Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation

Abstract

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in $\dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)$ with $s > 3$ and for any $0< \tau <1$. Such a result improves upon the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to $H^{20}(\mathbb {R}^2)$. More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in $H^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)$ with $s > 3$ and $0< \tau <1$. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity $\Vert u_2(t)\Vert _{L^\infty (\mathbb {R}^2)}$ for initial data only in $\dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)$ with $s >3$.

1 Introduction

The Incompressible Porous Media (IPM) system in two space dimensions is an active scalar equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \eta + {\textbf {u}} \cdot \nabla \eta =0,\\ {\textbf {u}}=- \kappa \nabla P + {\textbf {g}}\eta , \qquad {\textbf {g}}=(0,-g)^T, \qquad \text {(Darcy law)} \\ \nabla \cdot {\textbf {u}}=0, \end{array}\right. } \end{aligned}$$

(1)

modeling the dynamics of a fluid of density $\eta =\eta (t, x, y):\mathbb {R}^+\times \mathbb {R}^2 \rightarrow \mathbb {R}$ through a porous medium according to the Darcy law, where $\kappa >0$ and $g>0$ are the permeability coefficient and the gravity acceleration respectively, which hereafter are assumed to be $\kappa =g=1$. We refer to [6] and references therein for further explanations on the physics and the applications of the model. The active scalar velocity ${\textbf {u}}=(u_1, u_2)$ of system (1) can be reformulated in terms of a singular integral operator of degree 0

$$\begin{aligned} {\textbf {u}}={(-\mathcal {R}_2, \mathcal {R}_1) \mathcal {R}_1 \rho }, \end{aligned}$$

(2)

where $(\mathcal {R}_1, \mathcal {R}_2)$ is the two-dimensional Riesz transform, i.e.

$$\begin{aligned} \mathcal {R}_1=(-\Delta )^{-1/2} \partial _x, \qquad \mathcal {R}_2=(-\Delta )^{-1/2} \partial _y. \end{aligned}$$

(3)

We are interested in the stability properties of the stratified steady state ${\overline{\rho }}_{\text {eq}}(y)=\rho _0-y$, where $\rho _0>0$ is a constant averaged density. Introducing the perturbed unknown $\rho =\rho (t,x,y)$ such that $\eta (t,x,y)=\overline{\rho }_{\textrm{eq}}(y)+\rho (t,x,y)$, the perturbation $\rho$ satisfies

$$\begin{aligned} \partial _t \rho - \mathcal {R}_1^2 \rho = (\mathcal {R}_2 \mathcal {R}_1\rho , -\mathcal {R}_1^2 \rho ) \cdot \nabla \rho . \end{aligned}$$

(IPM-diss)

The nonlinear asymptotic stability of the stratified steady state $\overline{\rho }_{\text {eq}}(y)=\rho _0-y$ in the whole space $\mathbb {R}^2$ has been first established by Elgindi [16], for initial data at least in $H^{20}(\mathbb {R}^2).$ The analogous result but in the periodic finite channel $\mathbb {T}\times [-\pi ,\pi ]$ is due to Castro et al. [6] under slightly less restrictive regularity assumptions. We remark that the linear operator

$$\begin{aligned} \partial _t \rho -\mathcal {R}_1^2 \rho =0 \end{aligned}$$

in frequency space $\xi =(\xi _1, \xi _2)\in \mathbb {R}^2$ is dissipative everywhere but the hyperplane $\xi _1=0$ thanks to the monotonicity ${\bar{\rho }}_{\text {eq}}'(y)<0$ (stable stratification, see [15, 19]). Therefore, it appears natural to approximate (IPM-diss) by a system with a (partially) dissipative linear part. In this regard, it turns out that the two-dimensional Boussinesq equations in vorticity form with strongly damped vorticity is a good approximation to (IPM-diss) under a suitable scaling of time and unknowns. To the best of our knowledge, the approximation of (IPM-diss) via the Boussinesq system with damped vorticity is new and establishing its rigorous validity in the sense of strong convergence of solutions is in the scope of the present work.

The two-dimensional Boussinesq equations with damped velocity with gravity acceleration $g=1$ and background density profile $\overline{\rho }_{\text {eq}}(y)=\rho _0-y$ read as follows:

$$\begin{aligned} \left\{ \begin{aligned} \partial _t b- u_2&=-(u_1\partial _x b + u_2\partial _y b), \\ \partial _t u_1 + \partial _x P&=-\frac{u_1}{\varepsilon }-(u_1\partial _x u_1 + u_2\partial _y u_1), \\ \partial _t u_2 + \partial _y P&=-b-\frac{u_2}{\varepsilon }-(u_1\partial _x u_2 + u_2\partial _y u_2), \\ \partial _x u_1 + \partial _y u_2&=0. \end{aligned} \right. \end{aligned}$$

(4)

Here ${\textbf {u}}=(u_1,u_2) \in \mathbb {R}^2$ is the velocity field, b is the buoyancy term and P is the incompressible pressure. This system is obtained by a linearization of the density-dependent incompressible Euler equations with damped velocity around the hydrostatic stratified steady state

$$\begin{aligned} ({\overline{\rho }}_{\textrm{eq}} (y), (0,0), {\overline{P}}_{\textrm{eq}}(y)) \quad \text {with} \quad {\overline{P}}_{\textrm{eq}}'(y)=- \overline{\rho }_{\textrm{eq}} (y). \end{aligned}$$

(5)

Several mathematical works in the existing literature have been devoted to this system. Besides explaining the applications of (4) to electrocapillarity, Castro et al. [7] (see also references therein) establish the global-in-time well-posedness of system (4) in the bounded domain $\mathbb {T} \times [0,1]$. In the whole space, global smooth solutions have been first provided by Wan [35] by means of Green function analysis and energy methods.

