[0,T[\,\times \Omega$ (for some $T>0$), related to the smooth initial datum $(\rho _{0},u_{0})$.

The first energy estimate, involving $E_\varepsilon$, is obtained in a standard way.

Proposition 3.1

Let $(\rho ,u)$ be a smooth solution to system (2) in $[0,T[\,\times \Omega$, with initial datum $(\rho _0,u_0)$, for some positive time $T>0$.

Then, for all $\varepsilon >0$ and all $t\in [0,T[\,$, one has

$$\begin{aligned} \frac{d}{dt}E_\varepsilon [\rho ,u]+\nu \int _\Omega \rho \,|Du|^2\,dx=0. \end{aligned}$$

Proof

First of all, we multiply the second relation in system (2) by u: by use of the mass equation and due to the fact that $e^3\times \rho u$ is orthogonal to u, we arrive at the identity:

$$\begin{aligned}&\frac{1}{2}\,\frac{d}{dt}\int _\Omega \rho \,|u|^2\,dx+ \frac{1}{\varepsilon ^2}\int _\Omega P'(\rho )\,\nabla \rho \cdot u\,dx\\&\quad +\nu \,\int _\Omega \rho \,Du\,:\,\nabla u\,dx+ \frac{1}{2\,\varepsilon ^{2(1-\alpha )}}\,\frac{d}{dt}\int _\Omega |\nabla \rho |^2\,dx=0. \end{aligned}$$

On the one hand, we have the identity $Du:\nabla u=|Du|^2$; on the other hand, multiplying the equation for $\rho$ by $h'(\rho )/\varepsilon ^2$ gives

$$\begin{aligned} \frac{1}{\varepsilon ^2}\int _\Omega P'(\rho )\,\nabla \rho \cdot u\,dx= \frac{1}{\varepsilon ^2}\,\frac{d}{dt}\int _\Omega h(\rho )\,dx. \end{aligned}$$

Putting this relation into the previous one concludes the proof of the proposition. $\square$

Let us now consider the function $F_\varepsilon$: we have the following estimate.

Proposition 3.2

Let $(\rho ,u)$ be a smooth solution to system (2) in $[0,T[\,\times \Omega$, with initial datum $(\rho _0,u_0)$, for some positive time $T>0$.

Then there exists a “universal” constant $C>0$ such that, for all $t\in [0,T[\,$, one has

$$\begin{aligned}&\frac{1}{2}\int _\Omega \rho (t)\,\left| u(t)+\nu \,\nabla \log \rho (t)\right| ^2\,dx+ \frac{\nu }{\varepsilon ^{2(1-\alpha )}}\int ^t_0\!\!\int _\Omega \left| \nabla ^2\rho \right| ^2\,dx\,d\tau \,+ \nonumber \\&\quad +\,\frac{4\nu }{\varepsilon ^2}\int ^t_0\!\!\int _\Omega P'(\rho )\,\left| \nabla \sqrt{\rho }\right| ^2\,dx\,d\tau \,\le \,C(F_\varepsilon [\rho _0]+E_\varepsilon [\rho _0,u_0])\nonumber \\&\quad +\frac{\nu }{\varepsilon }\left| \int ^t_0\!\!\int _\Omega e^3\times u\cdot \nabla \rho \,dx\,d\tau \right| . \end{aligned}$$

(13)

Proof

We will argue as in Section 3 of [5]. First, by Lemma 2 of that paper we have the identity

$$\begin{aligned} \frac{1}{2}\,\frac{d}{dt}\int _\Omega \rho \,\left| \nabla \log \rho \right| ^2+\int _\Omega \nabla \mathrm{div}\,u\cdot \nabla \rho +\int _\Omega \rho \,Du:\nabla \log \rho \otimes \nabla \log \rho =0. \end{aligned}$$

(14)

Next, we multiply the momentum equation by $\nu \,\nabla \rho /\rho$ and we integrate over $\Omega$: we find

$$\begin{aligned}&\nu \int _\Omega \left( \partial _tu+u\cdot \nabla u\right) \cdot \nabla \rho + \nu ^2\int _\Omega Du:\left( \nabla ^2\rho -\frac{1}{\rho }\nabla \rho \otimes \nabla \rho \right) \\&\quad +\,\frac{\nu }{\varepsilon }\int _\Omega e^3\times u\cdot \nabla \rho + \frac{\nu }{\varepsilon ^{2(1-\alpha )}}\int _\Omega \left| \nabla ^2\rho \right| ^2+ \frac{4\,\nu }{\varepsilon ^2}\int _\Omega P'(\rho )\left| \nabla \sqrt{\rho }\right| ^2=0. \end{aligned}$$

Now we add (14), multiplied by $\nu ^2$, to this last relation, getting

$$\begin{aligned}&\frac{\nu ^2}{2}\,\frac{d}{dt}\int _\Omega \rho \left| \nabla \log \rho \right| ^2+ \frac{\nu }{\varepsilon ^{2(1-\alpha )}}\int _\Omega \left| \nabla ^2\rho \right| ^2+ \frac{4\nu }{\varepsilon ^2}\int _\Omega P'(\rho )\left| \nabla \sqrt{\rho }\right| ^2\\&\quad \qquad \quad +\frac{\nu }{\varepsilon }\int _\Omega e^3\times u\cdot \nabla \rho \\&\quad =\,-\,\nu \int _\Omega \partial _tu\cdot \nabla \rho \,-\,\nu ^2\int _\Omega \nabla \mathrm{div}\,u\cdot \nabla \rho \,-\, \nu \int _\Omega \left( u\cdot \nabla u\right) \cdot \nabla \rho \,\\&\quad \quad \qquad -\,\nu ^2\int _\Omega Du:\nabla ^2\rho . \end{aligned}$$

Using the mass equation and the identities

$$\begin{aligned}&-\int _\Omega u\cdot \nabla \mathrm{div}\,(\rho u)\,-\,\int _\Omega \left( u\cdot \nabla u\right) \cdot \nabla \rho = \int _\Omega \rho \nabla u:\,^t\nabla u \\&\quad -\int _\Omega \nabla \mathrm{div}\,u\cdot \nabla \rho \,-\,\int _\Omega Du:\nabla ^2\rho = 0 \end{aligned}$$

we end up with the equality

$$\begin{aligned}&\frac{d}{dt}F_\varepsilon + \frac{\nu }{\varepsilon ^{2(1-\alpha )}}\int _\Omega \left| \nabla ^2\rho \right| ^2\,dx+ \frac{4\nu }{\varepsilon ^2}\int _\Omega P'(\rho )\,\left| \nabla \sqrt{\rho }\right| ^2\,dx \\&\qquad \quad +\,\frac{\nu }{\varepsilon }\int _\Omega e^3\times u\cdot \nabla \rho \,dx= -\,\nu \,\frac{d}{dt}\int _\Omega u\cdot \nabla \rho \,dx+\nu \int _\Omega \rho \,\nabla u\,:\,^t\nabla u\,dx. \end{aligned}$$

We notice that this relation can be rewritten in the following way:

$$\begin{aligned}&\frac{1}{2}\,\frac{d}{dt}\int _\Omega \rho \,\left| u\!+\!\nu \,\nabla \log \rho \right| ^2\,dx\!+\! \frac{\nu }{\varepsilon ^{2(1-\alpha )}}\int _\Omega \left| \nabla ^2\rho \right| ^2\,dx\!+\! \frac{4\nu }{\varepsilon ^2}\int _\Omega P'(\rho )\,\left| \nabla \sqrt{\rho }\right| ^2\,dx \\&\quad +\,\frac{\nu }{\varepsilon }\int _\Omega e^3\times u\cdot \nabla \rho \,dx= \frac{1}{2}\,\frac{d}{dt}\int _\Omega \rho \,|u|^2\,dx+\nu \int _\Omega \rho \,\nabla u\,:\,^t\nabla u\,dx. \end{aligned}$$

Now we integrate with respect to time and we use Proposition 3.1. $\square$

Observe that, writing $e^3\times u\cdot \nabla \rho =2\,e^3\times \left( \sqrt{\rho }u\right) \cdot \nabla \sqrt{\rho }$ and using Young’s inequality and Proposition 3.1, one can control the last term in (13) and bound the quantity

$$\begin{aligned} F_\varepsilon [\rho ](t)+\frac{\nu }{\varepsilon ^{2(1-\alpha )}}\int ^t_0\int _\Omega \left| \nabla ^2\rho \right| ^2\,dx\,d\tau + \frac{\nu }{\varepsilon ^2}\int ^t_0\int _\Omega P'(\rho )\,\left| \nabla \sqrt{\rho }\right| ^2\,dx\,d\tau . \end{aligned}$$

Such a bound is enough to get additional regularity for the sequence of smooth approximate densities when constructing a weak solution, but it is not uniform with respect to $\varepsilon$: so it is not suitable to fully exploit the BD entropy structure of the system in our study.

Nonetheless, in Proposition 3.3 below we are going to show that, under our assumptions, it is possible to control the right-hand side of (13) in a uniform way with respect to $\varepsilon$. This is a key point in order to prove our results.

3.2 Uniform bounds

Now, we are going to establish uniform properties the family $(\rho _\varepsilon ,u_\varepsilon )$ satisfies.

First of all, by Proposition 3.1 and Remark 2.3, we infer the following properties:

• $\left( \varepsilon ^{-2}\,h(\rho _\varepsilon )\right) _\varepsilon \,\subset \,L^\infty (\mathbb {R}_+;L^1(\Omega ))$ is bounded;

• $\left( \sqrt{\rho _\varepsilon }\,u_\varepsilon \right) _\varepsilon$ is bounded in $L^\infty (\mathbb {R}_+;L^2(\Omega ))$;

• $\left( \sqrt{\rho _\varepsilon }\,Du_\varepsilon \right) _\varepsilon$ is a bounded subset of $L^2(\mathbb {R}_+;L^2(\Omega ))$;

• $\left( \nabla \rho _\varepsilon \right) _\varepsilon \,\subset \,L^\infty (\mathbb {R}_+;L^2(\Omega ))$, with

  $$\begin{aligned} \left\| \nabla \rho _\varepsilon \right\| _{L^\infty (\mathbb {R}_+;L^2(\Omega ))}\,\le \,C\,\varepsilon ^{1-\alpha }, \end{aligned}$$

  for some positive constant C.

Furthermore, arguing as in the proof to Lemma 2 of [19], from the uniform bound on the internal energy h we infer the control

$$\begin{aligned} \left\| \rho _\varepsilon \,-\,1\right\| _{L^\infty (\mathbb {R}_+;L^\gamma (\Omega ))}\,\le \,C\,\varepsilon . \end{aligned}$$

In particular, under our assumptions on $\alpha$ and $\gamma$, we always find (see point (i) of Lemma 7.3)

$$\begin{aligned} \left\| \rho _\varepsilon \,-\,1\right\| _{L^\infty (\mathbb {R}_+;L^2(\Omega ))}\,\le \,C\,\varepsilon . \end{aligned}$$

(15)

Now, as announced above, we are going to show how to derive, under our assumptions, BD entropy bounds which are uniform in $\varepsilon$. Of course, all the computations are justified for smooth functions: however, by a standard approximation procedure, the final bounds will be fulfilled also by the family of weak solutions.

Proposition 3.3

Let $(\rho _{0,\varepsilon },u_{0,\varepsilon })_\varepsilon$ be a family of initial data satisfying the assumptions (i)–(ii) of Sect. 2.1, and let $(\rho _\varepsilon ,u_\varepsilon )_\varepsilon$ be a family of corresponding weak solutions.

Then there exist an $\varepsilon _0>0$ and a constant $C>0$ (depending just on the constant $K_0$ of Remark 2.3 and on the viscosity coefficient $\nu )$ such that the inequality

$$\begin{aligned}&F_\varepsilon [\rho _\varepsilon ](t)+\dfrac{\nu }{\varepsilon ^2}\int ^t_0\int _\Omega P'(\rho _\varepsilon )\,|\nabla \sqrt{\rho _\varepsilon }|^2\,dx\,d\tau + \dfrac{\nu }{\varepsilon ^{2(1-\alpha )}}\int ^t_0\int _\Omega \left| \nabla ^2\rho _\varepsilon \right| ^2\,dx\,d\tau \,\\&\quad \le \,C\,(1+t) \end{aligned}$$

holds true for any $t>0$ and for all $0<\varepsilon \le \varepsilon _0$.

Proof

Our starting point is the inequality stated in Proposition 3.2: we have to control the last term in its right-hand side. For convenience, let us omit for a while the index $\varepsilon$ in the notation.

First of all, we can write

$$\begin{aligned} \int _\Omega e^3\times u\cdot \nabla \rho= & {} \int _\Omega \rho ^{(\gamma -1)/2}\,e^3\times u\cdot \nabla \rho + \int _\Omega \left( 1-\rho ^{(\gamma -1)/2}\right) \,e^3\times u\cdot \nabla \rho \\= & {} 2\int _\Omega e^3\times \left( \sqrt{\rho }\,u\right) \cdot \nabla \sqrt{\rho }\;\rho ^{(\gamma -1)/2}+ \int _\Omega \left( 1-\rho ^{(\gamma -1)/2}\right) \,e^3\\&\times u\cdot \nabla \rho . \end{aligned}$$

Now we focus on the last term: integrating by parts we get

$$\begin{aligned}&\int _\Omega \left( 1-\rho ^{(\gamma -1)/2}\right) \,e^3\times u\cdot \nabla \rho =\int _\Omega \rho \,\omega ^3\,\left( 1-\rho ^{(\gamma -1)/2}\right) \\&\quad +\frac{\gamma -1}{2}\int _\Omega \rho ^{(\gamma -1)/2}\,e^3\times u\cdot \nabla \rho , \end{aligned}$$

where we denoted by $\omega =\nabla \times u$ the vorticity of the fluid. Therefore, we find

$$\begin{aligned} \int _\Omega e^3\times u\cdot \nabla \rho \!=\! (\gamma +1)\int _\Omega e^3\times \left( \sqrt{\rho }\,u\right) \cdot \nabla \sqrt{\rho }\;\rho ^{(\gamma -1)/2}\!+\! \int _\Omega \rho \,\omega ^3\,\left( 1\!-\!\rho ^{(\gamma -1)/2}\right) . \end{aligned}$$

Let us deal with the first term: we have

$$\begin{aligned} \frac{\nu }{\varepsilon }\left| \int ^t_0\int _\Omega e^3\times \left( \sqrt{\rho }\,u\right) \cdot \nabla \sqrt{\rho }\;\rho ^{(\gamma -1)/2}\right|\le & {} \frac{\nu }{\varepsilon }\int ^t_0\left\| \sqrt{\rho }\,u\right\| _{L^2}\,\left\| \rho ^{(\gamma -1)/2}\,\nabla \sqrt{\rho }\right\| _{L^2} \nonumber \\\le & {} C\,\nu \,t\,+\frac{\nu }{2\,\varepsilon ^2}\int ^t_0\left\| \rho ^{(\gamma -1)/2}\,\nabla \sqrt{\rho }\right\| ^2_{L^2},\nonumber \\ \end{aligned}$$

(16)

where we have used the uniform bounds for $(\sqrt{\rho _\varepsilon }\,u_\varepsilon )_\varepsilon$ and Young’s inequality. Notice that, as $P'(\rho )=\rho ^{\gamma -1}$, the last term can be absorbed in the left-hand side of (13).

Now we consider the term involving the vorticity. Notice that, since $0<(\gamma -1)/2\le 1/2$, we can bound $|\rho ^{(\gamma -1)/2}-1|$ with $|\rho -1|$; then, using also the established uniform bounds, we get

$$\begin{aligned} \frac{\nu }{\varepsilon }\left| \int ^t_0\int _\Omega \rho \,\omega ^3\,\left( 1-\rho ^{(\gamma -1)/2}\right) \right|\le & {} \frac{\nu }{\varepsilon }\int ^t_0\Vert \rho -1\Vert _{L^2}\, \left\| \sqrt{\rho }\,Du\right\| _{L^2}\,\Vert \rho \Vert ^{1/2}_{L^\infty } \\\le & {} C\,\nu \left( \int ^t_0\Vert \rho \Vert _{L^\infty }\right) ^{1/2}. \end{aligned}$$

In order to control the $L^\infty$ norm of the density, we write $\rho =1+(\rho -1)$: for the second term we use Lemma 7.3 with $p=2$ and $\delta =1/2$. Keeping in mind also estimate (15), we get

$$\begin{aligned} C\,\nu \left( \int ^t_0\Vert \rho \Vert _{L^\infty }\right) ^{1/2}\le & {} C\,\nu \left( \int ^t_0\left( 1+\left\| \nabla ^2\rho \right\| _{L^2}\right) \right) ^{1/2} \\\le & {} \frac{C\,\nu }{2}\,(1+t)+\frac{C\,\nu }{2}\,\int ^t_0\left\| \nabla ^2\rho \right\| _{L^2} \\\le & {} C'\,\nu \,(1+t)+\frac{\nu }{4}\,\int ^t_0\left\| \nabla ^2\rho \right\| ^2_{L^2}, \end{aligned}$$

where we used twice Young’s inequality. Hence, in the end we obtain

$$\begin{aligned} \frac{\nu }{\varepsilon }\left| \int ^t_0\int _\Omega \rho \,\omega ^3\,\left( 1-\rho ^{(\gamma -1)/2}\right) \right| \,\le \,C\,\nu \,(1+t)+ C_\varepsilon \,\frac{\nu }{\varepsilon ^{2(1-\alpha )}}\,\left\| \nabla ^2\rho \right\| ^2_{L^2_t(L^2)}, \end{aligned}$$

(17)

with $C_\varepsilon =\varepsilon ^{2(1-\alpha )}/4$. Then, for any $\alpha \in [0,1]$ we can absorb the last term of this estimate into the left-hand side of (13).

Therefore, thanks to inequalities (16) and (17), combined with (13), we get the result. $\square$

Remark 3.4

The approach we followed seems to suggest that having $\Vert \rho _\varepsilon -1\Vert _{L^\infty _T(L^2)}\,\sim \,O(\varepsilon )$ is necessary to control the term coming from rotation in (13) and so to close the estimates (see in particular the bounds for the term involving vorticity).

This is the only (technical) reason for which we assumed $\gamma =2$ when (low capillarity limit), while for $\alpha =0$ (constant capillarity case) we can take more general pressure laws, namely any $1<\gamma \le 2$, since we still have inequality (15).

By the bounds established in Proposition 3.3, we infer also the following properties:

• $\left( \sqrt{\rho _\varepsilon }\,\,\nabla \log \rho _\varepsilon \right) _\varepsilon$ is bounded in the space $L^\infty _{loc}(\mathbb {R}_+;L^2(\Omega ))$;

• $\left( \varepsilon ^{-1}\,\rho _\varepsilon ^{(\gamma -1)/2}\,\nabla \sqrt{\rho }_\varepsilon \right) _\varepsilon \,\subset \,L^2_{loc}(\mathbb {R}_+;L^2(\Omega ))$ bounded;

• $\left( \varepsilon ^{-(1-\alpha )}\,\nabla ^2\rho _\varepsilon \right) _\varepsilon$ is bounded in $L^2_{loc}(\mathbb {R}_+;L^2(\Omega ))$.

In particular, from the last fact combined with estimate (15) and Lemma 7.3, we also gather that, for any fixed positive time T,

$$\begin{aligned} \left\| \rho _\varepsilon \,-\,1\right\| _{L^2([0,T];L^\infty (\Omega ))}\,\le \,C_T\,\varepsilon ^{1-\alpha }, \end{aligned}$$

(18)

where we denote by $C_T$ a quantity proportional (for some “universal” constant) to $1+T$.

Note that, thanks to the equality $\sqrt{\rho }\,\,\nabla \log \rho =2\,\nabla \sqrt{\rho }$, from the previous bounds we get also that $\left( \nabla \sqrt{\rho _\varepsilon }\right) _\varepsilon$ is bounded in $L^\infty ([0,T];L^2(\Omega ))$, for any $T>0$ fixed.

Let us also remark that we have a nice decay of the first derivatives of $\rho _\varepsilon$ even in the low capillarity regime: namely, for (and then $\gamma =2$), one has

$$\begin{aligned} \left\| \nabla \rho _\varepsilon \right\| _{L^2_T(L^2)}\,\le \,C_T\,\varepsilon \end{aligned}$$

(19)

for any $T>0$ fixed (see the second point of the previous list of bounds). Notice that the constant $C_T$ does not depend on $\alpha$ (recall Proposition 3.3).

Finally, let us state an important property on the quantity $D(\rho ^{3/2}_\varepsilon \,u_\varepsilon )$. First of all, we write

$$\begin{aligned} D(\rho ^{3/2}_\varepsilon \,u_\varepsilon )= & {} \rho _\varepsilon \,\sqrt{\rho _\varepsilon }\,Du_\varepsilon +\frac{3}{2}\,\sqrt{\rho _\varepsilon }\,u_\varepsilon \, D\rho _\varepsilon \nonumber \\= & {} \sqrt{\rho _\varepsilon }\,Du_\varepsilon +\left( \rho _\varepsilon \,-\,1\right) \sqrt{\rho _\varepsilon }\,Du_\varepsilon + \frac{3}{2}\,\sqrt{\rho _\varepsilon }\,u_\varepsilon \,D\rho _\varepsilon . \end{aligned}$$

(20)

The first term in the right-hand side clearly belongs to $L^2_T(L^2)$, while, by uniform bounds and Sobolev embeddings, the second and the third ones are uniformly bounded in $L^2_T(L^{3/2})$. Therefore, we infer that $(D(\rho ^{3/2}_\varepsilon \,u_\varepsilon ))_\varepsilon$ is a uniformly bounded family in $L^2_T(L^2+L^{3/2})$.

3.3 Constraints on the limit

As specified in the introduction, we want to study the weak limit of the family $(\rho _\varepsilon ,u_\varepsilon )_\varepsilon$, i.e. we want to pass to the limit for $\varepsilon \rightarrow 0$ in Eqs. (8)–(9) when computed on $(\rho _\varepsilon ,u_\varepsilon )$.

The present paragraph is devoted to establishing some properties the weak limit has to satisfy.

By uniform bounds, seeing $L^\infty$ as the dual of $L^1$, we infer, up to extraction of subsequences, the weak convergences

$$\begin{aligned} \sqrt{\rho _\varepsilon }\,u_\varepsilon \,\mathop {\rightharpoonup }\limits ^{*}\,u&\quad \text{ in } \quad&L^\infty (\mathbb {R}_+;L^2(\Omega )) \\ \sqrt{\rho _\varepsilon }\,Du_\varepsilon \,\rightharpoonup \,U&\quad \text{ in } \quad&L^2(\mathbb {R}_+;L^2(\Omega )). \end{aligned}$$

Here $\mathop {\rightharpoonup }\limits ^{*}$ denotes the weak-$*$ convergence in $L^\infty (\mathbb {R}_+;L^2(\Omega ))$.

On the other hand, thanks to the estimates for the density, we immediately deduce that $\rho _\varepsilon \,\rightarrow \,1$ (strong convergence) in $L^\infty (\mathbb {R}_+;L^2(\Omega ))$, with convergence rate of order $\varepsilon$. So, we can write $\rho _\varepsilon =1+\varepsilon \,r_\varepsilon$, with the family $(r_\varepsilon )_\varepsilon$ bounded in $L^\infty (\mathbb {R}_+;L^2(\Omega ))$, and then (up to an extraction) weakly convergent to some r in this space.

Notice that, in the case $\alpha =0$, we know that actually $(\rho _\varepsilon )_\varepsilon$ strongly converges to 1 in the space $L^\infty (\mathbb {R}_+;H^1(\Omega ))\,\cap \,L^2_{loc}(\mathbb {R}_+;H^2(\Omega ))$, still with rate $O(\varepsilon )$. Then we infer also that

$$\begin{aligned} \quad r_\varepsilon \,\rightharpoonup \,r\quad \text{ in } \quad L^\infty (\mathbb {R}_+;H^1(\Omega ))\,\cap \,L^2_{loc}(\mathbb {R}_+;H^2(\Omega )). \end{aligned}$$

(21)

In the case , thanks to (19) we gather instead that

$$\begin{aligned} \quad r_\varepsilon \,\rightharpoonup \,r \quad \text{ in } \quad L^2_{loc}(\mathbb {R}_+;H^1(\Omega )). \end{aligned}$$

(22)

Notice also that, as expected, one has $U=Du$, and then $u\in L^2(\mathbb {R}_+;H^1(\Omega ))$. As a matter of fact, consider Eq. (20): using again the trick $\rho _\varepsilon =1+\left( \rho _\varepsilon -1\right)$ together with (15), it is easy to check that

$$\begin{aligned} D(\rho _\varepsilon ^{3/2}\,u_\varepsilon )\,\longrightarrow \,Du\quad \text{ in } \quad \mathcal {D}'. \end{aligned}$$

On the other hand, the bounds (15) and (19) imply that the right-hand side of (20) weakly converges to U, and this proves our claim.

Let us also point out that

$$\begin{aligned} \rho _\varepsilon \,u_\varepsilon \,\rightharpoonup \,u\quad \text{ in } \quad L^2([0,T];L^2(\Omega )). \end{aligned}$$

(23)

In fact, we can write $\rho _\varepsilon u_\varepsilon =\sqrt{\rho _\varepsilon }u_\varepsilon +\left( \sqrt{\rho _\varepsilon }-1\right) \sqrt{\rho _\varepsilon }u_\varepsilon$. By $\left| \sqrt{\rho _\varepsilon }-1\right| \le \left| \rho _\varepsilon -1\right|$ and Sobolev embeddings, we get that the second term in the right-hand side converges strongly to 0 in $L^\infty ([0,T];L^1(\Omega )\cap L^{3/2}(\Omega ))\,\cap \,L^2([0,T];L^2(\Omega ))$.

Exactly in the same way, we find that

$$\begin{aligned} \quad \rho _\varepsilon \,Du_\varepsilon \,\rightharpoonup \,Du\quad \text{ in } \quad L^1([0,T];L^2(\Omega ))\,\cap \,L^2([0,T];L^1(\Omega )\cap L^{3/2}(\Omega )). \end{aligned}$$

(24)

We conclude this part by proving the following proposition, which can be seen as the analogue of the Taylor–Proudman theorem in our context.

Proposition 3.5

Let $(\rho _\varepsilon ,u_\varepsilon )_\varepsilon$ be a family of weak solutions (in the sense of Definition 2.2 above) to system (2)–(3), with initial data $(\rho _{0,\varepsilon },u_{0,\varepsilon })$ satisfying the hypotheses fixed in Sect. 2. Let us define $r_\varepsilon :=\varepsilon ^{-1}\left( \rho _\varepsilon -1\right)$, and let (r, u) be a limit point of the sequence $(r_\varepsilon ,u_\varepsilon )_\varepsilon$.

Then $r=r(x^h)$ and $u=(u^h(x^h),0)$, with $\mathrm{div}\,_{\!h}u^h=0$. Moreover, r and u are linked by the relation

$$\begin{aligned} \left\{ \begin{array}{ll} u^h\;=\,\nabla _h^\perp r &{} \quad \mathrm{if }\quad 0\,<\,\alpha \,\le \,1 \\ u^h\;=\,\nabla _h^\perp (\mathrm{Id}\,-\Delta )r &{} \quad \mathrm{if }\quad \alpha =0. \end{array}\right. \end{aligned}$$

Proof

Let us consider first the mass equation in the (classical) weak formulation, i.e. (8): writing $\rho _\varepsilon =1+\varepsilon \,r_\varepsilon$ as above, for any $\phi \in \mathcal {D}([0,T[\,\times \Omega )$ we have

$$\begin{aligned} -\,\varepsilon \int ^T_0\int _\Omega r_\varepsilon \,\partial _t\phi \,-\,\int ^T_0\int _\Omega \rho _\varepsilon \,u_\varepsilon \cdot \nabla \phi = \varepsilon \int _\Omega r_{0,\varepsilon }\,\phi (0). \end{aligned}$$

Letting $\varepsilon \rightarrow 0$, we deduce that $\int ^T_0\int _\Omega u\cdot \nabla \phi =0$, which implies

$$\begin{aligned} \mathrm{div}\,\,u\;\equiv \,0\quad \text{ almost } \text{ everywhere } \text{ in } \quad [0,T]\times \Omega . \end{aligned}$$

(25)

After that, we turn our attention to the (modified) weak formulation of the momentum equation, given by (9): we multiply it by $\varepsilon$ and we pass to the limit $\varepsilon \rightarrow 0$. By uniform bounds, it is easy to see that the only integrals which do not go to 0 are the ones involving the pressure, the rotation and the capillarity: let us analyse them carefully.

First of all, let us deal with the pressure term: we rewrite it as

$$\begin{aligned} \frac{1}{\varepsilon }\int ^T_0\int _\Omega \nabla P(\rho _\varepsilon )\cdot \rho _\varepsilon \,\psi= & {} \frac{1}{\varepsilon }\int ^T_0\int _\Omega \nabla P(\rho _\varepsilon )\cdot \left( \rho _\varepsilon -1\right) \,\psi + \frac{1}{\varepsilon }\int ^T_0\int _\Omega \nabla P(\rho _\varepsilon )\cdot \psi \\= & {} \int ^T_0\int _\Omega r_\varepsilon \,\rho _\varepsilon ^{\gamma -1}\,\nabla \rho _\varepsilon \cdot \psi + \frac{1}{\varepsilon }\int ^T_0\int _\Omega \nabla P(\rho _\varepsilon )\cdot \psi . \end{aligned}$$

Using the boundedness of $(r_\varepsilon )_\varepsilon$ in $L^\infty _T(L^2)$ and the strong convergence of $\nabla \rho _\varepsilon \rightarrow 0$ in $L^\infty _T(L^2)$ and (as $0<\gamma -1\le 1$) of $\rho _\varepsilon ^{\gamma -1}\rightarrow 1$ in $L^{2}_T(L^\infty )$, one infers that the former term of the last equality goes to 0. The latter, instead, can be rewritten in the following way:

$$\begin{aligned} \frac{1}{\varepsilon }\int ^T_0\!\!\!\int _\Omega \nabla P(\rho _\varepsilon )\cdot \psi= & {} -\frac{1}{\varepsilon }\int ^T_0\!\!\!\int _\Omega (P(\rho _\varepsilon )-P(1)-P'(1)\left( \rho _\varepsilon -1\right) )\mathrm{div}\,\psi \\&+\frac{1}{\varepsilon }\,P'(1)\int ^T_0\!\!\!\int _\Omega \nabla \rho _\varepsilon \cdot \psi . \end{aligned}$$

Notice that the quantity $P(\rho _\varepsilon )-P(1)-P'(1)\left( \rho _\varepsilon -1\right)$ coincides, up to a factor $1/(\gamma -1)$, with the internal energy $h(\rho _\varepsilon )$: since $h(\rho _\varepsilon )/\varepsilon ^2$ is bounded in $L^\infty _T(L^1)$ (see Sect. 3.2), the first integral tends to 0 for $\varepsilon \rightarrow 0$. Finally, thanks also to bounds (21) and (22), we find

$$\begin{aligned} \frac{1}{\varepsilon }\int ^T_0\int _\Omega \nabla P(\rho _\varepsilon )\cdot \rho _\varepsilon \,\psi \;\longrightarrow \; \int ^T_0\int _\Omega \nabla r\cdot \psi . \end{aligned}$$

We now consider the rotation term: by (23) and the strong convergence $\rho _\varepsilon \rightarrow 1$ in $L^\infty _T(L^2)$, we immediately get

$$\begin{aligned} \int ^T_0\int _\Omega e^3\times \rho _\varepsilon ^2\,u_\varepsilon \cdot \psi \;\longrightarrow \;\int ^T_0\int _\Omega e^3\times u\cdot \psi . \end{aligned}$$

Finally, we deal with the capillarity terms: on the one hand, thanks to the uniform bounds for $(\varepsilon ^{-(1-\alpha )}\,\nabla \rho _\varepsilon )_\varepsilon$ and $(\varepsilon ^{-(1-\alpha )}\,\nabla ^2\rho _\varepsilon )_\varepsilon$, we get

$$\begin{aligned} \frac{2\,\varepsilon }{\varepsilon ^{2(1-\alpha )}}\int ^T_0\int _\Omega \rho _\varepsilon \,\Delta \rho _\varepsilon \,\nabla \rho _\varepsilon \cdot \psi \;\longrightarrow \;0. \end{aligned}$$

On the other hand, splitting $\rho ^2_\varepsilon =1+\left( \rho _\varepsilon -1\right) \left( \rho _\varepsilon +1\right)$ and using uniform bounds again, one easily gets that the quantity

$$\begin{aligned} \frac{\varepsilon }{\varepsilon ^{2(1-\alpha )}}\int ^T_0\int _\Omega \rho _\varepsilon ^2\,\Delta \rho _\varepsilon \,\mathrm{div}\,\psi = \frac{\varepsilon ^\alpha }{\varepsilon ^{1-\alpha }}\int ^T_0\int _\Omega \rho _\varepsilon ^2\,\Delta \rho _\varepsilon \,\mathrm{div}\,\psi \end{aligned}$$

converges to 0 in the case , while it converges to $\int ^T_0\int _\Omega \Delta r\,\mathrm{div}\,\psi$ in the case $\alpha =0$.

Let us now restrict for a while to the case $\alpha =0$. To sum up, in the limit $\varepsilon \rightarrow 0$, the equation for the velocity field (tested against $\varepsilon \,\rho _\varepsilon \,\psi$) gives the constraint

$$\begin{aligned} e^3\,\times \,u+\nabla \widetilde{r}=0,\quad \text{ with } \quad \widetilde{r}\,:=\,r\,-\,\Delta r. \end{aligned}$$

(26)

This means that

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _1\widetilde{r}=\,u^2 \\ \partial _2\widetilde{r}=-\,u^1 \\ \partial _3\widetilde{r}=0, \end{array}\right. } \end{aligned}$$

which immediately implies that $\widetilde{r}=\widetilde{r}(x^h)$ depends just on the horizontal variables. From this, it follows that also $u^h=u^h(x^h)$.

Moreover, from the previous system we easily deduce that

$$\begin{aligned} \mathrm{div}\,_{\!h}\,u^h=0, \end{aligned}$$

(27)

which, together with (25), entails that $\partial _3u^3\equiv 0$. Due to the complete slip boundary conditions, we then infer that $u^3\equiv 0$ almost everywhere in $[0,T]\times \Omega$.

In the end, we have proved that the limit velocity field u is two-dimensional, horizontal and divergence-free.

Finally, let us come back to r: by what we have said before, $\partial _3r$ fulfills the elliptic equation

$$\begin{aligned} -\,\Delta \partial _3r+\partial _3r=0\quad \text{ in } \quad \Omega . \end{aligned}$$

By passing to Fourier transform in $\mathbb {R}^2\times \mathbb {T}^1$, or by energy methods (because $\partial _3r\in L^\infty _T(H^1)$), or by spectral theory (since the Laplace operator has only positive eigenvalues), we find that

$$\begin{aligned} \partial _3r\,\equiv \,0\quad \Longrightarrow \quad r=r(x^h). \end{aligned}$$

(28)

The same arguments as above also apply when $0<\alpha \le 1$, working with r itself instead of $\widetilde{r}$. Notice that the property $r=r(x^h)$ is then straightforward, because of the third equation in (26).

The proposition is now completely proved. $\square$

4 Vanishing capillarity limit: the case $\alpha =1$

In this section we restrict our attention to the vanishing capillarity limit, and we prove Theorem 2.4 in the special (and simpler) case $\alpha =1$. In fact, when the system presents an anisotropy in $\varepsilon$, which requires a modification of the arguments of the proof: we refer to Sect. 6 for the analysis.

We first study the propagation of acoustic waves, from which we infer (by use of the RAGE theorem) the strong convergence of the quantities $(r_\varepsilon )_\varepsilon$ and $(\rho ^{3/2}_\varepsilon \,u_\varepsilon )_\varepsilon$ in $L^2_T(L^2_{loc}(\Omega ))$. We are then able to pass to the limit in the weak formulation (8)–(9), and to identify the limit system.

4.1 Analysis of the acoustic waves

The present paragraph is devoted to the analysis of the acoustic waves. The main goal is to apply the well-known RAGE theorem to prove dispersion of the components of the solutions which are orthogonal to the kernel of the singular perturbation operator.

We shall follow the analysis performed in [11].

4.1.1 The acoustic propagator

First of all, we rewrite system (2) in the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon \,\partial _tr_\varepsilon +\mathrm{div}\,\,V_\varepsilon =0 \\ \varepsilon \,\partial _tV_\varepsilon +(e^3\times V_\varepsilon +\nabla r_\varepsilon )=\varepsilon \,f_\varepsilon , \end{array}\right. } \end{aligned}$$

(29)

where we have defined $V_\varepsilon \,:=\,\rho _\varepsilon \,u_\varepsilon$ and

$$\begin{aligned} f_\varepsilon:= & {} -\,\mathrm{div}\,\left( \rho _\varepsilon u_\varepsilon \otimes u_\varepsilon \right) +\nu \,\mathrm{div}\,\left( \rho _\varepsilon Du_\varepsilon \right) \nonumber \\&-\,\frac{1}{\varepsilon ^2}\nabla (P(\rho _\varepsilon )-P(1)-P'(1)\left( \rho _\varepsilon -1\right) )+ \rho _\varepsilon \,\nabla \Delta \rho _\varepsilon . \end{aligned}$$

(30)

Of course, system (29) has to be read in the weak sense specified by Definition 2.2: for any scalar $\phi \in \mathcal {D}([0,T[\,\times \Omega )$ one has

$$\begin{aligned} -\,\varepsilon \int ^T_0\int _\Omega r_\varepsilon \,\partial _t\phi \,dx\,dt\,-\,\int ^T_0\int _\Omega V_\varepsilon \cdot \nabla \phi \,dx\,dt= \varepsilon \int _\Omega r_{0,\varepsilon }\,\phi (0)\,dx, \end{aligned}$$

and, for any $\psi \in \mathcal {D}([0,T[\,\times \Omega )$ with values in $\mathbb {R}^3$,

$$\begin{aligned}&\int ^T_0\!\!\int _\Omega (-\,\varepsilon \,V_\varepsilon \cdot \partial _t(\rho _\varepsilon \psi )+\rho _\varepsilon e^3\times V_\varepsilon \cdot \psi \,-\, r_\varepsilon \,\mathrm{div}\,\!(\rho _\varepsilon \psi ))\\&\quad =\varepsilon \int _\Omega \rho _{0,\varepsilon }^2\,u_{0,\varepsilon }\,\psi (0)+\varepsilon \int ^T_0\langle f_\varepsilon ,\rho _\varepsilon \psi \rangle , \end{aligned}$$

where we have set

$$\begin{aligned} \langle f_\varepsilon ,\zeta \rangle:= & {} \int _\Omega (\rho _\varepsilon u_\varepsilon \otimes u_\varepsilon :\nabla \zeta \,-\, \nu \,\rho _\varepsilon Du_\varepsilon :\nabla \zeta \,-\,\Delta \rho _\varepsilon \,\nabla \rho _\varepsilon \cdot \zeta \,- \\&-\,\rho _\varepsilon \,\Delta \rho _\varepsilon \,\mathrm{div}\,\zeta + \frac{1}{\varepsilon ^2}(P(\rho _\varepsilon )-P(1)-P'(1)\left( \rho _\varepsilon -1\right) )\mathrm{div}\,\zeta )dx \\= & {} \int _\Omega (f^1_\varepsilon :\nabla \zeta +f^2_\varepsilon :\nabla \zeta +f^3_\varepsilon \cdot \zeta +f^4_\varepsilon \,\mathrm{div}\,\zeta + f^5_\varepsilon \,\mathrm{div}\,\zeta )\,dx. \end{aligned}$$

Since $(\sqrt{\rho _\varepsilon }u_\varepsilon )_\varepsilon \subset L^\infty _T(L^2)$ and $f^5_\varepsilon \sim h(\rho _\varepsilon )$, uniform bounds imply that $(f^1_\varepsilon )_\varepsilon$ and $(f^5_\varepsilon )_\varepsilon$ are uniformly bounded in $L^\infty _T(L^1)$. Since $\nabla \rho _\varepsilon \in L^\infty _T(L^2)$ and $\nabla ^2\rho _\varepsilon \in L^2_T(L^2)$ are uniformly bounded, we get that $(f^3_\varepsilon )_\varepsilon \subset L^2_T(L^1)$ is bounded, and so is $(f^4_\varepsilon )_\varepsilon$ in $L^2_T(L^2+L^1)$ (write $\rho _\varepsilon =1+(\rho _\varepsilon -1)$ and use (15) for the second term). Finally, by writing $\rho _\varepsilon =\sqrt{\rho _\varepsilon }+(\sqrt{\rho _\varepsilon }-1)\sqrt{\rho _\varepsilon }$ and arguing similarly as for (24), we discover that $(f^2_\varepsilon )_\varepsilon$ is bounded in $L^2_T(L^2+L^1)$.