We refer to [3, 7] for a (formal) derivation of (4) by a linearization of the density-dependent incompressible Euler equations around the hydrostatic stratified steady state under the strong Boussinesq approximation $\rho \sim \rho _0$ [15].

For our scopes, it is convenient to rewrite the system in vorticity $\omega$ and stream function $\phi$ formulation where ${\textbf {u}}=\nabla ^\perp \phi$ with the sign convention (as in [16]) $\nabla ^\perp =(\partial _y, -\partial _x)$, which gives

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t b - (-\Delta )^{-1} \partial _x \omega = - {\textbf {u}} \cdot \nabla b, \\&\partial _t \omega + \frac{\omega }{\varepsilon } - \partial _x b = - {\textbf {u}} \cdot \nabla \omega ,\\&\Delta \phi =\omega . \end{aligned} \right. \end{aligned}$$

(2D-Bouss)

Our first goal is to establish a systematic and improved proof of the global-in-time existence of smooth solutions for small data to (2D-bouss) (in Theorem 2.1), exploiting the anisotropic nature of the system by means of anisotropic Littlewood-Paley decomposition of the Fourier space $(\xi _1, \xi _2) \in \mathbb {R}^2$ as introduced in [8, 21, 33]. Since the linear and undamped approximation of the system supports the propagation of anisotropic waves of dispersion relation ${\pm \xi _1}/{|\xi |}= \mp i\widehat{\mathcal {R}_1}$ (see [2, 15]), it is not surprising that the horizontal anisotropic decomposition of the phase space plays a key role in our refined analysis. In our case, the linear part of system (2D-bouss) is dissipative provided that $\xi _1 \ne 0$, therefore it is natural to build an energy functional with an anisotropic Fourier multiplier as a weight (multiplier method). This idea allows us to prove that $\Vert u_2(t)\Vert _{L^\infty (\mathbb {R}^2)}$ is integrable in time without assuming any additional $L^1(\mathbb {R}^2)$ integrability of the initial data. More precisely, the control of $\Vert u_2\Vert _{L^1_T L^\infty (\mathbb {R}^2)}$ without $L^1$-integrability and high regularity assumptions is new and this is a striking point to provide a substantially improved global-in-time well-posedness of (2D-bouss) for small data only in $\dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)$ with $0< \tau <1$ and $s \geqq 3+\tau$.

For convenience of the reader, we point out that our small-data global-in-time existence in Theorem 2.1 could be reformulated as a result of nonlinear asymptotic stability of the hydrostatic steady state (5) with $\overline{\rho }_{\textrm{eq}}(y)=\rho _0-y$, under the evolution of the Boussinesq system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t \eta + {\textbf {u}}\cdot \nabla \eta =0, \\ \partial _t {\textbf {u}} + {\textbf {u}} \cdot \nabla {\textbf {u}} + \nabla P =\eta {\textbf {g}}, \qquad {\textbf {g}}=(0,-g),\\ \nabla \cdot {\textbf {u}}=0.\qquad \qquad \qquad \qquad \qquad \qquad \end{array}\right. } \end{aligned}$$

(E)

This reformulation holds for system (E) with initial density $\eta _{\textrm{in}}(x,y)$ such that $\Vert \eta _{\textrm{in}}-\overline{\rho }_{\textrm{eq}}\Vert _X \ll 1$ for a suitable functional space X, in the same spirit of [7].

Our second and main goal is to rigorously justify the relaxation limit of the two-dimensional Boussinesq equations with damped vorticity (2D-bouss) towards (IPM-diss) under a suitable scaling that we introduce later on. To the best of our knowledge, this relaxation approximation is new.

Finally, as a byproduct of the relaxation limit and the global well-posedness of (2D-bouss), we provide a new proof of existence of global smooth solutions in $\mathbb {R}^2$ to the equation (IPM-diss) for the perturbation $\rho$, with small initial data $\rho (0,x,y)=\rho _{\textrm{in}}(x,y) \in \dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s(\mathbb {R}^2)$ with $0< \tau <1$ and $s \geqq 3+\tau$. Again, this result can be reformulated in terms of the solution $\eta$ to (1), for an initial datum $\eta _{\textrm{in}}(x,y)=\overline{\rho }_{\textrm{eq}}(y)+\rho _{\textrm{in}}(x,y)$ such that $\Vert \eta _{\textrm{in}}(x,y)-\overline{\rho }_{\textrm{eq}}(y)\Vert _X =\Vert \rho _{\textrm{in}}(x,y)\Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \ll 1$. Such reformulation yields the nonlinear asymptotic stability of equation (1) around the stratified steady state $\overline{\rho }_{\textrm{eq}}(y)$, namely the setting of Elgindi [16] that proves the result in $H^{20}(\mathbb {R}^2)$, while in our Theorem 2.3 we only need that the initial perturbation $\rho _\text {in}(x,y) \in \dot{H}^{1-\tau }(\mathbb {R}^2) \cap \dot{H}^s$ with $s>3$.