Then we get that $(f_\varepsilon )_\varepsilon$ is bounded in $L^2_T(W^{-1,2}(\Omega )+W^{-1,1}(\Omega ))$.

This having been established, let us turn our attention to the acoustic propagator, i.e. the operator $\mathcal {A}$ defined by

$$\begin{aligned} \begin{array}{lccc} \mathcal {A}\,: &{} L^2(\Omega )\;\times \;L^2(\Omega ) &{} \longrightarrow &{} H^{-1}(\Omega )\;\times \;H^{-1}(\Omega ) \\ &{} (r\;,\;V) &{} \mapsto &{} (\mathrm{div}\,V\;,\;e^3\times V\,+\nabla r). \end{array} \end{aligned}$$

We remark that $\mathcal {A}$ is skew-adjoint with respect to the $L^2(\Omega )\times L^2(\Omega )$ scalar product.

Notice that, by Proposition 3.5, any limit point $\left( r,u\right)$ of the sequence of weak solutions has to belong to $\mathrm{Ker}\,\mathcal {A}$.

Moreover, the following proposition holds true. For the proof, see Subsection 3.1 of [11].

Proposition 4.1

Let us denote by $\sigma _p(\mathcal {A})$ the point spectrum of $\mathcal {A}$. Then $\sigma _p(\mathcal {A})=\{0\}$.

In particular, if we define by $\mathrm{Eigen}\,\mathcal {A}$ the space spanned by the eigenvectors of $\mathcal {A}$, we have $\mathrm{Eigen}\,\mathcal {A}\;\equiv \;\mathrm{Ker}\,\mathcal {A}$.

4.1.2 Application of the RAGE theorem

Let us first recall the RAGE theorem and some of its consequences. The present form is the same used in [11] (see [9], Theorem 5.8).

Theorem 4.2

Let $\mathcal H$ be a Hilbert space and $\mathcal {B}\,:\,D(\mathcal {B})\subset \mathcal H\,\longrightarrow \,\mathcal H$ a self-adjoint operator. Denote by $\Pi _\mathrm{cont}$ the orthogonal projection onto the subspace $\mathcal H_\mathrm{cont}$, where

$$\begin{aligned} \mathcal H\;=\;\mathcal H_\mathrm{cont}\,\oplus \,\overline{\mathrm{Eigen}\,(\mathcal {B})} \end{aligned}$$

and $\overline{\Theta }$ is the closure of a subset $\Theta$ in $\mathcal H$. Finally, let $\mathcal {K}\,:\,\mathcal H\,\longrightarrow \,\mathcal H$ be a compact operator.

Then, in the limit for $T\rightarrow +\infty$ one has

$$\begin{aligned} \left\| \frac{1}{T}\,\int ^T_0e^{-i\,t\,\mathcal {B}}\;\mathcal {K}\;\Pi _\mathrm{cont}\;e^{i\,t\,\mathcal {B}}\;dt\right\| _{\mathcal {L}(\mathcal H)}\, \longrightarrow \,0. \end{aligned}$$

Exactly as in [11], from the previous theorem we infer the following properties.

Corollary 4.3

Under the hypothesis of Theorem 4.2, suppose moreover that $\mathcal {K}$ is self-adjoint, with $\mathcal {K}\ge 0$.

Then there exists a function $\mu$, with $\mu (\varepsilon )\rightarrow 0$ for $\varepsilon \rightarrow 0$, such that:

• (1) for any $Y\in \mathcal H$ and any $T>0$, one has

  $$\begin{aligned} \frac{1}{T}\int ^T_0\left\| \mathcal {K}^{1/2}\,e^{i\,t\,\mathcal {B}/\varepsilon }\,\Pi _\mathrm{cont}Y\right\| ^2_{\mathcal H}\,dt\,\le \, \mu (\varepsilon )\,\Vert Y\Vert _{\mathcal H}^2\,; \end{aligned}$$

• (2) for any $T>0$ and any $X\in L^2([0,T];\mathcal H)$, one has

  $$\begin{aligned} \frac{1}{T^2}\left\| \mathcal {K}^{1/2}\,\Pi _\mathrm{cont}\int ^t_0e^{i\,(t-\tau )\,\mathcal {B}/\varepsilon }\,X(\tau )\,d\tau \right\| ^2_{L^2([0,T];\mathcal H)}\,\le \, \mu (\varepsilon )\,\left\| X\right\| ^2_{L^2([0,T];\mathcal H)}. \end{aligned}$$

We now come back to our problem. For any fixed $M>0$, define the Hilbert space $H_M$ by

$$\begin{aligned}&H_M\,:=\,\left\{ (r,V)\in L^2(\Omega )\times L^2(\Omega )\;|\; \widehat{r}(\xi ^h,k)\equiv 0\; \text{ and } \;\widehat{V}(\xi ^h,k)\equiv 0\quad \right. \nonumber \\&\quad \left. \text{ if }\quad |\xi ^h|+|k|>M\right\} , \end{aligned}$$

(31)

and let $P_M\,:\,L^2(\Omega )\times L^2(\Omega )\,\longrightarrow \,H_M$ be the orthogonal projection onto $H_M$. For a fixed $\theta \in \mathcal {D}(\Omega )$ such that $0\le \theta \le 1$, we also define the operator

$$\begin{aligned} \mathcal {K}_{M,\theta }(r,V)\,:=\,P_M(\theta \,P_M(r,V)) \end{aligned}$$

acting on $H_M$. Note that $\mathcal {K}_{M,\theta }$ is self-adjoint and positive; moreover, it is also compact by Rellich–Kondrachov theorem, since its range is included in the set of functions having compact spectrum.

We want to apply the RAGE theorem to

$$\begin{aligned} \mathcal H=H_M\;,\quad \mathcal {B}=i\,\mathcal {A}\;,\quad \mathcal {K}=\mathcal {K}_{M,\theta }\quad \text{ and } \quad \Pi _\mathrm{cont}=Q^\perp , \end{aligned}$$

where Q and $Q^\perp$ are the orthogonal projections onto respectively $\mathrm{Ker}\,\mathcal {A}$ and $(\mathrm{Ker}\,\mathcal {A})^\perp$.

Let us set $\left( r_{\varepsilon ,M},V_{\varepsilon ,M}\right) \,:=\,P_M(r_\varepsilon ,V_\varepsilon )$, and note that, thanks to a priori bounds, for any M it makes sense to apply the term $f_\varepsilon$ to any element of $H_M$. Hence, from system (29) we get

$$\begin{aligned} \varepsilon \,\frac{d}{dt}(r_{\varepsilon ,M},\,V_{\varepsilon ,M})+\mathcal {A}(r_{\varepsilon ,M},\,V_{\varepsilon ,M})= \varepsilon \,(0,\,f_{\varepsilon ,M}), \end{aligned}$$

(32)

where $(0,\,f_{\varepsilon ,M})\,\in \,H_M^*\cong H_M$ is defined by

$$\begin{aligned} \langle (0,f_{\varepsilon ,M}),\,\left( s,P_M(\rho _\varepsilon \,W)\right) \rangle _{H_M}:= & {} \int _\Omega (f^1_\varepsilon :\nabla P_M(\rho _\varepsilon W)+ f^2_\varepsilon :\nabla P_M(\rho _\varepsilon W)\\&\; +f^3_\varepsilon \cdot P_M(\rho _\varepsilon W)+ f^4_\varepsilon \,\mathrm{div}\,P_M(\rho _\varepsilon W)\\&+f^5_\varepsilon \,\mathrm{div}\,P_M(\rho _\varepsilon W))dx \end{aligned}$$

for any $(s,W)\in H_M$. Moreover, by Bernstein inequalities (due to the localization in the phase space) it is easy to see that, for any $T>0$ fixed and any $W\in H_M$,

$$\begin{aligned} \left\| P_M(\rho _\varepsilon W)\right\| _{L^2_T(W^{1,\infty }\cap H^1)}\le & {} C(M)\,\left\| \rho _\varepsilon W\right\| _{L^2_T(L^2)} \\\le & {} C(M)\,\left( \Vert W\Vert _{L^2_T(L^2)}+\Vert \rho _\varepsilon -1\Vert _{L^\infty _T(L^2)}\,\Vert W\Vert _{L^2_T(L^\infty )}\right) , \end{aligned}$$

for some constant C(M) depending only on M. This fact, combined with the uniform bounds we established on $f_\varepsilon$, entails

$$\begin{aligned} \left\| (0,\,f_{\varepsilon ,M})\right\| _{L^2_T(H_M)}\,\le \,C(M). \end{aligned}$$

(33)

By use of Duhamel’s formula, solutions to Eq. (32) can be written as

$$\begin{aligned} (r_{\varepsilon ,M},\,V_{\varepsilon ,M})(t)=e^{i\,t\,\mathcal {B}/\varepsilon }(r_{\varepsilon ,M},\,V_{\varepsilon ,M})(0)+ \int ^t_0e^{i\,(t-\tau )\,\mathcal {B}/\varepsilon }\,(0,\,f_{\varepsilon ,M})\,d\tau . \end{aligned}$$

(34)

Note that, by definition (and since $[P_M,Q]=0$),

$$\begin{aligned} \left\| \left( \mathcal {K}_{M,\theta }\right) ^{1/2}\,Q^\perp (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\right\| ^2_{H_M}= \int _\Omega \theta \,\left| Q^\perp (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\right| ^2\,dx. \end{aligned}$$

Therefore, a straightforward application of Corollary 4.3 (recalling also Proposition 4.1) gives that, for $T>0$ fixed and for $\varepsilon$ going to 0,

$$\begin{aligned} Q^\perp (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\,\longrightarrow \,0\quad \text{ in } \quad L^2([0,T]\times K) \end{aligned}$$

(35)

for any fixed $M>0$ and any compact set $K\subset \Omega$.

On the other hand, applying operator Q to Eq. (34) and differentiating in time, by use also of bounds (33) we discover that (for any $M>0$ fixed) the family $(\partial _tQ(r_{\varepsilon ,M},\,V_{\varepsilon ,M}))_\varepsilon$ is uniformly bounded (with respect to $\varepsilon$) in the space $L^2_T(H_M)$. Moreover, as $H_M\hookrightarrow H^m$ for any $m\in \mathbb {N}$, we infer also that it is compactly embedded in $L^2(K)$ for any $M>0$ and any compact subset $K\subset \Omega$. Hence, Ascoli–Arzelà theorem implies that, for $\varepsilon \rightarrow 0$,

$$\begin{aligned} Q(r_{\varepsilon ,M},\,V_{\varepsilon ,M})\,\longrightarrow \,(r_M,\,u_M)\quad \text{ in } \quad L^2([0,T]\times K). \end{aligned}$$

(36)

4.2 Passing to the limit

In the present subsection we conclude the proof of Theorem 2.4 when $\alpha =1$. First of all, we show strong convergence of the $r_\varepsilon$’s and the velocity fields; then we pass to the limit in the weak formulation of the equations, and we identify the limit system.

4.2.1 Strong convergence of the velocity fields

The goal of the present paragraph is to prove the following proposition, which will allow us to pass to the limit in the weak formulation (8)–(9) of our system.

Proposition 4.4

Let $\alpha =1$ and $\gamma =2$. For any $T>0$, for $\varepsilon \rightarrow 0$ one has, up to extraction of a subsequence, the strong convergences

$$\begin{aligned} r_\varepsilon \,\longrightarrow \,r\quad \text{ and } \quad \rho _\varepsilon ^{3/2}\,u_\varepsilon \,\longrightarrow \,u\quad \text{ in } \quad L^2([0,T];L^2_{loc}(\Omega )). \end{aligned}$$

Proof

We start by decomposing $\rho ^{3/2}_\varepsilon \,u_\varepsilon$ into low and high frequencies: namely, for any $M>0$ fixed, we can write

$$\begin{aligned} \rho ^{3/2}_\varepsilon \,u_\varepsilon =P_M(\rho ^{3/2}_\varepsilon \,u_\varepsilon )+\left( \mathrm{Id}\,-P_M\right) (\rho ^{3/2}_\varepsilon \,u_\varepsilon ). \end{aligned}$$

Let us consider the low frequencies term first: again, it can be separated into the sum of two pieces, namely

$$\begin{aligned} P_M(\rho ^{3/2}_\varepsilon \,u_\varepsilon )=\varepsilon \,P_M(\varepsilon ^{-1}\left( \sqrt{\rho _\varepsilon }-1\right) \,\rho _\varepsilon \,u_\varepsilon )+ P_M(\rho _\varepsilon \,u_\varepsilon ). \end{aligned}$$

By uniform bounds (recall also Sect. 3.3), we have that $(\rho _\varepsilon \,u_\varepsilon )_\varepsilon$ is bounded in $L^2_T(L^2)$, while $(\varepsilon ^{-1}\left( \sqrt{\rho _\varepsilon }-1\right) )_\varepsilon$ is clearly bounded in $L^\infty _T(L^2)$. Then, using also Bernstein’s inequalities, we infer that the former item in the previous equality goes to 0 in $L^2_T(L^2)$, in the limit for $\varepsilon \rightarrow 0$.

On the other hand, by properties (35) and (36) we immediately get that $P_M(\rho _\varepsilon \,u_\varepsilon )$ converges to $u_M=P_M(u)$ strongly $L^2_T(L^2_{loc})$. Recall that u is the limit velocity field identified in Sect. 3.3, and which has to satisfy, together with r, the constraints given in Proposition 3.5.

We deal now with the high frequencies term. Recall that, by decomposition (20), we have already deduced the uniform inclusion $(D(\rho _\varepsilon ^{3/2}\,u_\varepsilon ))_\varepsilon \,\subset \,L^2_T(L^2+L^{3/2})$. Then by Lemma 7.3 and Proposition 7.4, we get

$$\begin{aligned} \left\| \left( \mathrm{Id}\,-P_M\right) (\rho ^{3/2}_\varepsilon \,u_\varepsilon )\right\| _{L^2_T(L^2)}\,\le \,c_M, \end{aligned}$$

for some constant $c_M$, depending just on M (and not on $\varepsilon$) and which tends to 0 for $M\rightarrow +\infty$.

For the convergence of $r_\varepsilon$ to r one can argue in an analogous way. The control of the high frequency part is actually easier, thanks to (22). For the low frequencies, we decompose again $P_Mr_\varepsilon =QP_Mr_\varepsilon +Q^\perp P_Mr_\varepsilon$, for which we use (36) and (35) respectively.

The proposition is then proved. $\square$

4.2.2 The limit system

Thanks to the convergence properties established in Sect. 3.3 and by Proposition 4.4, we can pass to the limit in the weak formulation (8)–(9). For this, we evaluate the equations on an element which already belongs to $\mathrm{Ker}\,\mathcal {A}$.

So, let us take $\phi \in \mathcal {D}([0,T[\,\times \Omega )$, with $\phi =\phi (x^h)$, and use $\psi =\left( \nabla ^\perp _h\phi ,0\right)$ as a test function in Eq. (9): since $\mathrm{div}\,\psi =0$, we get

$$\begin{aligned}&\int ^T_0\int _\Omega (-\rho _\varepsilon ^2\,u_\varepsilon \cdot \partial _t\psi \,-\,\rho _\varepsilon u_\varepsilon \otimes \rho _\varepsilon u_\varepsilon :\nabla \psi \nonumber \\&\quad \quad +\rho _\varepsilon ^2\left( u_\varepsilon \cdot \psi \right) \mathrm{div}\,u_\varepsilon +\frac{1}{\varepsilon }\,e^3\times \rho _\varepsilon ^2u_\varepsilon \cdot \psi \,+ \nonumber \\&\quad \quad +\,\nu \rho _\varepsilon Du_\varepsilon :\rho _\varepsilon \nabla \psi +\nu \rho _\varepsilon Du_\varepsilon :\left( \psi \otimes \nabla \rho _\varepsilon \right) + 2\,\rho _\varepsilon \Delta \rho _\varepsilon \nabla \rho _\varepsilon \cdot \psi )\,dx\,dt\nonumber \\&\quad =\int _\Omega \rho _{0,\varepsilon }^2\,u_{0,\varepsilon }\cdot \psi (0)\,dx. \end{aligned}$$

(37)

Now, we rewrite the rotation term in the following way:

$$\begin{aligned} \frac{1}{\varepsilon }\int ^T_0\!\!\int _\Omega e^3\times \rho _\varepsilon ^2u_\varepsilon \cdot \psi= & {} \frac{1}{\varepsilon }\int ^T_0\!\!\int _\Omega \rho _\varepsilon ^2u^h_\varepsilon \cdot \nabla _h\phi \\= & {} \frac{1}{\varepsilon }\int ^T_0\!\!\int _\Omega \rho _\varepsilon u^h_\varepsilon \cdot \nabla _h\phi +\int ^T_0\!\!\int _\Omega r_\varepsilon \rho _\varepsilon u^h_\varepsilon \cdot \nabla _h\phi \\= & {} -\,\int _\Omega r_{0,\varepsilon }\,\phi (0)\,-\,\int ^T_0\!\!\int _\Omega r_\varepsilon \,\partial _t\phi +\int ^T_0\!\!\int _\Omega r_\varepsilon \,\rho _\varepsilon \, u^h_\varepsilon \cdot \nabla _h\phi , \end{aligned}$$

where the last equality comes from the mass equation (8) tested on $\phi$. Notice that the last term in the right-hand side converges, due to (23) and the strong convergence of $r_\varepsilon$ in $L^2_T(L^2)$ (which is guaranteed by Proposition 4.4).

Using again the trick $\rho _\varepsilon =1+(\rho _\varepsilon -1)$, we can also write

$$\begin{aligned} 2\int ^T_0\int _\Omega \rho _\varepsilon \Delta \rho _\varepsilon \nabla \rho _\varepsilon \cdot \psi \!=\!2\int ^T_0\int _\Omega \Delta \rho _\varepsilon \, \nabla \rho _\varepsilon \cdot \psi \!+\!2\int ^T_0\int _\Omega \left( \rho _\varepsilon -1\right) \,\Delta \rho _\varepsilon \,\nabla \rho _\varepsilon \cdot \psi \,: \end{aligned}$$

by uniform bounds and (19), it is easy to see that both terms goes to 0 for $\varepsilon \rightarrow 0$.

Putting these last two relations into (37) and using convergence properties established above in order to pass to the limit, we arrive at the equation

$$\begin{aligned}&\int ^T_0\int _\Omega (-\,u\cdot \partial _t\psi \,-\,u\otimes u:\nabla \psi \,-\,r\,\partial _t\phi \nonumber \\&\quad +\,r\,u^h\cdot \nabla _h\phi +\nu Du:\nabla \psi )\,dx\,dt= \int _\Omega (u_0\cdot \psi (0)+r_0\,\phi (0))\,dx. \end{aligned}$$

(38)

Now we use that $\psi =(\nabla _h^\perp \phi ,0)$ and that, by Proposition 3.5, $u=(\nabla ^\perp _hr,0)$. Keeping in mind that all these functions don’t depend on $x^3$, by integration by parts we get

$$\begin{aligned} -\int ^T_0\int _\Omega u\cdot \partial _t\psi \,dx\,dt= & {} \int ^T_0\int _{\mathbb {R}^2}\Delta _hr\,\partial _t\phi \,dx^h\,dt \\ -\int ^T_0\int _\Omega u\otimes u:\nabla \psi \,dx\,dt= & {} -\int ^T_0\int _{\mathbb {R}^2}\nabla ^\perp _hr\cdot \nabla _h\Delta _hr\, \phi \,dx^h\,dt \\ \nu \int ^T_0\int _\Omega Du:\nabla \psi \,dx\,dt= & {} \frac{\nu }{2}\int ^T_0\int _{\mathbb {R}^2}\Delta _h^2r\,\phi \,dx^h\,dt. \end{aligned}$$

In the same way, one can show that the following identity holds true:

$$\begin{aligned} \int ^T_0\int _\Omega r\,u^h\cdot \nabla _h\phi \,dx\,dt=\int ^T_0\int _{\mathbb {R}^2}\nabla _hr\cdot \nabla ^\perp _hr\,\phi \,dx^h\,dt=0. \end{aligned}$$

Putting all these equalities together completes the proof of Theorem 2.4 in the case $\alpha =1$.

5 The case $\alpha =0$

We consider now the case of constant capillarity coefficient, i.e. $\alpha =0$: the present section is devoted to the proof of Theorem 2.6.

The main issue of the analysis here is that, now, the singular perturbation operator becomes

$$\begin{aligned} \mathcal {A}_0\,: L^2(\Omega )\;\times \;L^2(\Omega )&\longrightarrow H^{-1}(\Omega )\;\times \;H^{-3}(\Omega ) \nonumber \\ (\,r\;,\;V\,)&\mapsto (\mathrm{div}\,V,\;e^3\times V\,+\nabla (\mathrm{Id}\,-\Delta )r), \end{aligned}$$

(39)

which is no more skew-adjoint with respect to the usual $L^2$ scalar product. Nonetheless, it is possible to symmetrize our system, i.e. it is possible to find a scalar product on the space $H_M$, defined in (31), with respect to which the operator $\mathcal {A}_0$ becomes skew-adjoint.

Indeed, passing in Fourier variables, one can easily compute a positive self-adjoint $4\times 4$ matrix $\mathcal S$ such that $\mathcal S\,\mathcal {A}_0=-\,\mathcal {A}_0^*\,\mathcal S$, which is exactly the condition for $\mathcal {A}_0$ to be skew-adjoint with respect to the scalar product defined by $\mathcal S$. Hence, one can apply the RAGE theorem to $\mathcal {A}_0$, acting on $H_M$ endowed with the scalar product $\mathcal S$.

After this brief introduction, let us go back to the proof of Theorem 2.6. As before, we first analyse the acoustic waves, proving strong convergence for the velocity fields, and then we will pass to the limit.

5.1 Propagation of acoustic waves

In the case $\alpha =0$, the equation for acoustic waves is the same as (29), with operator $\mathcal {A}$ replaced by $\mathcal {A}_0$: namely, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon \,\partial _tr_\varepsilon +\mathrm{div}\,\,V_\varepsilon =0 \\ \varepsilon \,\partial _tV_\varepsilon +(e^3\times V_\varepsilon +\nabla (\mathrm{Id}\,-\Delta )r_\varepsilon )=\varepsilon \,\widetilde{f}_\varepsilon , \end{array}\right. } \end{aligned}$$

(40)

where $\widetilde{f}_\varepsilon$ is analogous to $f_\varepsilon$, which was defined in Sect. 4.1.1, but with the last term of formula (30) replaced by

$$\begin{aligned} \frac{1}{\varepsilon ^2}\,(\rho _\varepsilon -1)\,\nabla \Delta \rho _\varepsilon . \end{aligned}$$

In particular, we have

$$\begin{aligned} \langle \widetilde{f}_\varepsilon ,\phi \rangle:= & {} \int _\Omega (\rho _\varepsilon u_\varepsilon \otimes u_\varepsilon :\nabla \phi \,-\, \nu \,\rho _\varepsilon Du_\varepsilon :\nabla \phi \,-\,\frac{1}{\varepsilon ^2}\,\Delta \rho _\varepsilon \,\nabla \rho _\varepsilon \cdot \phi \, \\&-\,\frac{1}{\varepsilon ^2}\,(\rho _\varepsilon -1)\,\Delta \rho _\varepsilon \,\mathrm{div}\,\phi + \frac{1}{\varepsilon ^2}(P(\rho _\varepsilon )-P(1)-P'(1)\left( \rho _\varepsilon -1\right) )\mathrm{div}\,\phi )dx. \end{aligned}$$

Exactly as before, by uniform bounds we get that $(\widetilde{f}_\varepsilon )_\varepsilon$ is bounded in $L^2_T(W^{-1,2}(\Omega )+W^{-1,1}(\Omega ))$.

Recall also that, as in the previous case, Eq. (40) hold true when computed on test functions of the form $(\varphi ,\,\rho _\varepsilon \,\psi )$.

Let us turn our attention to the acoustic propagator $\mathcal {A}_0$, defined in (39). We have the following statement, which is the analogous of Proposition 4.1.

Proposition 5.1

One has $\sigma _p(\mathcal {A}_0)=\{0\}$. In particular, $\mathrm{Eigen}\,\mathcal {A}_0\;\equiv \;\mathrm{Ker}\,\mathcal {A}_0$.

Proof

We have to look for $\lambda \in \mathbb {C}$ for which the following system

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm{div}\,V\;=\;\lambda \,r \\ e^3\times V+\nabla (r\,-\,\Delta r)\;=\;\lambda \,V \end{array}\right. } \end{aligned}$$

has non-trivial solutions $(r,V)\ne (0,0)$.

Denoting by $\widehat{v}$ the Fourier transform of a function v in the domain $\mathbb {R}^2\times \mathbb {T}^1$, defined for any $(\xi ^h,k)\in \mathbb {R}^2\times \mathbb {Z}$ by the formula

$$\begin{aligned} \widehat{v}(\xi ^h,k)\,:=\,\frac{1}{\sqrt{2}}\int _{-1}^{1}\int _{\mathbb {R}^2}e^{-i\,x^h\cdot \xi ^h}\,v(x^h,x^3)\,dx^h\;e^{-i\,x^3k}\,dx^3, \end{aligned}$$

we can write the previous system in the equivalent way

$$\begin{aligned} {\left\{ \begin{array}{ll} i\left( \xi ^h\cdot \widehat{V}^h+k\,\widehat{V}^3\right) \;=\;\lambda \,\widehat{r} \\ -\widehat{V}^2+i\,\xi ^1\left( 1+\left| \xi ^h\right| ^2+k^2\right) \widehat{r}\;=\;\lambda \,\widehat{V}^1 \\ \widehat{V}^1+i\,\xi ^2\left( 1+\left| \xi ^h\right| ^2+k^2\right) \widehat{r}\;=\;\lambda \,\widehat{V}^2 \\ \,i\,k\left( 1+\left| \xi ^h\right| ^2+k^2\right) \widehat{r}\;=\;\lambda \,\widehat{V}^3, \end{array}\right. } \end{aligned}$$

where $\left| \xi ^h\right| ^2=\left| \xi ^1\right| ^2+\left| \xi ^2\right| ^2$. For notation convenience, let us set $\zeta (\xi ^h,k)=\left| \xi ^h\right| ^2+k^2$: after easy computations, we arrive to the following equation for $\lambda$,

$$\begin{aligned} \lambda ^4+(1+\zeta (\xi ^h,k)+\zeta ^2(\xi ^h,k))\,\lambda ^2+ k^2\,(1+\zeta (\xi ^h,k))=0, \end{aligned}$$

from which we immediately infer that

$$\begin{aligned} \lambda ^2=-\,\frac{1}{2}\left( 1+\zeta +\zeta ^2\pm \sqrt{\left( 1+\zeta +\zeta ^2\right) ^2\,-\,4\,k^2\,(1+\zeta )}\right) . \end{aligned}$$

To have $\lambda$ in the discrete spectrum of $\mathcal {A}_0$, we need to delete its dependence on $\xi ^h$: since $1+\zeta >0$, the only way to do it is to have $k=0$, for which $\lambda =0$. $\square$

Now, for any fixed $M>0$, we consider the space $H_M$, which was defined in (31), endowed with the scalar product

$$\begin{aligned} \langle (r_1,V_1),\,(r_2,V_2)\rangle _{H_M}\,:=\,\langle r_1,\,(\mathrm{Id}\,-\Delta )r_2\rangle _{L^2}+\langle V_1,\,V_2\rangle _{L^2}. \end{aligned}$$

(41)

In fact, it is easy to verify that the previous bilinear form is symmetric and positive definite. Moreover, we observe that $\Vert (r,V)\Vert ^2_{H_M}=\left\| (\mathrm{Id}\,-\Delta )^{1/2}r\right\| ^2_{L^2}+\left\| V\right\| ^2_{L^2}$. Straightforward computations also show that $\mathcal {A}_0$ is skew-adjoint with respect to this scalar product, namely

$$\begin{aligned} \langle \mathcal {A}_0(r_1,V_1),\,(r_2,V_2)\rangle _{H_M}=-\,\langle (r_1,V_1),\,\mathcal {A}_0(r_2,V_2)\rangle _{H_M}. \end{aligned}$$

Now, let us set $P_M\,:\,L^2(\Omega )\times L^2(\Omega )\,\longrightarrow \,H_M$ to be the orthogonal projection onto $H_M$, as in Sect. 4.1.2. This time, for a fixed $\theta \in \mathcal {D}(\Omega )$ such that $0\le \theta \le 1$, we define the operator

$$\begin{aligned} \widetilde{\mathcal {K}}_{M,\theta }(r,V)\,:=\,((\mathrm{Id}\,-\Delta )^{-1}P_M(\theta \,P_Mr),\,P_M(\theta \,P_MV)). \end{aligned}$$

Note that $\widetilde{\mathcal {K}}_{M,\theta }$ is self-adjoint and positive with respect to the scalar product $\langle \,\cdot ,\,\cdot \,\rangle _{H_M}$. Moreover, as before, it is compact by Rellich–Kondrachov theorem.

Now, exactly as done in Sect. 4.1.2, we apply the RAGE theorem to

$$\begin{aligned} \mathcal H=H_M\;,\quad \mathcal {B}=i\,\mathcal {A}_0\;,\quad \mathcal {K}=\widetilde{\mathcal {K}}_{M,\theta }\quad \text{ and } \quad \Pi _\mathrm{cont}=Q^\perp , \end{aligned}$$

where Q and $Q^\perp$ are still the orthogonal projections onto respectively $\mathrm{Ker}\,\mathcal {A}_0$ and $(\mathrm{Ker}\,\mathcal {A}_0)^\perp$.

Since, with our definitions, we still have

$$\begin{aligned} \left\| \left( \widetilde{\mathcal {K}}_{M,\theta }\right) ^{1/2}\,Q^\perp (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\right\| ^2_{H_M}= \int _\Omega \theta \,\left| Q^\perp (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\right| ^2\,dx, \end{aligned}$$

a direct application of the RAGE theorem (or better of Corollary 4.3) immediately gives us

$$\begin{aligned} Q^\perp (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\,\longrightarrow \,0\quad \text{ in } \quad L^2([0,T]\times K) \end{aligned}$$

(42)

for any fixed $M>0$ and any compact $K\subset \Omega$.

On the other hand, exactly as we did in Sect. 4.1.2, by Ascoli–Arzelà theorem we can deduce the strong convergence

$$\begin{aligned} Q(r_{\varepsilon ,M},\,V_{\varepsilon ,M})\,\longrightarrow \,(r_M,\,u_M)\quad \text{ in } \quad L^2([0,T]\times K). \end{aligned}$$

(43)

5.2 Passing to the limit

Thanks to relations (42) and (43), Proposition 4.4 still holds true: namely, we have the strong convergences of

$$\begin{aligned} r_\varepsilon \,\longrightarrow \,r\quad \text{ and } \quad \rho _\varepsilon ^{3/2}\,u_\varepsilon \;\longrightarrow \;u\quad \text{ in } \quad L^2([0,T];L^2_{loc}(\Omega )), \end{aligned}$$

and this allows us to pass to the limit in the non-linear terms. Note that we get in particular the strong convergence of $(\nabla r_\varepsilon )_\varepsilon$ in $L^2_T(H^{-1}_{loc})$ (up to extraction of a subsequence); on the other hand, by uniform bounds we know that this family is bounded in $L^2_T(H^1)$. Then, by interpolation we have also the strong convergence in all the intermediate spaces, and especially

$$\begin{aligned} \nabla r_\varepsilon \,\longrightarrow \,\nabla r\quad \text{ in } \quad L^2([0,T];L^2_{loc}(\Omega )). \end{aligned}$$

(44)

In order to compute the limit system, let us take $\phi \in \mathcal {D}([0,T[\,\times \Omega )$, with $\phi =\phi (x^h)$, and use $\psi =\left( \nabla ^\perp _h\phi ,0\right)$ as a test function in Eq. (9). Since $\mathrm{div}\,\psi =0$, as before we get

$$\begin{aligned}&\int ^T_0\!\!\!\int _\Omega (-\rho _\varepsilon ^2\,u_\varepsilon \cdot \partial _t\psi \,-\,\rho _\varepsilon u_\varepsilon \otimes \rho _\varepsilon u_\varepsilon :\nabla \psi + \rho _\varepsilon ^2\left( u_\varepsilon \cdot \psi \right) \mathrm{div}\,u_\varepsilon +\frac{1}{\varepsilon }\,e^3\times \rho _\varepsilon ^2u_\varepsilon \cdot \psi \nonumber \\&\qquad +\,\nu \rho _\varepsilon Du_\varepsilon :\rho _\varepsilon \nabla \psi +\nu \rho _\varepsilon Du_\varepsilon :\left( \psi \otimes \nabla \rho _\varepsilon \right) + \frac{2}{\varepsilon ^2}\,\rho _\varepsilon \Delta \rho _\varepsilon \nabla \rho _\varepsilon \cdot \psi )\,dx\,dt\nonumber \\&\quad =\int _\Omega \rho _{0,\varepsilon }^2\,u_{0,\varepsilon }\cdot \psi (0)\,dx. \end{aligned}$$

(45)

Also in this case, we rewrite the rotation term by using the mass Eq. (8):

$$\begin{aligned} \frac{1}{\varepsilon }\int ^T_0\!\!\int _\Omega e^3\times \rho _\varepsilon ^2u_\varepsilon \cdot \psi= & {} \frac{1}{\varepsilon }\int ^T_0\!\!\int _\Omega \rho _\varepsilon u^h_\varepsilon \cdot \nabla _h\phi +\frac{1}{\varepsilon }\int ^T_0\!\!\int _\Omega \left( \rho _\varepsilon -1\right) \rho _\varepsilon u^h_\varepsilon \cdot \nabla _h\phi \\= & {} -\,\int _\Omega r_{0,\varepsilon }\,\phi (0)\,-\,\int ^T_0\!\!\int _\Omega r_\varepsilon \,\partial _t\phi +\int ^T_0\!\!\int _\Omega r_\varepsilon \,\rho _\varepsilon \, u^h_\varepsilon \cdot \nabla _h\phi . \end{aligned}$$

Again, the last term in the right-hand side converges, due to (23) and the strong convergence of $r_\varepsilon$ in $L^2_T(L^2)$.

For analysing the capillarity term, we write

$$\begin{aligned}&\frac{2}{\varepsilon ^2}\int ^T_0\int _\Omega \rho _\varepsilon \Delta \rho _\varepsilon \nabla \rho _\varepsilon \cdot \psi = \frac{2}{\varepsilon ^2}\int ^T_0\int _\Omega \Delta \rho _\varepsilon \,\nabla \rho _\varepsilon \cdot \psi \\&\quad +\frac{2}{\varepsilon ^2}\int ^T_0\int _\Omega \left( \rho _\varepsilon -1\right) \,\Delta \rho _\varepsilon \,\nabla \rho _\varepsilon \cdot \psi . \end{aligned}$$

By uniform bounds, we gather that the second term goes to 0; on the other hand, combining (44) with the weak convergence of $\Delta r_\varepsilon$ in $L^2_T(L^2)$ implies that also the first term converges for $\varepsilon \rightarrow 0$.

Putting these last two relations into (45) and using convergence properties established above in order to pass to the limit, we arrive at the equation

$$\begin{aligned}&\int ^T_0\int _\Omega (-\,u\cdot \partial _t\psi \,-\,u\otimes u:\nabla \psi \,-\,r\,\partial _t\phi \, \nonumber \\&\quad +\,r\,u^h\cdot \nabla _h\phi +\nu Du:\nabla \psi +2\,\Delta r\,\nabla r\cdot \psi )\,dx\,dt\nonumber \\&\quad =\int _\Omega (u_0\cdot \psi (0)+r_0\,\phi (0))\,dx. \end{aligned}$$

(46)

Now we use that $\psi =(\nabla _h^\perp \phi ,0)$ and that, by Proposition 3.5, $u=(\nabla ^\perp _h\widetilde{r},0)$, where we have set $\widetilde{r}\,:=\,\left( \mathrm{Id}\,-\Delta \right) r$. Keeping in mind that all these functions do not depend on $x^3$, by integration by parts we get

$$\begin{aligned} -\int ^T_0\int _\Omega u\cdot \partial _t\psi \,dx\,dt= & {} \int ^T_0\int _{\mathbb {R}^2}\Delta _h\widetilde{r}\,\partial _t\phi \,dx^h\,dt \\ -\int ^T_0\int _\Omega u\otimes u:\nabla \psi \,dx\,dt= & {} -\int ^T_0\int _{\mathbb {R}^2}\nabla ^\perp _h\widetilde{r}\cdot \nabla _h\Delta _h\widetilde{r}\, \phi \,dx^h\,dt \\= & {} -\int ^T_0\int _{\mathbb {R}^2}\nabla ^\perp _hr\cdot \nabla _h\Delta _hr\,\phi \,dx^h\,dt\,+ \\&+\,\int ^T_0\int _{\mathbb {R}^2}\nabla ^\perp _hr\cdot \nabla _h\Delta ^2_hr\,\phi \,dx^h\,dt\,\\&-\, \int ^T_0\int _{\mathbb {R}^2}\nabla ^\perp _h\Delta _hr\cdot \nabla _h\Delta ^2_hr\,\phi \,dx^h\,dt \\ \nu \int ^T_0\int _\Omega Du:\nabla \psi \,dx\,dt= & {} \frac{\nu }{2}\int ^T_0\int _{\mathbb {R}^2}\Delta _h^2\widetilde{r}\,\phi \,dx^h\,dt. \end{aligned}$$

Moreover, it is also easy to see that the following identities hold true:

$$\begin{aligned} \int ^T_0\int _\Omega r\,u^h\cdot \nabla _h\phi \,dx\,dt= & {} -\int ^T_0\int _{\mathbb {R}^2}\nabla ^\perp _hr\cdot \nabla _h\Delta _h r\,\phi \,dx^h\,dt \\ 2\int ^T_0\int _\Omega \Delta r\,\nabla r\cdot \psi \,dx\,dt= & {} 2\int ^T_0\int _{\mathbb {R}^2}\nabla ^\perp _hr\cdot \nabla _h\Delta _h r\,\phi \,dx^h\,dt. \end{aligned}$$

Then, these terms, together with the first one coming from the transport part $u\otimes u$, cancel out.

Hence, putting all these equalities together gives us the quasi-geostrophic type equation stated in Theorem 2.6, which is now completely proved.

6 Vanishing capillarity limit: anisotropic scaling

In this section we complete the proof of Theorem 2.4, focusing on the remaining cases . The results of Sect. 3 still holding true, we just have to analyse the propagation of acoustic waves and to prove strong convergence of the velocity fields.

First of all, let us write system (2) in the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon \,\partial _tr_\varepsilon +\mathrm{div}\,\,V_\varepsilon =0 \\ \varepsilon \,\partial _tV_\varepsilon +(e^3\times V_\varepsilon +\nabla r_\varepsilon )=\varepsilon \,f_{\varepsilon ,\alpha }+ \varepsilon ^{\alpha }\,g_\varepsilon . \end{array}\right. } \end{aligned}$$

Here, like in the previous section, $f_{\varepsilon ,\alpha }$ is obtained from $f_\varepsilon$ of Sect. 4.1.1, by replacing the last term of formula (30) with

$$\begin{aligned} \frac{1}{\varepsilon ^{2(1-\alpha )}}\,(\rho _\varepsilon -1)\,\nabla \Delta \rho _\varepsilon ; \end{aligned}$$

moreover, we have defined

$$\begin{aligned} g_\varepsilon \,:=\,\frac{1}{\varepsilon ^{1-\alpha }}\,\nabla \Delta (\rho _\varepsilon -1). \end{aligned}$$

Notice that $(f_{\varepsilon ,\alpha })_\varepsilon$ is bounded in $L^2_T(W^{-1,2}(\Omega )+W^{-1,1}(\Omega ))$ for any $\alpha$, while uniform bounds imply that $(g_\varepsilon )_\varepsilon$ is bounded in $L^2_T(W^{-1,2}(\Omega ))$.

For , we remark that the term $g_\varepsilon$ is of higher order than $f_{\varepsilon ,\alpha }$: then, we cannot treat it as a remainder. Then, the first step is to put it on the left-hand side of the equation, and to read it as a small perturbation of the acoustic propagator $\mathcal {A}$ (defined in Sect. 4.1.1). Hence, we are led to consider a one-parameter continuous family of operators, each one of which admits a symmetrizer on the Hilbert space $H_M$.

Roughly speaking, all these operators have the same point spectrum (we will be much more precise below, see Sect. 6.2): the idea is then to apply a sort of RAGE theorem for one-parameter family of operators and metrics, in order to prove dispersion of the components of the solutions orthogonal to the kernels of the acoustic propagators.

In what follows, first of all we will set up the problem in an abstract way, showing a RAGE-type theorem for families of operators and metrics. This having been done, we will apply the general theory to our particular case, and this will complete the proof of Theorem 2.4 for $\alpha \,\in \,]0,1[$.

6.1 RAGE theorem depending on a parameter

As just said, we want to extend the RAGE theorem to the case when both operators and metrics depend on a small parameter $\eta$ (for us, $\eta =\varepsilon ^{2\alpha }$).

For the sake of completeness, we start by presenting some variants of the Wiener theorem, which is the basis to prove the RAGE theorem.

6.1.1 Variants of the Wiener theorem

First of all, some definitions are in order.

Definition 6.1

Given two positive measures $\mu$ and $\nu$ defined on a measurable space $(X,\Sigma )$, we say $\;\mu \,\le \,\nu \;$ if $\mu (A)\le \nu (A)$ for any measurable set $A\in \Sigma$.

Definition 6.2

Let $(\mu _{\eta })_{\eta }$ be a one-parameter family of positive measures on a measurable space $(X,\Sigma )$. We say that it is a continuous family (with respect to $\eta$) if, for any $A\in \Sigma$, the map $\eta \,\mapsto \,\mu _\eta (A)$ is continuous from [0, 1] to $\mathbb {R}_+$.

The notion of continuity we adopt corresponds then to the strong topology in the space of measures on $(X,\Sigma )$. Notice that this notion requires no uniformity with respect to $A\in \Sigma$.

The first result is a very simple adaptation of the original Wiener theorem, which can be found e.g. in [28] (see the Appendix to Section XI.17). Its proof goes along the lines of the original one: for later use, however, we give the most of the details.

Proposition 6.3

Let $(\mu _\eta )_{\eta \in [0,1]}$ be a family of finite Baire measures on $\mathbb {R}$, such that

$$\begin{aligned} \mu _{\eta _1}\,\le \,\mu _{\eta _2}\quad \forall \quad 0\,\le \,\eta _1\,\le \,\eta _2\,\le \,1. \end{aligned}$$

(47)

For any $\eta \in [0,1]$, let us define the Fourier transform of $\mu _\eta$ by the formula

$$\begin{aligned} F_\eta (t)\,:=\,\int _\mathbb {R}e^{-ixt}\,d\mu _\eta (x). \end{aligned}$$

Then one has

$$\begin{aligned} \lim _{\eta \rightarrow 0}\,\lim _{T\rightarrow +\infty }\,\frac{1}{2T}\int ^T_{-T}\left| F_\eta (t)\right| ^2\,dt= \sum _{x\in \mathbb {R}}|\mu _0\left( \{x\}\right) |^2. \end{aligned}$$

In particular, if $\mu _0$ has no pure points, then the limit is 0.

Proof

Like in the proof of the original statement, for any fixed $\eta$ we can write

$$\begin{aligned} \frac{1}{2T}\int ^T_{-T}\left| F_\eta (t)\right| ^2\,dt= & {} \int _\mathbb {R}d\mu _\eta (x)\left( \int _\mathbb {R}d\mu _\eta (y) \left( \frac{1}{2T}\int ^T_{-T}e^{-i(x-y)t}\,dt\right) \right) \\= & {} \int _\mathbb {R}d\mu _\eta (x)\left( \int _\mathbb {R}d\mu _\eta (y)\,\frac{\sin (T(x-y))}{T(x-y)}\right) \end{aligned}$$

by Fubini’s theorem. Let us now define

$$\begin{aligned} H_\eta (T,x)\,:=\,\int _\mathbb {R}\frac{\sin (T(x-y))}{T(x-y)}\,d\mu _\eta (y)\!: \end{aligned}$$

the integrand in $H_\eta$ is pointwise bounded by 1; moreover, for $T\longrightarrow +\infty$, it converges to 0 if $y\ne x$, and to 1 if $y=x$. Hence, by dominated convergence theorem we have

$$\begin{aligned} \lim _{T\rightarrow +\infty }H_\eta (T,x)=\mu _\eta \left( \{x\}\right) . \end{aligned}$$

Moreover, $|H_\eta (T,x)|\,\le \,\mu _\eta (\mathbb {R})$; then, by dominated convergence theorem again we infer that

$$\begin{aligned} \lim _{T\rightarrow +\infty }\int _\mathbb {R}d\mu _\eta (x)\left( \int _\mathbb {R}d\mu _\eta (y)\,\frac{\sin (T(x-y))}{T(x-y)}\right) = \sum _{x\in \mathbb {R}}|\mu _\eta \left( \{x\}\right) |^2. \end{aligned}$$

Finally, we take the limit for $\eta \longrightarrow 0$ and we apply the monotone convergence theorem. $\square$

Remark 6.4

Note that, if monotonicity hypothesis (47) is not fulfilled, then one gets

$$\begin{aligned} \lim _{\eta \rightarrow 0}\,\lim _{T\rightarrow +\infty }\,\frac{1}{2T}\int ^T_{-T}\left| F_\eta (t)\right| ^2\,dt= \lim _{\eta \rightarrow 0}\,\sum _{x\in \mathbb {R}}|\mu _\eta \left( \{x\}\right) |^2. \end{aligned}$$

In particular, if $\mu _\eta$ has no pure points for any $\eta$, then still the limit is 0.

We are now interested in linking the parameters $\eta$ and T together, and in performing the two limits at the same time. In this case, we can no more apply the dominated convergence theorem, as the measures themselves change when T increases. However, the next statement says that the previous result still holds true.

Theorem 6.5

Let $\sigma :[0,1]\longrightarrow [0,1]$ be a continuous increasing function, such that $\sigma (0)=0$ and $\sigma (1)=1$. Let $(\mu _{\sigma (\varepsilon )})_{\varepsilon \in [0,1]}$ be a family of finite Baire measures on $\mathbb {R}$, such that one of the two following conditions holds true:

• $(\mu _{\eta })_{\eta \in [0,1]}$ is monotone increasing in the sense of inequality (47);

• $(\mu _{\eta })_{\eta \in [0,1]}$ is a continuous family, in the sense of Definition 6.2.

For any $\varepsilon \in [0,1]$, let us denote by $F_\varepsilon$ the Fourier transform of the measure $\mu _{\sigma (\varepsilon )}$.

Then we have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\,\frac{1}{2T}\int ^T_{-T}\left| F_\varepsilon (t/\varepsilon )\right| ^2\,dt= \sum _{x\in \mathbb {R}}|\mu _0\left( \{x\}\right) |^2. \end{aligned}$$

In particular, the limit is 0 if $\mu _0$ has no pure points.

Proof

First of all, by the change of variable $\tau =t/\varepsilon$, we are reconducted to prove that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\,\frac{\varepsilon }{2T}\int ^{T/\varepsilon }_{-T/\varepsilon }\left| F_\varepsilon (t)\right| ^2\,dt= \sum _{x\in \mathbb {R}}|\mu _0\left( \{x\}\right) |^2. \end{aligned}$$

Next, as done in the previous proof, the following equalities hold true:

$$\begin{aligned} \frac{\varepsilon }{2T}\int ^{T/\varepsilon }_{-T/\varepsilon }\left| F_\varepsilon (t)\right| ^2\,dt= & {} \int d\mu _{\sigma (\varepsilon )}(x)\left( \int d\mu _{\sigma (\varepsilon )}(y)\,\frac{\sin (T(x-y)/\varepsilon )}{T(x-y)/\varepsilon }\right) \\= & {} \int d\mu _{\sigma (\varepsilon )}(x)\left( \mu _{\sigma (\varepsilon )}(\{x\})\right. \\&\left. + \int _{y\ne x} d\mu _{\sigma (\varepsilon )}(y)\,\frac{\sin (T(x-y)/\varepsilon )}{T(x-y)/\varepsilon }\right) \\= & {} \sum _{x\in \mathbb {R}}|\mu _{\sigma (\varepsilon )}\left( \{x\}\right) |^2\\&+\int d\mu _{\sigma (\varepsilon )}(x)\left( \int _{y\ne x} d\mu _{\sigma (\varepsilon )}(y)\,\frac{\sin (T(x-y)/\varepsilon )}{T(x-y)/\varepsilon }\right) . \end{aligned}$$

By monotone convergence theorem (in the case of a monotone family) or dominated convergence theorem (in the case of a continuous family), the first term on the right-hand side converges to the same quantity computed in $\varepsilon =0$. So, we have just to prove that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\,\int d\mu _{\sigma (\varepsilon )}(x)\left( \int _{y\ne x}d\mu _{\sigma (\varepsilon )}(y)\, \left| \frac{\sin (T(x-y)/\varepsilon )}{T(x-y)/\varepsilon }\right| \right) =0. \end{aligned}$$

(48)

We first consider the case when the family of measures is monotone decreasing.

Let us fix a $\delta >0$, and let $\varepsilon _\delta$ be such that $\varepsilon /T\,\le \,\delta$ for all $\varepsilon \le \varepsilon _\delta$. Moreover, let us define the sets $Y_\le \,:=\,\left\{ y\ne x\,|\,|x-y|\le g(\delta )\right\}$ and $Y_\ge \,:=\,\left\{ y\ne x\,|\,|x-y|> g(\delta )\right\}$, for a suitable continuous function $g(\delta )$, going to 0 for $\delta \rightarrow 0$, to be determined later.

Then we can split the second integral in (48) into the sum of the integrals on $Y_\le$ and $Y_\ge$, and elementary inequalitites give us

$$\begin{aligned} \int _{y\ne x}d\mu _{\sigma (\varepsilon )}(y)\,\left| \frac{\sin (T(x-y)/\varepsilon )}{T(x-y)/\varepsilon }\right| \,\le \, \int _{Y_\le }d\mu _{\sigma (\varepsilon )}(y)+ \int _{Y_\ge }\frac{\varepsilon }{T\,|x-y|}\,d\mu _{\sigma (\varepsilon )}(y). \end{aligned}$$

We first focus on the former term: we have

$$\begin{aligned} \int d\mu _{\sigma (\varepsilon )}(x)\int _{Y_\le }d\mu _{\sigma (\varepsilon )}(y)\le & {} \int \mu _{\sigma (\varepsilon )}([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\mu _{\sigma (\varepsilon )}(x) \\\le & {} \int \mu _{1}([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\mu _{\sigma (\varepsilon )}(x) \\\le & {} \int \mu _{1}([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\mu _{1}(x), \end{aligned}$$

where the last inequality follow from the monotonicity property of the family of measures (and from the fact that the integrand does not depend on $\varepsilon$ anymore). From dominated convergence theorem, one can show that

$$\begin{aligned} \lim _{\delta \rightarrow 0}\int \mu _{1}([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\mu _{1}(x)=0 \end{aligned}$$

(recall that $g(\delta )\rightarrow 0$ for $\delta \rightarrow 0$), and from this fact we deduce

$$\begin{aligned} \int d\mu _{\sigma (\varepsilon )}(x)\int _{Y_\le }d\mu _{\sigma (\varepsilon )}(y)\,\le \,C_\delta , \end{aligned}$$

(49)

for some suitable $C_\delta$ converging to 0 for $\delta \rightarrow 0$.

We now consider the integral over $Y_\ge$. By definition of $\delta$, for any $\varepsilon \le \varepsilon _\delta$ one has

$$\begin{aligned} \int d\mu _{\sigma (\varepsilon )}(x)\left( \int _{Y_\ge }\frac{\varepsilon }{T\,|x-y|}\,d\mu _{\sigma (\varepsilon )}(y)\right)\le & {} \frac{\delta }{g(\delta )}\, \int \mu _{\sigma (\varepsilon )}(\mathbb {R})\,d\mu _{\sigma (\varepsilon )}(x) \\\le & {} \frac{\delta }{g(\delta )}\,|\mu _{\sigma (\varepsilon )}\left( \mathbb {R}\right) |^2. \end{aligned}$$

The term on the right-hand side can be bounded by the same quantity computed in $\mu _1$; therefore, if we take for instance $g(\delta )=\sqrt{\delta }$, we find, for all $\varepsilon \le \varepsilon _\delta$,

$$\begin{aligned} \int d\mu _{\sigma (\varepsilon )}(x)\left( \int _{Y_\ge }\frac{\varepsilon }{T\,|x-y|}\,d\mu _{\sigma (\varepsilon )}(y)\right) \,\le \,C\,\sqrt{\delta }. \end{aligned}$$

(50)

In the end, putting inequalities (50) and (49) together gives us relation (48), and this completes the proof of the theorem.

Let us now prove (48) in the case of a continuous family of measures. As before, we split the domain of the second integral into $Y_\le$ and $Y_\ge$.

The control of the integral over $Y_\ge$ can be performed exactly as done above:

$$\begin{aligned} \int d\mu _{\sigma (\varepsilon )}(x)\left( \int _{Y_\ge }\frac{\varepsilon }{T\,|x-y|}\,d\mu _{\sigma (\varepsilon )}(y)\right) \,\le \, \frac{\delta }{g(\delta )}\,|\mu _{\sigma (\varepsilon )}\left( \mathbb {R}\right) |^2\;\le \;C\,\frac{\delta }{g(\delta )} \end{aligned}$$

for any $\varepsilon \le \varepsilon _\delta$. Again, the choice $g(\delta )=\sqrt{\delta }$ gives us (49).

For the integral over $Y_\le$, we still have

$$\begin{aligned} \int d\mu _{\sigma (\varepsilon )}(x)\int _{Y_\le }d\mu _{\sigma (\varepsilon )}(y)\,\le \, \int \mu _{\sigma (\varepsilon )}([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\mu _{\sigma (\varepsilon )}(x). \end{aligned}$$

The term (say) I on the right-hand side of the previous inequality can be written as the sum of three terms: $I=I_1+I_2+I_3$, where we have defined

$$\begin{aligned} I_1:= & {} \int \left( \mu _{\sigma (\varepsilon )}\,-\,\mu _0\right) ([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\mu _{\sigma (\varepsilon )}(x) \\ I_2:= & {} \int \mu _0([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\left( \mu _{\sigma (\varepsilon )}(x)-\mu _0(x)\right) \\ I_3:= & {} \int \mu _0([x-g(\delta ),x+g(\delta )]{\setminus }\{x\})\,d\mu _0(x). \end{aligned}$$

By dominated convergence theorem, one can easily check that $I_3\rightarrow 0$ for $\delta \rightarrow 0$. Moreover, the following controls hold true:

$$\begin{aligned} \left| I_1\right| \,\le \,\mu _{\sigma (\varepsilon )}(\mathbb {R})\,\left| \mu _{\sigma (\varepsilon )}\,-\,\mu _0\right| (\mathbb {R})\quad \text{ and } \quad \left| I_2\right| \,\le \,\mu _{0}(\mathbb {R})\,\left| \mu _{\sigma (\varepsilon )}\,-\,\mu _0\right| (\mathbb {R}). \end{aligned}$$

By continuity hypothesis on the family of measures, hence, we have that for any $\varepsilon \le \,\varepsilon _\delta '$, the quantity $\left| I_1\right| +\left| I_2\right|$ is bounded by $C\,\sqrt{\delta }$, where C is a universal constant, independent of $\delta$ and $\varepsilon$.

In the end, we infer that there exists a $C>0$ such that, for all $\delta >0$ fixed and for any $\varepsilon \le \min \left\{ \varepsilon _\delta ,\varepsilon _\delta '\right\}$, one has

$$\begin{aligned} \int d\mu _{\sigma (\varepsilon )}(x)\left( \int _{y\ne x}d\mu _{\sigma (\varepsilon )}(y)\, \left| \frac{\sin (T(x-y)/\varepsilon )}{T(x-y)/\varepsilon }\right| \right) \,\le \,C\,\sqrt{\delta }, \end{aligned}$$

which completes the proof of (48) and of the theorem. $\square$

6.1.2 RAGE-type theorems

We are now ready to prove some results in the same spirit as the RAGE theorem, for families of operators and metrics.

Despite our attempt of generality, we have to make very precise assumptions for such families, which are modelled on our problem issued from the Navier–Stokes–Korteweg system. On the other hand, these hypothesis seem to us to be important in order to prove our result: we will point out where they will be used.

First of all, let us introduce some notations.

We are going to work in a fixed space $\mathcal {H}$; we will consider in $\mathcal H$ a continuous family of scalar products $(\mathcal S_\eta )_{\eta \in [0,1]}$, each one of which induces a Hilbert structure on $\mathcal H$. In general, we will write $(\mathcal H,\mathcal S_\eta )$ if we consider the Hilbert structure on $\mathcal H$ induced by the scalar product $\mathcal S_\eta$; if we do not specify the scalar product (for instance, in speaking of a self-adjoint operator), we mean we are referring to $\mathcal S_0$. In fact, $\mathcal S_0$ will be a sort of “reference metric” for us, and we will consider the $\mathcal S_\eta$’s like perturbations of it.

Moreover, we will use equivalently the notations $\mathcal S_\eta (X,Y)=\langle X,Y\rangle _\eta$, and we will denote by $\Vert \,\cdot \,\Vert _\eta$ the induced norm. We will also write $X\perp _\eta Y$ if X and Y are orthogonal with respect to $\mathcal S_\eta$; equally, given two subspaces $\mathcal E_1,\mathcal E_2\,\subset \,\mathcal H$, we write $\mathcal E_1\oplus _\eta \mathcal E_2$ if they are orthogonal with respect to $\mathcal S_\eta$. For a linear operator $\mathcal P$ defined on $\mathcal H$, we will set $\Vert \mathcal P\Vert _{\mathcal L(\eta )}$ its operator norm with respect to the scalar product $\mathcal S_\eta$; for $\eta =0$ we will use the notations $\Vert \,\cdot \,\Vert _{\mathcal L(0)}$ and $\Vert \,\cdot \,\Vert _{\mathcal L(\mathcal H)}$ in an equivalent way. Finally, the adjoint of $\mathcal P$ with respect to $\mathcal S_\eta$ will be denoted by $\mathcal P^{*(\eta )}$.

In the same time, we will consider a one-parameter family of operators $(\mathcal B_\eta )_{\eta \in [0,1]}$, and we will see each $\mathcal B_\eta$ as a perturbation of a self-adjoint operator $\mathcal B_0$ (recall that we mean self-adjoint with respect to $\mathcal S_0$).

From the original statement (see Theorem 4.2 above), we immediately infer the following one-parameter variant of the RAGE theorem.

Proposition 6.6

Let $(\mathcal H,\mathcal S_0)$ be a Hilbert space, and let $(\mathcal S_\eta )_{\eta \in [0,1]}$ be a one-parameter family of scalar products on $\mathcal H$, and suppose that they induces equivalent metrics, independently of $\eta$.

Let $(\mathcal B_\eta )_{\eta \in [0,1]}$ be a family of operators on $\mathcal H$ such that $\mathcal B_\eta$ is self-adjoint with respect to the inner product $\mathcal S_\eta$ for all $\eta \in [0,1]$.

Let $\Pi _\mathrm{cont,\eta }$ the orthogonal (with respect to $\mathcal S_\eta$) projection onto $\mathcal H_{\mathrm{cont},\eta }$, where we defined

$$\begin{aligned} \mathcal {H}\;=\;\mathcal H_\mathrm{cont, \eta }\;\oplus _\eta \;\overline{\mathrm{Eigen}\,(\mathcal {B}_\eta )}. \end{aligned}$$

Then, for any family of compact operators $(\mathcal K_\eta )_\eta$ on $\mathcal H$, one has

$$\begin{aligned} \lim _{\eta \rightarrow 0}\,\lim _{T\rightarrow +\infty }\,\left\| \frac{1}{T}\int ^T_0e^{-it\mathcal B_\eta }\,\mathcal K_\eta \, \Pi _\mathrm{cont,\eta }\,e^{it\mathcal B_\eta }\,dt\right\| _{\mathcal L(\eta )}=0. \end{aligned}$$

As a matter of fact, by Theorem 4.2 the limit in $T\rightarrow +\infty$ is 0 at any $\eta$ fixed.

Remark that, for simplicity, we assumed that all the scalar products $\mathcal S_\eta$ are equivalent to each other. However, such a hypothesis is not really needed at this level: it is enough to suppose that each operator $\mathcal K_\eta$ on $\mathcal H$ is compact with respect to the topology induced by $\mathcal S_\eta$.

In view of the application to the Navier–Stokes–Korteweg system, we are interested now in linking the parameters $\eta$ and T together. In this case, in order to prove a result in the same spirit of the RAGE theorem we need some additional hypotheses.

More precisely, we suppose that both $(\mathcal S_\eta )_\eta$ and $(\mathcal B_\eta )_\eta$ are defined by use of a family of automorphisms $(\Lambda _\eta )_\eta$ of $\mathcal H$ (again, we mean here that they are bounded with respect to the reference metric $\mathcal S_0$). Let us make an important remark.

Remark 6.7

We will always suppose that the family of automorphisms $(\Lambda _\eta )_\eta$ is (real) bounded-holomorphic in the sense of [20], Chapter VII (see Section 1). This will be important to have series expansions in $\eta$ for $\Lambda _\eta$ and its inverse $\Lambda ^{-1}_\eta$ (see also [20], Chapter VII, Paragraph 6.2).

Note however that the situation we consider in Sect. 6.2 will be much simpler: we will have $\Lambda _\eta =1+\eta \Delta$, and everything will be explicit.

We aim at proving the following statement.

Theorem 6.8

Let $(\mathcal H,\mathcal S_0)$ be a Hilbert space, and $\mathcal B_0\in \mathcal L(\mathcal H)$ be a self-adjoint operator. Let $(\Lambda _\eta )_{\eta \in [0,1]}$ be a bounded-holomorphic family of automorphisms of $\mathcal H$, with $\Lambda _0=\mathrm{Id}\,$, such that each $\Lambda _\eta$ is self-adjoint and such that the monotonicity property

$$\begin{aligned} \Lambda _{\eta _1}\,\le \,\Lambda _{\eta _2}\quad \forall \quad 0\le \eta _1\le \eta _2\le 1 \end{aligned}$$

(in the sense of self-adjoint operators) is verified. For any $\eta \in [0,1]$, let $\mathcal S_\eta$ be the scalar product on $\mathcal H$ induced by $\Lambda _\eta$: for all $X,Y\in \mathcal H$, we set $\mathcal S_\eta (X,Y)\,:=\,\langle X,\Lambda _\eta Y\rangle _0$. Define also $\mathcal B_\eta \,:=\,\mathcal B_0\circ \Lambda _\eta$, and suppose that $\sigma _p(\mathcal B_\eta )=\{0\}$ for all $\eta$. Let $\mathcal H_\mathrm{cont,\eta }$ the orthogonal complement of $\mathrm{Ker}\,\mathcal B_\eta$ in $\mathcal H$ with respect to $\mathcal S_\eta$:

$$\begin{aligned} \mathcal {H}\;=\;\mathcal H_\mathrm{cont, \eta }\;\oplus _\eta \;\mathrm{Ker}\,\mathcal B_\eta , \end{aligned}$$

(51)

and let $\Pi _\mathrm{cont,\eta }$ be the orthogonal (with respect to $\mathcal S_\eta$) projection onto $\mathcal H_\mathrm{cont,\eta }$. Let us now take $\eta =\sigma (\varepsilon )$, where $\sigma :[0,1]\longrightarrow [0,1]$ is a continuous increasing function such that $\sigma (0)=0$ and $\sigma (1)=1$.

Then, for any compact operator $\mathcal K$ on $\mathcal H$ and any $T>0$ fixed, defining $\mathcal K_{\sigma (\varepsilon )}=\Lambda _{\sigma (\varepsilon )}^{-1}\mathcal K$, one has that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\,\left\| \frac{1}{T}\int ^T_0\exp \left( -i\,\frac{t}{\varepsilon }\,\mathcal B_{\sigma (\varepsilon )}\right) \,\mathcal K_{\sigma (\varepsilon )}\, \Pi _{\mathrm{cont},\sigma (\varepsilon )}\,\exp \left( i\,\frac{t}{\varepsilon }\,\mathcal B_{\sigma (\varepsilon )}\right) \,dt\right\| _{\mathcal L(\varepsilon )}=0. \end{aligned}$$

Before proving the theorem, let us make some comments.

Remark 6.9

• (i) Notice that, by definitions of $\mathcal S_\eta$ and $\mathcal B_\eta$, it immediately follows that each operator $\mathcal B_\eta$ is self-adjoint with respect to the scalar product $\mathcal S_\eta$. Then, the orthogonal decomposition (51) and the definition of the semigroup $\exp (it\mathcal B_\eta )$ make sense.

• (ii) Decomposition (51) is based on the hypothesis $\sigma _p(\mathcal B_\eta )=\{0\}$ for all $\eta$. Such a spectral condition is important for stating Lemma 6.10 and deriving Corollary 6.11, which will be used in the proof.

• (iii) On the other hand, the hypothesis $\sigma _p(\mathcal B_\eta )=\{0\}$ for all $\eta$ looks quite strong, but it actually applies to the problem we want to deal with (see Proposition 6.16). We will not pursue here the issue of weakening this condition; moreover, in Proposition 6.15 we will give a sufficient condition in order to guarantee it (again, such a condition applies to our case, see also Remark 6.17).

• (iv) The monotonocity of the family of automorphisms $(\Lambda _\eta )_\eta$ implies an analogous property for the scalar products $(\mathcal S_\eta )_\eta$; moreover, since $\Lambda _1$ is in particular continuous on $(\mathcal H,\mathcal S_0)$, we have also $\Vert \,\cdot \,\Vert _1\,\le \,C\,\Vert \,\cdot \,\Vert _0$. Then, the metrics (and so the topologies) induced by the $\mathcal S_\eta$’s are all equivalent: hence, saying that an operator $\mathcal K$ is compact, without any other specification, makes sense in this context.

• (v) The fact that the compact operators $\mathcal K_\eta$ depend on $\eta$ is important for us, because in the end we want to obtain an analogue of Corollary 4.3 (the compact operator has to be self-adjoint with respect to each scalar product we consider). However, working with $\mathcal K$ (independent of $\eta$) would not have been really useful: in the proof we will need to compute its adjoint with respect to $\mathcal S_\eta$, and there a dependence on $\eta$ would arise in any case.

• (vi) We also remark the following points. On the one hand, the fact that the $\mathcal K_\eta$’s are perturbations of a fixed compact operator $\mathcal K$ allows us to reduce the proof to the case of an operator of rank 1 (as in the original RAGE theorem, see [9]): indeed, we need that the approximation by finite rank operators is, in some sense, uniform in $\eta$. On the other hand, in the proof we will exploit also the particular form $\mathcal K_\eta =\Lambda _\eta ^{-1}\mathcal K$ of the perturbations: it allows us to “play” with the special definition of the scalar products $\mathcal S_\eta$. Notice that such a hypothesis is well-adapted to the case we want to consider (see Sect. 6.2).

This having been pointed out, some preliminary results are in order.

Lemma 6.10

Under the hypotheses of Theorem 6.8, for all $\eta \in [0,1]$ one has the equality $\mathrm{Ker}\,\mathcal B_\eta =\Lambda _\eta ^{-1}\,\mathrm{Ker}\,\mathcal B_0$. In particular, $\mathcal H_\mathrm{cont, \eta }\,\equiv \,\mathcal H_\mathrm{cont,0}$ for all $\eta$.

Proof

Let $X\in \mathrm{Ker}\,\mathcal B_0$. Then, by definition of $\mathcal B_\eta =\mathcal B_0\circ \Lambda _\eta$, one immediately has $\mathcal B_\eta \Lambda _\eta ^{-1}X=0$, and hence $\Lambda _\eta ^{-1}\mathrm{Ker}\,\mathcal B_0\subset \mathrm{Ker}\,\mathcal B_\eta$.

On the other hand, if $Y\in \mathrm{Ker}\,\mathcal B_\eta$, the element $X:=\Lambda _\eta Y$ belongs to $\mathrm{Ker}\,\mathcal B_0$. So $Y=\Lambda _\eta ^{-1}X$, which proves the other inclusion $\mathrm{Ker}\,\mathcal B_\eta \subset \Lambda _\eta ^{-1}\mathrm{Ker}\,\mathcal B_0$.

Let us now work with the orthogonal complements of the kernels.

Fix $E\in \mathcal H_\mathrm{cont,\eta }$: we want to prove $\langle E,X\rangle _0=0$ for all $X\in \mathrm{Ker}\,\mathcal B_0$. In fact, from writing $X=\Lambda _\eta Y$, with $Y\in \mathrm{Ker}\,\mathcal B_\eta$, one infers

$$\begin{aligned} \langle E,X\rangle _0=\langle E,\Lambda _\eta Y\rangle _0=\langle E,Y\rangle _\eta =0. \end{aligned}$$

Then $\mathcal H_\mathrm{cont,\eta }\subset \mathcal H_\mathrm{cont,0}$.

The reverse inclusion is obtained in a totally analogous way. $\square$

Notice that, a priori, the previous proposition does not tell us anything about the orthogonal projections onto these subspaces. For instance, if $\Pi _{K,\eta }$ denotes the orthogonal (with respect to $\mathcal S_\eta$) projection onto $\mathrm{Ker}\,\mathcal B_\eta$, we cannot infer that $\Pi _{K,\eta }=\Lambda _\eta ^{-1}\Pi _{K,0}$.

Nonetheless, we can state the following corollary.

Corollary 6.11

For all $\eta \in [0,1]$, we have

$$\begin{aligned} \Pi _\mathrm{cont,\eta }=\Pi _\mathrm{cont,0}+\eta \,\mathcal R_\eta \;,\quad \text{ with } \quad \sup _{\eta \in [0,1]}\left\| \mathcal R_\eta \right\| _{\mathcal L(\mathcal H)}\,\le \,C. \end{aligned}$$

Proof

First of all, since $\mathcal H_\mathrm{cont,\eta }\equiv \mathcal H_\mathrm{cont,0}$ by Lemma 6.10, we infer

$$\begin{aligned} \Pi _\mathrm{cont,\eta }\circ \Pi _\mathrm{cont,0}=\Pi _\mathrm{cont,0}. \end{aligned}$$

Now, any $X\in \mathcal H$ can be decomposed into $X=\Pi _\mathrm{cont,0}X+\Pi _{K,0}X$. Hence, from the previous equality we get

$$\begin{aligned} \Pi _\mathrm{cont,\eta }X=\Pi _\mathrm{cont,0}X+\Pi _\mathrm{cont,\eta }\Pi _{K,0}X. \end{aligned}$$

Then, we have just to understand the action of $\Pi _\mathrm{cont,\eta }$ on $\mathrm{Ker}\,\mathcal B_0$.

Let $Z\in \mathrm{Ker}\,B_0$. By Lemma 6.10 we know that $Z_\eta :=\Lambda _\eta ^{-1}Z\in \mathrm{Ker}\,B_\eta$. On the other hand, by hypothesis on the family $(\Lambda _\eta )_\eta$, we can write $\Lambda _\eta =\mathrm{Id}\,+\eta \,\mathcal {D}_\eta$, for a suitable bounded family of self-adjoint operators $(\mathcal {D}_\eta )_\eta \,\subset \,\mathcal L(\mathcal H)$. Then one gathers

$$\begin{aligned} \Pi _\mathrm{cont,\eta }Z= & {} \Pi _\mathrm{cont,\eta }\Lambda _\eta Z_\eta \;=\;\Pi _\mathrm{cont,\eta }Z_\eta + \eta \,\Pi _\mathrm{cont,\eta }\,\mathcal {D}_\eta \,Z_\eta \\= & {} \eta \,\Pi _\mathrm{cont,\eta }\,\mathcal {D}_\eta \,Z_\eta , \end{aligned}$$

where the last equality follows from the fact that $Z_\eta \in \mathrm{Ker}\,B_\eta$.

To complete the proof of the corollary, we have just to show that

$$\begin{aligned} \left\| \Pi _\mathrm{cont,\eta }\,\mathcal {D}_\eta \,Z_\eta \right\| _0\,\le \,C\,\Vert Z\Vert _0, \end{aligned}$$

(52)

for a constant $C>0$ independent of $\eta$.

We already know that $\sup _{\eta }\left\| \mathcal D_\eta \right\| _{\mathcal L(\mathcal H)}\,\le \,C$. So, let us estimate $\left\| \Pi _\mathrm{cont,\eta }\right\| _{\mathcal L(\mathcal H)}$. For all $Y\in \mathcal H$, we have

$$\begin{aligned} \left\| \Pi _\mathrm{cont,\eta }\,Y\right\| _0^2= & {} \langle \Pi _\mathrm{cont,\eta }\,Y,\,\Pi _\mathrm{cont,\eta }\,Y\rangle _0 \;=\;\langle \Pi _\mathrm{cont,\eta }\,Y,\,\Lambda _\eta ^{-1}\,\Pi _\mathrm{cont,\eta }\,Y\rangle _\eta \\\le & {} \left\| \Pi _\mathrm{cont,\eta }\,Y\right\| _\eta ^2\,\left\| \Lambda _\eta ^{-1}\right\| _{\mathcal L(\eta )}. \end{aligned}$$

For the former term, we use that $\Pi _\mathrm{cont,\eta }$ is an orthogonal projection with respect to the scalar product $\mathcal S_\eta$, so its $\eta$-norm is bounded by 1: using then the monotonicity property of the $\Lambda _\eta$’s and the continuity of $\Lambda _1$ with respect to $\mathcal S_0$, we finally get

$$\begin{aligned} \left\| \Pi _\mathrm{cont,\eta }\,Y\right\| _\eta ^2\,\le \,\left\| Y\right\| _\eta ^2\,\le \,C\,\left\| Y\right\| _0^2. \end{aligned}$$

For the latter term, we argue exactly as above: for all $Y\in \mathcal H$,

$$\begin{aligned} \left\| \Lambda ^{-1}_{\eta }\,Y\right\| _\eta ^2= & {} \langle \Lambda ^{-1}_{\eta }\,Y,\,\Lambda ^{-1}_{\eta }\,Y\rangle _\eta \;=\; \langle \Lambda ^{-1}_{\eta }\,Y,\,Y\rangle _0 \\\le & {} C\,\Vert Y\Vert _0^2\;\le \;C\,\Vert Y\Vert _\eta ^2, \end{aligned}$$

where the last estimate comes from the monotonicity hypothesis. Combining these two last inequalities together, we easily deduce (52), from which the corollary follows. $\square$

We need also the following simple lemma.

Lemma 6.12

Under the hypotheses of Theorem 6.8, let us fix a $X\in \mathcal H$ and consider the spectral measure $\mu _\eta$ associated to the element $\Pi _\mathrm{cont,\eta }\,\Lambda ^{-1}_\eta \,\Pi _\mathrm{cont,0}\,X$.

Then one has $\mu _\eta (\mathbb {R})\,\longrightarrow \,\mu _0(\mathbb {R})$ for $\eta \rightarrow 0$.

Proof

By definition of spectral measure and the spectral theorem, one has

$$\begin{aligned} \mu _\eta (\mathbb {R})= & {} \int _\mathbb {R}d\mu _\eta (x)\;=\;\left\| \Pi _\mathrm{cont,\eta }\,\Lambda ^{-1}_\eta \,\Pi _\mathrm{cont,0}\,X\right\| ^2_{\eta } \\= & {} \langle \Pi _\mathrm{cont,\eta }\,\Lambda ^{-1}_\eta \,\Pi _\mathrm{cont,0}\,X,\,\Lambda ^{-1}_\eta \,\Pi _\mathrm{cont,0}\,X\rangle _\eta . \end{aligned}$$

Now, by definition of $\mathcal S_\eta$ and Corollary 6.11, we have

$$\begin{aligned} \mu _\eta (\mathbb {R})=\langle \Pi _\mathrm{cont,\eta }\,\Lambda ^{-1}_\eta \,\Pi _\mathrm{cont,0}\,X,\,\Pi _\mathrm{cont,0}\,X\rangle _0= \langle \Lambda ^{-1}_\eta \,\Pi _\mathrm{cont,0}\,X,\,\Pi _\mathrm{cont,0}\,X\rangle _0+O(\eta ), \end{aligned}$$

where we used also that $\Pi _\mathrm{cont,0}$ is self-adjoint with respect to $\mathcal S_0$ and $\Pi _\mathrm{cont,0}^2=\Pi _\mathrm{cont,0}$. At this point, thanks to the hypothesis over the family $(\Lambda _\eta )_\eta$, we can write $\Lambda ^{-1}_\eta =\mathrm{Id}\,+\eta \,\widetilde{\mathcal D}_\eta$, for a suitable bounded family of self-adjoint operators $(\widetilde{\mathcal D}_\eta )_\eta \subset \mathcal L(\mathcal H)$ (see also Chapter VII of [20], in particular Paragraph 3.2). Then the previous relation becomes

$$\begin{aligned} \mu _\eta (\mathbb {R})=\langle \Pi _\mathrm{cont,0}\,X,\,\Pi _\mathrm{cont,0}\,X\rangle _0+O(\eta )= \mu _0(\mathbb {R})+O(\eta ), \end{aligned}$$

and this proves the claim of the lemma. $\square$

We can finally prove Theorem 6.8. We will follow the main lines of the proof given in [9] (see Theorem 5.8, Chapter 5).

Proof of Theorem 6.8 First of all, we notice that, up to perform the change of variable $\tau =t/\varepsilon$, our claim is equivalent to show that

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\,\left\| \frac{\varepsilon }{T}\int ^{T/\varepsilon }_0\exp \left( -i\,t\,\mathcal B_{\sigma (\varepsilon )}\right) \,\mathcal K_{\sigma (\varepsilon )}\, \Pi _{\mathrm{cont},\sigma (\varepsilon )}\,\exp \left( i\,t\,\mathcal B_{\sigma (\varepsilon )}\right) \,dt\right\| _{\mathcal L(\varepsilon )}=0. \end{aligned}$$

For notation convenience, for the moment we keep writing $\eta$ instead of $\sigma (\varepsilon )$.

Since a compact operator can be approximated (in the norm topology) by finite rank operators, and each finite rank operator can be written as a finite sum of operators of rank 1, it is enough to restrict to the case of $\mathrm{rk}\,\mathcal K=1$.

Recall here point (vi) of Remark 6.9: approximating $\mathcal K$ gives the “same approximation” for all $\mathcal K_\eta$ (up to the isomorphism $\Lambda _\eta$). We are not able to exploit this reduction to rank 1 operators if the approximation itself depended on the particular compact operator $\mathcal K_\eta$, with no relations between them.

Since $\mathrm{rk}\,\mathcal K=1$, we can represent $\mathcal K$ with respect to the reference scalar product $\mathcal S_0$ in the form $\mathcal K\,\varphi =\langle X,\,\varphi \rangle _0\;Y$, for suitable $X,Y\in \mathcal H$. Hence, by definitions of $\mathcal K_\eta$ and $\mathcal S_\eta$, we have

$$\begin{aligned} \mathcal K_\eta \,\varphi \;=\;\Lambda _\eta ^{-1}\,\mathcal K\,\varphi= & {} \langle X,\,\varphi \rangle _0\;\Lambda _\eta ^{-1}\,Y\;=\; \langle \Lambda _\eta ^{-1}\,X,\,\varphi \rangle _\eta \;\Lambda _\eta ^{-1}\,Y \\= & {} \langle X_\eta ,\,\varphi \rangle _\eta \;Y_\eta , \end{aligned}$$

where we have denoted $X_\eta =\Lambda _\eta ^{-1}\,X$ and $Y_\eta =\Lambda _\eta ^{-1}\,Y$. Therefore, its adjoint $\mathcal K_\eta ^{*(\eta )}$ (with respect to the scalar product $\mathcal S_\eta$) is given by

$$\begin{aligned} \mathcal K_\eta ^{*(\eta )}\,\varphi \;=\;\langle Y_\eta ,\,\varphi \rangle _\eta \;X_\eta . \end{aligned}$$

Now, as in [9], for any $\varepsilon \in [0,1]$ fixed and denoting again $\eta =f(\varepsilon )$, we define the operator

$$\begin{aligned} Q_\varepsilon (T):= & {} \frac{\varepsilon }{T}\int ^{T/\varepsilon }_0e^{-it\mathcal B_\eta }\,\mathcal K_\eta \,\Pi _\mathrm{cont,\eta }\,e^{it\mathcal B_\eta }\,dt \\= & {} \frac{\varepsilon }{T}\,\int ^{T/\varepsilon }_0\langle e^{-it\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,X_\eta ,\;\cdot \;\rangle _\eta \, e^{-it\mathcal B_\eta }\,Y_\eta \,dt \end{aligned}$$

and its adjoint (again, with respect to $\mathcal S_\eta$)

$$\begin{aligned} Q^{*(\eta )}_\varepsilon (T)\;=\;\frac{\varepsilon }{T}\int ^{T/\varepsilon }_0\langle e^{-it\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,Y_\eta ,\;\cdot \;\rangle _\eta \, e^{-it\mathcal B_\eta }\,X_\eta \,dt. \end{aligned}$$

Then, for all $\varphi \in \mathcal H$, the following identity holds true:

$$\begin{aligned}&Q_\varepsilon (T)\,Q^{*(\eta )}_\varepsilon (T)\,\varphi \;=\;\frac{\varepsilon }{T}\int ^{T/\varepsilon }_0\langle e^{-it\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,X_\eta ,\, Q^{*(\eta )}_\varepsilon (T)\,\varphi \rangle _\eta \,e^{-it\mathcal B_\eta }\,Y_\eta \,dt \\&\quad =\;\frac{\varepsilon ^2}{T^2}\int ^{T/\varepsilon }_0\int ^{T/\varepsilon }_0\langle e^{-it\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,X_\eta ,\, e^{-is\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,X_\eta \rangle _\eta \,\\&\quad \quad \times \langle e^{is\mathcal B_\eta }\,Y_\eta ,\,\varphi \rangle _\eta \,e^{-it\mathcal B_\eta }\,Y_\eta \,ds\,dt. \end{aligned}$$

Therefore, we can write

$$\begin{aligned}&\left\| \frac{\varepsilon }{T}\int ^{T/\varepsilon }_0e^{-it\mathcal B_\eta }\,\mathcal K_\eta \,\Pi _\mathrm{cont,\eta }\,e^{it\mathcal B_\eta }\,dt\right\| ^2_{\mathcal L(\eta )}\\&\quad = \left\| Q_\varepsilon (T)\right\| ^2_{\mathcal L(\eta )}\;=\;\left\| Q_\varepsilon (T)\;Q^{*(\eta )}_\varepsilon (T)\right\| _{\mathcal L(\eta )} \\&\quad \le \;\frac{\varepsilon ^2}{T^2}\,\left\| Y_\eta \right\| ^2_\eta \,\int ^{T/\varepsilon }_0\int ^{T/\varepsilon }_0\left| \langle e^{-it\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,X_\eta ,\, e^{-is\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,X_\eta \rangle _\eta \right| \,ds\,dt. \end{aligned}$$

By definitions and the continuity of the map $\eta \,\mapsto \,\Lambda _\eta$, we infer

$$\begin{aligned} \left\| Y_\eta \right\| ^2_\eta =\langle \Lambda _\eta ^{-1}Y,\,\Lambda _\eta ^{-1}Y\rangle _\eta = \langle \Lambda _\eta ^{-1}Y,\,Y\rangle _0\,\le \,C\,\Vert Y\Vert _0^2, \end{aligned}$$

and applying the Cauchy–Schwarz inequality we arrive at

$$\begin{aligned}&\left\| \frac{\varepsilon }{T}\int ^{T/\varepsilon }_0e^{-it\mathcal B_\eta }\,\mathcal K_\eta \,\Pi _\mathrm{cont,\eta }\,e^{it\mathcal B_\eta }\,dt\right\| ^2_{\mathcal L(\eta )} \\&\quad \le \;C\,\Vert Y\Vert _0^2\,\left( \frac{\varepsilon ^2}{T^2}\int ^{T/\varepsilon }_0\int ^{T/\varepsilon }_0\left| \langle \Pi _\mathrm{cont,\eta }\,X_\eta ,\, e^{i(t-s)\mathcal B_\eta }\,\Pi _\mathrm{cont,\eta }\,X_\eta \rangle _\eta \right| ^2\,ds\,dt\right) ^{\!1/2}. \end{aligned}$$

We now focus on the integral term on the right-hand side of the previous inequality. Notice that, since $\Lambda _\eta ^{-1}\,\Pi _{K,0}X\in \mathrm{Ker}\,\mathcal B_\eta$, one can write

$$\begin{aligned} \Pi _\mathrm{cont,\eta }\,X_\eta =\Pi _\mathrm{cont,\eta }\,\Lambda _\eta ^{-1}\,X= \Pi _\mathrm{cont,\eta }\,\Lambda ^{-1}_\eta \,\Pi _\mathrm{cont,0}\,X. \end{aligned}$$

(53)

Then, coming back to the notation $\eta =\sigma (\varepsilon )$, let us consider the quantity

$$\begin{aligned}&\mathcal J_\varepsilon \,:=\,\frac{\varepsilon ^2}{T^2}\int ^{T/\varepsilon }_0\!\!\int ^{T/\varepsilon }_0\\&\quad \left| \langle \Pi _{\mathrm{cont},\sigma (\varepsilon )}\,\Lambda ^{-1}_{\sigma (\varepsilon )}\,\Pi _\mathrm{cont,0}\,X,\, e^{i(t-s)\mathcal B_{\sigma (\varepsilon )}}\,\Pi _{\mathrm{cont},\sigma (\varepsilon )}\,\Lambda ^{-1}_{\sigma (\varepsilon )}\, \Pi _\mathrm{cont,0}\,X\rangle _{\sigma (\varepsilon )}\right| ^2\,ds\,dt. \end{aligned}$$

We denote by $\mu _{\sigma (\varepsilon )}$ the spectral measure associated to $\Pi _{\mathrm{cont},\sigma (\varepsilon )}\,\Lambda ^{-1}_{\sigma (\varepsilon )}\,\Pi _\mathrm{cont,0}X$. Therefore, repeating the computations in [9], it follows that

$$\begin{aligned} \mathcal J_\varepsilon= & {} \frac{\varepsilon ^2}{T^2}\int ^{T/\varepsilon }_0\int ^{T/\varepsilon }_0\left( \int _\mathbb {R}\int _\mathbb {R}\exp (i\,(t-s)\,(x-y))\, d\mu _{\sigma (\varepsilon )}(x)\,d\mu _{\sigma (\varepsilon )}(y)\right) ds\,dt \\\le & {} \int _\mathbb {R}\int _\mathbb {R}\left( \frac{\sin ((x-y)T/(2\varepsilon ))}{(x-y)T/(2\varepsilon )}\right) ^2\, d\mu _{\sigma (\varepsilon )}(x)\,d\mu _{\sigma (\varepsilon )}(y). \end{aligned}$$

Now, in order to estimate the double integral on the right-hand side of the previous relation, it is just a matter of reproducing the proof of Theorem 6.5 in the case of continuous dependence on a parameter. Notice that, here, the family of measures does not depend continuously on $\varepsilon$: as a matter of facts, we are in the continuous part of the spectrum, which is highly unstable (see also [20], Chapter X). Nonetheless, it is quite easy to see that Lemma 6.12, combined with the fact that $\mu _{\sigma (\varepsilon )}$ has no pure points for any $\varepsilon$, is enough to make the arguments work, and to prove that the previous quantity goes to 0 for $\varepsilon \rightarrow 0$.

This concludes the proof of Theorem 6.8. $\square$

Remark 6.13

We proved the previous theorem by direct computations. Notice that one could also compare the two propagators, related to $\mathcal B_0$ and to $\mathcal B_{\sigma (\varepsilon )}$, and use properties from perturbation theory of semigroups: we refer e.g. to Theorem 2.19 of [20], Chapter IX (see also Theorem 13.5.8 of [17]). However, these results fail to provide uniform bounds on time intervals $[0,T/\varepsilon ]$ when $\varepsilon \rightarrow 0$: this is why we preferred to prove estimates “by hands”.

Alternatively, one could use the Baker–Campbell–Hausdorff formula (see for instance [13, 23]) in order to write the propagator $\exp (it\mathcal B_{\sigma (\varepsilon )})$ as the propagator $\exp (it\mathcal B_0)$ related to the unperturbed operator, plus a uniformly bounded remainder of order $\sigma (\varepsilon )$.

Also in this case, we have the analogue of Corollary 4.3.

Corollary 6.14

Under the hypotheses of Theorem 6.8, suppose moreover that $\mathcal {K}$ is self-adjoint, with $\mathcal {K}\ge 0$.

Then there exist a constant $C>0$ and a function $\mu$, with $\mu (\varepsilon )\rightarrow 0$ for $\varepsilon \rightarrow 0$, such that:

• (1) for any $Y\in \mathcal H$ and any $T>0$, one has

  $$\begin{aligned} \frac{1}{T}\int ^T_0\left\| \mathcal {K}_{\sigma (\varepsilon )}^{1/2}\,e^{i\,t\,\mathcal {B}_{\sigma (\varepsilon )}/\varepsilon }\, \Pi _{\mathrm{cont},\sigma (\varepsilon )}Y\right\| ^2_{\sigma (\varepsilon )}\,dt\,\le \,C\,\mu (\varepsilon )\,\Vert Y\Vert _{0}^2\,; \end{aligned}$$

• (2) for any $T>0$ and any $X\in L^2([0,T];\mathcal H)$, one has

  $$\begin{aligned}&\frac{1}{T^2}\left\| \mathcal {K}_{\sigma (\varepsilon )}^{1/2}\,\Pi _{\mathrm{cont},\sigma (\varepsilon )}\int ^t_0e^{i\,(t-\tau )\,\mathcal {B}_{\sigma (\varepsilon )}/\varepsilon }\, X(\tau )\,d\tau \right\| ^2_{L^2([0,T];(\mathcal H,\mathcal S_{\sigma (\varepsilon )}))}\,\\&\quad \le \,C\,\mu (\varepsilon )\,\left\| X\right\| ^2_{L^2([0,T];\mathcal H)}. \end{aligned}$$

Let us conclude this part giving a sufficient condition in order to guarantee the spectral property $\sigma _p(\mathcal B_\eta )=\{0\}$ at least for $\eta$ close to 0: we need 0 to be an isolated eigenvalue ¹ of $\mathcal B_0$.

Proposition 6.15

Let $(\mathcal H,\mathcal S_0)$ be a Hilbert space. Let $\mathcal B_0\in \mathcal L(\mathcal H)$ be a self-adjoint operator such that $\sigma _p(\mathcal B_0)=\{0\}$, and suppose also that 0 is an isolated eigenvalue.

Let $(\Lambda _\eta )_{\eta \in [0,1]}$ be a bounded-holomorphic family of automorphisms of $\mathcal H$, with $\Lambda _0=\mathrm{Id}\,$, such that each $\Lambda _\eta$ is self-adjoint. For any $\eta \in [0,1]$, define the operator $\mathcal B_\eta \,:=\,\mathcal B_0\circ \Lambda _\eta$. Then $\sigma _p(\mathcal B_\eta )=\{0\}$ for $\eta$ small enough.

Proof

By hypothesis, we know that there exists $v\in \mathcal H$, $v\ne 0$, such that $\mathcal B_0\,v=0$. We want to solve the equation

$$\begin{aligned} \mathcal B_\eta \,v_\eta =\lambda _\eta \,v_\eta \end{aligned}$$

(54)

and show that $\lambda _\eta =0$, at least for small $\eta$.

As done above (see Corollary 6.11), by hypothesis we can write $\Lambda _\eta =\mathrm{Id}\,+\eta \,\mathcal {D}_\eta$, for a bounded family $(\mathcal {D}_\eta )_\eta$ of self-adjoint operators.

Moreover, since 0 is an isolated eigenvalue of $\mathcal {B}_0$, by perturbation theory (see [20], Theorem 3.16 of Chapter IV and Theorems 1.7 and 1.8 of Chapter VII), for suitably small $\eta$ we also have

$$\begin{aligned} \lambda _\eta =\lambda _0+\eta \,\widetilde{\lambda }_{\eta }\quad \text{ and } \quad v_\eta =v_0+\eta \,\widetilde{v}_\eta , \end{aligned}$$

where the families of remainders $(\widetilde{\lambda }_\eta )_\eta \subset \mathbb {R}$ and $(\widetilde{v}_\eta )_\eta \subset \mathcal H$ are bounded.

We now insert the previous expansions into (54), getting

$$\begin{aligned} \mathcal B_0\,v_0+\eta \,\mathcal B_0\,\widetilde{v}_\eta +\eta \,\mathcal B_0\,\mathcal D_\eta \,v_0+ \eta ^2\,\mathcal B_0\,\mathcal D_\eta \,\widetilde{v}_\eta =\lambda _0\,v_0+\eta \,\lambda _0\,\widetilde{v}_\eta + \eta \,\widetilde{\lambda }_\eta \,v_0+\eta ^2\,\widetilde{\lambda }_\eta \,\widetilde{v}_\eta , \end{aligned}$$

and we compare the terms with the same power of $\eta$.

From the 0-th order part, we obviously get that $\lambda _0=0$ and $v_0=v$. Then, the equality involving the terms of order $\eta$ reduces to

$$\begin{aligned} \mathcal B_0\,\widetilde{v}_\eta +\mathcal B_0\,\mathcal D_\eta \,v=\widetilde{\lambda }_\eta \,v. \end{aligned}$$

Now, we take the $\mathcal S_0$-scalar product of both sides with v: since $\mathcal B_0$ is self-adjoint and $v\in \mathrm{Ker}\,\mathcal B_0$, we immediately get

$$\begin{aligned} \widetilde{\lambda }_\eta \,\Vert v\Vert _0^2=0. \end{aligned}$$

Using that $v\ne 0$, we infer that $\widetilde{\lambda }_\eta =0$, which in turn implies $\lambda _\eta =0$ (for $\eta$ small enough). $\square$

6.2 Application to the vanishing capillarity limit

Let us apply now the previous results to our case. As pointed out at the beginning of Sect. 6, for a fixed $\alpha \,\in \,]0,1[\,$ we rewrite system (2) in the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \varepsilon \,\partial _tr_\varepsilon +\mathrm{div}\,\,V_\varepsilon =0 \\ \varepsilon \,\partial _tV_\varepsilon +(e^3\times V_\varepsilon +\nabla (\mathrm{Id}\,\,-\,\varepsilon ^{2\alpha }\,\Delta )r_\varepsilon )= \varepsilon \,f_{\varepsilon ,\alpha }, \end{array}\right. } \end{aligned}$$

(55)

where the family $(f_{\varepsilon ,\alpha })_\varepsilon$ is bounded in $L^2_T(W^{-1,2}(\Omega )+W^{-1,1}(\Omega ))$.

Then, we are led to study the family of operators $(\mathcal {A}^{(\alpha )}_\varepsilon )_\varepsilon$, defined by

$$\begin{aligned} \mathcal {A}^{(\alpha )}_\varepsilon \,:\quad (r\;,\;V)\quad \mapsto \quad (\mathrm{div}\,V\;,\;e^3\times V\,+\nabla (\mathrm{Id}\,\,-\,\varepsilon ^{2\alpha }\,\Delta )r). \end{aligned}$$

Notice that one has $\mathcal {A}^{(\alpha )}_0\equiv \mathcal {A}$ and $\mathcal {A}^{(\alpha )}_1\equiv \mathcal {A}_0$, where $\mathcal {A}$ is defined in Sect. 4.1.1 and $\mathcal {A}_0$ in formula (39).

As one can expect, one has the following result about the point spectrum of each operator.

Proposition 6.16

For any $0\le \varepsilon \le 1$, the point spectrum $\sigma _p\left( \mathcal {A}^{(\alpha )}_\varepsilon \right)$ contains only 0. In particular, $\mathrm{Eigen}\,\mathcal {A}^{(\alpha )}_\varepsilon \;\equiv \;\mathrm{Ker}\,\mathcal {A}^{(\alpha )}_\varepsilon$.

Proof

The same computations performed in the proof of Proposition 5.1 give us

$$\begin{aligned} \lambda ^2=-\,\frac{1}{2}\left( 1+(1+\varepsilon ^{2\alpha }\zeta )\zeta \pm \sqrt{\left( 1+(1+\varepsilon ^{2\alpha }\zeta )\zeta \right) ^2\,-\,4\, k^2\,(1+\varepsilon ^{2\alpha }\zeta )}\right) , \end{aligned}$$

where we recall that we have set $\zeta (\xi ^h,k)=\left| \xi ^h\right| ^2+k^2$.

As before, to have $\lambda$ in the discrete spectrum of $\mathcal {A}_0$, we need to delete its dependence on $\xi ^h$: since $1+\varepsilon ^{2\alpha }\zeta >0$, the only way to do it is to have $k=0$, for which $\lambda =0$. $\square$

Remark 6.17

Notice that simple computations show also that 0 is an isolated eigenvalue of the operator $\mathcal {A}^{(\alpha )}_0$ (here we have to use that in the space $H_M$ the frequencies are bounded). Then, one could alternatively apply Proposition 6.15.

As done in Sect. 5, it is easy to find a family of scalar products $(S^{(\alpha )}_\varepsilon )_\varepsilon$ on the space $H_M$ such that, for each $\varepsilon$, $S^{(\alpha )}_\varepsilon$ is a symmetrizer for the operator $\mathcal {A}^{(\alpha )}_\varepsilon$. Indeed, it is enough to define $S^{(\alpha )}_\varepsilon$ by a formula analogous to (41), where the operator $\mathrm{Id}\,-\Delta$ is replaced by $\mathrm{Id}\,-\varepsilon ^{2\alpha }\Delta$ in the first term on the right-hand side of the equality (see Eq. (56) below). Notice that $S^{(\alpha )}_0$ coincides with the usual $L^2$ scalar product, while $S^{(\alpha )}_1$ is exactly the inner product defined by formula (41).

Let us point out that each $\mathcal {A}^{(\alpha )}_\varepsilon$ can be obtained composing the acoustic propagator $\mathcal {A}$ with an automorphism $\Lambda ^{(\alpha )}_\varepsilon$ of the Hilbert space $H_M$:

$$\begin{aligned} \mathcal {A}^{(\alpha )}_\varepsilon =\mathcal {A}\,\circ \,\Lambda ^{(\alpha )}_\varepsilon \;,\quad \Lambda ^{(\alpha )}_\varepsilon (r,\,V)\,:=\,((\mathrm{Id}\,\,-\,\varepsilon ^{2\alpha }\,\Delta )r,\,V). \end{aligned}$$

The same can be said also about the scalar products $S^{(\alpha )}_\varepsilon$: namely,

$$\begin{aligned} \langle (r_1,V_1),\,(r_2,V_2)\rangle _{S^{(\alpha )}_\varepsilon }:= & {} \langle r_1,\, (\mathrm{Id}\,-\varepsilon ^{2\alpha }\Delta )r_2\rangle _{L^2}+\langle V_1,\,V_2\rangle _{L^2} \nonumber \\= & {} \langle (r_1,V_1),\,\Lambda ^{(\alpha )}_\varepsilon \,(r_2,V_2)\rangle _{S^{(\alpha )}_0}. \end{aligned}$$

(56)

Now, we define the operator $\mathcal K_{M,\theta }$ as in Sect. 4.1.2:

$$\begin{aligned} \mathcal {K}_{M,\theta }(r,V)\,:=\,P_M(\theta \,P_M(r,V)), \end{aligned}$$

where $P_M\,:\,L^2(\Omega )\times L^2(\Omega )\,\longrightarrow \,H_M$ is the orthogonal projection onto the space $H_M$, defined by (31), and $\theta \in \mathcal {D}(\Omega )$ is such that $0\le \theta \le 1$. Recall that $\mathcal K_{M,\theta }$ is compact, self-adjoint and positive.

Following what we have done before, we want to apply Theorem 6.8 to

$$\begin{aligned} \mathcal H=H_M\;,\quad \Lambda _{\sigma (\varepsilon )}=\Lambda ^{(\alpha )}_\varepsilon \;,\quad \mathcal {B}_0=i\,\mathcal {A}^{(\alpha )}_0\;,\quad \mathcal {K}=\mathcal {K}_{M,\theta }\;,\quad \Pi _{\mathrm{cont},\sigma (\varepsilon )}=Q_{\varepsilon }^{\perp } \end{aligned}$$

(obviously, $\sigma (\varepsilon )=\varepsilon ^{2\alpha }$ here), where $Q_\varepsilon$ and $Q_\varepsilon ^\perp$ denote the orthogonal projections (orthogonal with respect to the scalar product $\mathcal S^{(\alpha )}_\varepsilon$) onto respectively $\mathrm{Ker}\,\mathcal {A}^{(\alpha )}_\varepsilon$ and $(\mathrm{Ker}\,\mathcal {A}^{(\alpha )}_\varepsilon )^\perp$.

We apply operator $P_M$ to the system for acoustic waves (55): adopting the same notations as in the previous sections, it becomes

$$\begin{aligned} \varepsilon \,\frac{d}{dt}(r_{\varepsilon ,M},\,V_{\varepsilon ,M})+\mathcal {A}^{(\alpha )}_\varepsilon (r_{\varepsilon ,M},\,V_{\varepsilon ,M})= \varepsilon \,(0,\,f_{\varepsilon ,\alpha ,M}), \end{aligned}$$

(57)

where uniform bounds give a control analogous to (33) also for $(0,\,f_{\varepsilon ,\alpha ,M})$. Notice that, all the scalar products being equivalent on $H_M$, it is enough to have the bound on the $\mathcal S^{(\alpha )}_0$ norm.

By use of Duhamel’s formula, solutions to the previous acoustic equation can be written as

$$\begin{aligned} (r_{\varepsilon ,M},\,V_{\varepsilon ,M})(t)=e^{i\,t\,\mathcal {B}_{\sigma (\varepsilon )}/\varepsilon }(r_{\varepsilon ,M},\,V_{\varepsilon ,M})(0)+ \int ^t_0e^{i\,(t-\tau )\,\mathcal {B}_{\sigma (\varepsilon )}/\varepsilon }\,(0,\,f_{\varepsilon ,\alpha ,M})\,d\tau . \end{aligned}$$

Again, by definition we have

$$\begin{aligned} \left\| \left( \left( \Lambda ^{(\alpha )}_\varepsilon \right) ^{-1}\circ \mathcal {K}_{M,\theta }\right) ^{1/2}\,Q^\perp _\varepsilon (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\right\| ^2_{\sigma (\varepsilon )}= \int _\Omega \theta \,\left| Q^\perp _\varepsilon (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\right| ^2\,dx. \end{aligned}$$

Therefore, a straightforward application of Corollary 6.14 implies that, for any $T>0$ fixed and for $\varepsilon$ going to 0,

$$\begin{aligned} Q^\perp _\varepsilon (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\,\longrightarrow \,0\quad \text{ in } \quad L^2([0,T]\times K) \end{aligned}$$

(58)

for any fixed $M>0$ and any compact $K\subset \Omega$.

On the other hand, applying operator $Q_\varepsilon$ to Eq. (57), we infer that, for any fixed $M>0$, the family $(\partial _tQ_\varepsilon (r_{\varepsilon ,M},\,V_{\varepsilon ,M}))_\varepsilon$ is bounded (uniformly in $\varepsilon$) in the space $L^2_T(H_M)$. Moreover, as $H_M\hookrightarrow H^m$ for any $m\in \mathbb {N}$, we infer also that it is compactly embedded in $L^2(K)$ for any $M>0$ and any compact subset $K\subset \Omega$. Hence, as in the previous sections, Ascoli–Arzelà theorem implies that, for $\varepsilon \rightarrow 0$,

$$\begin{aligned} Q_\varepsilon (r_{\varepsilon ,M},\,V_{\varepsilon ,M})\,\longrightarrow \,(r_M,\,u_M)\quad \text{ in } \quad L^2([0,T]\times K). \end{aligned}$$

(59)

Thanks to relations (58) and (59), the analogue of Proposition 4.4 still holds true: namely, we have the strong convergence (up to extraction of subsequences)

$$\begin{aligned} r_\varepsilon \,\longrightarrow \,r\quad \text{ and } \quad \rho _\varepsilon ^{3/2}\,u_\varepsilon \;\longrightarrow \;u\quad \text{ in } \quad L^2([0,T];L^2_{loc}(\Omega )), \end{aligned}$$

where r and u are the limits which have been identified in Sect. 3.3 and which have to satisfy the constraints given in Proposition 3.5.

The previous strong convergence properties allow us to pass to the limit in the non-linear terms. Then, the analysis of the limit system can be performed as in Sect. 4.2.2.

This concludes the proof of Theorem 2.4 in the remaining cases .

Appendix 1: A primer on Littlewood–Paley theory

Let us recall here the main ideas of Littlewood–Paley theory, which we exploited in the previous analysis. We refer e.g. to [1] (Chapter 2) and [24] (Chapters 4 and 5) for details.

For simplicity of exposition, let us deal with the $\mathbb {R}^d$ case; however, the construction can be adapted to the d-dimensional torus $\mathbb {T}^d$, and then also to the case of $\mathbb {R}^{d_1}\times \mathbb {T}^{d_2}$.

First of all, let us introduce the so called “Littlewood–Paley decomposition”, based on a non-homogeneous dyadic partition of unity with respect to the Fourier variable.

We, fix a smooth radial function $\chi$ supported in the ball B(0, 2),  equal to 1 in a neighborhood of B(0, 1) and such that $r\mapsto \chi (r\,e)$ is nonincreasing over $\mathbb {R}_+$ for all unitary vectors $e\in \mathbb {R}^d$. Set $\varphi \left( \xi \right) =\chi \left( \xi \right) -\chi \left( 2\xi \right)$ and $\varphi _j(\xi ):=\varphi (2^{-j}\xi )$ for all $j\ge 0$.

The dyadic blocks $(\Delta _j)_{j\in \mathbb {Z}}$ are defined by²

$$\begin{aligned} \Delta _j:=0\ \hbox { if }\ j\le -2,\quad \Delta _{-1}:=\chi (D)\quad \text{ and } \quad \Delta _j:=\varphi (2^{-j}D)\; \text{ if } \; j\ge 0. \end{aligned}$$

Throughout the paper we will use freely the following classical property: for any $u\in \mathcal {S}',$ the equality $u=\sum _{j}\Delta _ju$ holds true in $\mathcal {S}'$.

Let us also mention the so-called Bernstein’s inequalities, which explain the way derivatives act on spectrally localized functions.

Lemma 7.1

Let . A constant C exists so that, for any nonnegative integer k, any couple (p, q) in $[1,+\infty ]^2$ with $p\le q$ and any function $u\in L^p$, we have, for all $\lambda >0$,

$$\begin{aligned}&\mathrm{supp}\, \widehat{u} \subset B(0,\lambda R)\quad \Longrightarrow \quad \Vert \nabla ^k u\Vert _{L^q}\, \le \, C^{k+1}\,\lambda ^{k+d\left( \frac{1}{p}-\frac{1}{q}\right) }\,\Vert u\Vert _{L^p}\;;\\&\mathrm{supp}\, \widehat{u} \subset \{\xi \in \mathbb {R}^d\,|\, r\lambda \le |\xi |\le R\lambda \} \quad \Longrightarrow \quad C^{-k-1}\,\lambda ^k\Vert u\Vert _{L^p}\,\\&\quad \le \, \Vert \nabla ^k u\Vert _{L^p}\, \le \, C^{k+1} \, \lambda ^k\Vert u\Vert _{L^p}. \end{aligned}$$

By use of Littlewood–Paley decomposition, we can define the class of Besov spaces.

Definition 7.2

Let $s\in \mathbb {R}$ and $1\le p,r\le +\infty$. The non-homogeneous Besov space $B^{s}_{p,r}$ is defined as the subset of tempered distributions u for which

$$\begin{aligned} \Vert u\Vert _{B^{s}_{p,r}}\,:=\, \left\| \left( 2^{js}\,\Vert \Delta _ju\Vert _{L^p}\right) _{j\in \mathbb {N}}\right\| _{\ell ^r}\,<\,+\infty . \end{aligned}$$

Besov spaces are interpolation spaces between the Sobolev ones. In fact, for any $k\in \mathbb {N}$ and $p\in [1,+\infty ]$ we have the following chain of continuous embeddings:

$$\begin{aligned} B^k_{p,1}\hookrightarrow W^{k,p}\hookrightarrow B^k_{p,\infty }, \end{aligned}$$

where $W^{k,p}$ denotes the classical Sobolev space of $L^p$ functions with all the derivatives up to the order k in $L^p$. Moreover, for all $s\in \mathbb {R}$ we have the equivalence $B^s_{2,2}\equiv H^s$, with

$$\begin{aligned} \Vert f\Vert _{H^s}\,\sim \,\left( \sum _{j\ge -1}2^{2 j s}\,\Vert \Delta _jf\Vert ^2_{L^2}\right) ^{1/2}. \end{aligned}$$

Let us now collect some bounds which are straightforward consequences of Bernstein’s inequalities. The statements are not optimal: we limit to present the properties we used in our analysis.

Lemma 7.3

• (i) For $1\le p\le 2$, one has $\Vert f\Vert _{L^2}\,\le \,C(\Vert f\Vert _{L^p}+\Vert \nabla f\Vert _{L^2})$.

• (ii) For any $0<\delta \le 1/2$ and any $1\le p\le +\infty$, one has

  $$\begin{aligned} \Vert f\Vert _{L^\infty }\,\le \,C\left( \Vert f\Vert _{L^p}+\Vert \nabla f\Vert ^{(1/2)-\delta }_{L^2}\, \left\| \nabla ^2f\right\| ^{(1/2)+\delta }_{L^2}\,\right) . \end{aligned}$$

• (iii) Let $1\le p\le 2$ such that $1/p\,<\,1/d+1/2$. For any $j\in \mathbb {N}$, there exists a constant $C_j$, depending just on j, d and p, such that

  $$\begin{aligned} \left\| \left( \mathrm{Id}\,\,-\,S_j\right) f\right\| _{L^2}\,\le \,C_j\,\left\| \nabla f\right\| _{B^0_{p,\infty }}. \end{aligned}$$

  Moreover, denoting $\beta \,:=\,1\,-\,d(1/p\,-\,1/2)>0$, we have the explicit formula

  $$\begin{aligned} C_j=\left( \frac{1}{1-2^{-2\beta }}\right) ^{\!\!1/2}\;2^{-\beta (j-1)}. \end{aligned}$$

  In particular, if $\nabla f=\nabla f_1 + \nabla f_2$, with $\nabla f_1\in B^0_{2,\infty }$ and $\nabla f_2\in B^0_{p,\infty }$, then

  $$\begin{aligned} \left\| \left( \mathrm{Id}\,\,-\,S_j\right) f\right\| _{L^2}\,\le \,\widetilde{C}_j\,\left( \left\| \nabla f_1\right\| _{B^0_{2,\infty }}+ \left\| \nabla f_2\right\| _{B^0_{p,\infty }}\right) , \end{aligned}$$

  for a new constant $\widetilde{C}_j$ still going to 0 for $j\rightarrow +\infty$.

Proof

For the first inequality, it is enough to write $f=\Delta _{-1}f+(\mathrm{Id}\,-\Delta _{-1})f$. The former term can be controlled by $\Vert f\Vert _{L^p}$ by Bernstein’s inequalities; for the latter, instead we can write

$$\begin{aligned} \Vert (\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^2}\le & {} \sum _{k\ge 0}\Vert \Delta _k(\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^2} \\\le & {} C\,\sum _{k\ge 0}2^{-k}\,\Vert \Delta _k(\mathrm{Id}\,-\Delta _{-1})\nabla f\Vert _{L^2}\;\le \;C\,\Vert \nabla f\Vert _{L^2}, \end{aligned}$$

where we used again Bernstein’s inequalities and the characterization $L^2\equiv B^0_{2,2}$.

In order to prove the second estimate, we proceed exactly as before. Again, Bernstein’s inequalities allow us to bound low frequencies by $\Vert f\Vert _{L^p}$. Next we have:

$$\begin{aligned} \Vert (\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^\infty }\le & {} C\,\sum _{k\ge 0}2^{3k/2}\,\Vert \Delta _k(\mathrm{Id}\,-\Delta _{-1})f\Vert _{L^2} \\\le & {} C\,\sum _{k\ge 0}2^{-\delta k}\,\left\| |D|^{\delta +3/2}\Delta _k(\mathrm{Id}\,-\Delta _{-1})f\right\| _{L^2} \end{aligned}$$

(for any ), where we denoted |D| the Fourier multiplier having symbol equal to $|\xi |$. By interpolation we can write

$$\begin{aligned}&\left\| |D|^{\delta +3/2}\Delta _k(\mathrm{Id}\,-\Delta _{-1})f\right\| _{L^2}\,\le \,C\,\left\| \Delta _k(\mathrm{Id}\,-\Delta _{-1})\nabla f\right\| ^{\sigma }_{L^2}\,\\&\quad \left\| \Delta _k(\mathrm{Id}\,-\Delta _{-1})\nabla ^2f\right\| ^{1-\sigma }_{L^2}, \end{aligned}$$

for $\sigma \,\in \,]0,1[\,$ (actually, $\sigma =(1/2)-\delta$), and this immediately gives the conclusion.

Let us finally prove the third claim. By spectral localization we can write

$$\begin{aligned} \left\| \left( \mathrm{Id}\,-S_j\right) f\right\| ^2_{L^2}\le & {} \sum _{k\ge j-1}\Vert \Delta _kf\Vert ^2_{L^2}\;\le \; \sum _{k\ge j-1}2^{-2k}\Vert \nabla \Delta _kf\Vert ^2_{L^2} \\\le & {} \sum _{k\ge j-1}2^{2kd\left( 1/p\,-\,1/2\right) }\,2^{-2k}\,\Vert \nabla \Delta _kf\Vert ^2_{L^p}. \end{aligned}$$

Keeping in mind that, by hypothesis, $d\left( 1/p\,-\,1/2\right) -1=-\beta <0$, we infer the desired inequality and the explicit expression for $C_j$. $\square$

Finally, let us recall that one can rather work with homogeneous dyadic blocks $(\dot{\Delta }_j)_{j\in \mathbb {Z}}$, with

$$\begin{aligned} \dot{\Delta }_j:=\varphi (2^{-j}D)\quad \text{ for } \text{ all } \quad j\in \mathbb {Z}, \end{aligned}$$

and introduce the homogeneous Besov spaces $\dot{B}^s_{p,r}$, defined by the condition

$$\begin{aligned} \Vert u\Vert _{\dot{B}^{s}_{p,r}}\,:=\, \left\| \left( 2^{js}\,\Vert \dot{\Delta }_ju\Vert _{L^p}\right) _{\!j\in \mathbb {Z}}\,\right\| _{\ell ^r}\,<\,+\infty . \end{aligned}$$

We do not enter into the details here; we just limit ourselves to recall refined embeddings of homogeneous Besov spaces into Lebesgue spaces (see Theorem 2.40 of [1]).

Proposition 7.4

For any $2\le p<+\infty$, one has the continuous embeddings $\dot{B}^0_{p,2}\,\hookrightarrow \,L^p$ and $L^{p'}\,\hookrightarrow \,\dot{B}^0_{p',2}$.