1.1 A New Formulation

A first key element of our approach is the use of the symmetrized variables introduced in [3]. With the notation

$$\begin{aligned} \Lambda =(-\Delta )^{1/2}=\sqrt{\xi _1^2+\xi _2^2}, \end{aligned}$$

where $\xi =(\xi _1, \xi _2) \in \mathbb {R}^2$ denotes the frequency coordinate in Fourier space, we introduce the new unknown

$$\begin{aligned} \begin{aligned} \Omega := \Lambda ^{-1} \omega , \end{aligned} \end{aligned}$$

(6)

so that system (2D-bouss) can be as rewitten

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t b - \mathcal {R}_1 \Omega = (\mathcal {R}_2 \Omega , - \mathcal {R}_1 \Omega ) \cdot (\nabla b), \\&\partial _t \Omega + \frac{\Omega }{\varepsilon } - \mathcal {R}_1 b = \Lambda ^{-1} [ (\mathcal {R}_2 \Omega , - \mathcal {R}_1 \Omega ) \cdot (\nabla \Lambda \Omega )], \end{aligned} \right. \end{aligned}$$

(2D-B)

where $\mathcal {R}_j$, $j\in \{1,2\}$ are the components of the Riesz transform as in (3). Now, we introduce the auxiliary variable

$$\begin{aligned} z:= \Omega - \varepsilon \mathcal {R}_1 b. \end{aligned}$$

(7)

The system in (b, z) then reads as follows:

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t b - \varepsilon \mathcal {R}_1^2 b = \mathcal {R}_1 z + (\mathcal {R}_2 \Omega , - \mathcal {R}_1 \Omega ) \cdot \nabla b, \\&\partial _t z + \frac{z}{\varepsilon }={-}\varepsilon \mathcal {R}_1^2 \Omega {-} \varepsilon \mathcal {R}_1 [(\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot \nabla b] + \Lambda ^{-1} [(\mathcal {R}_2 \Omega , -\mathcal {R}_{1}\Omega ) \cdot (\nabla \Lambda \Omega )]. \end{aligned} \right. \end{aligned}$$

(8)

Such formulation partially diagonalizes the linear part of the system, except two linear terms in the right-hand side of (8) that in the energy estimates are absorbed by the left-hand side. This allows us to avoid hypocoercivity techniques [1, 12], obtaining a priori estimates simply based on the energy method. We point out that these a priori estimates are uniform in the vanishing parameter $\varepsilon$, which is a key point to justify the relaxation towards (IPM-diss).

The use of the good unknown $z=\Omega - \varepsilon \mathcal {R}_1 b$ is inspired both by the work of Hoff and Haspot in [22, 23], where the authors introduce the effective velocity for the compressible Navier–Stokes equations and the results of Crin-Barat and Danchin in [10, 11, 13] on partially dissipative hyperbolic systems. In the aforementioned works, this reformulation is only possible in some specific frequency regime (high frequencies for Navier–Stokes and low frequencies for partially dissipative systems) where the eigenvalues of the linear operator are real. In the present paper, the particular form of the Riesz transform (an operator of degree 0) allows us to use such diagonalization in all frequencies as the eigenvalues $\lambda _\pm$ of the linear part of system (2D-B )

$$\begin{aligned} \lambda _{\pm }=\frac{1}{2 \varepsilon } \pm \frac{1}{2 \varepsilon } \sqrt{1-\frac{4 \varepsilon ^2 \xi _1^2}{|\xi |^2}} \end{aligned}$$

are real in the whole frequency space for any $\varepsilon \leqq 1/2$, so that the variable z in (7) is a good unknown in all frequency regimes.

1.2 Formal Justification of the Relaxation Limit as $\varepsilon \rightarrow 0$

Taking inspiration from the theory of partially dissipative systems (see for instance [4, 9, 24, 25, 28, 31, 32, 36] and references therein), one can expect to rigorously justify the relaxation limit from (2D-B ) towards (IPM-diss) as $\varepsilon \rightarrow 0$, by applying the following scaling:

$$\begin{aligned} ({\widetilde{b}}^\varepsilon ,{\widetilde{\Omega }}^\varepsilon )(\tau ,x) \triangleq (b,\frac{\Omega }{\varepsilon })(t,x) \quad \!\text{ with }\!\quad \tau =\varepsilon t. \end{aligned}$$

(9)

The system (2D-B ) in the scaled unknowns $({\widetilde{b}}^\varepsilon ,{\widetilde{\Omega }}^\varepsilon )$ reads as follows:

$$\begin{aligned} \left\{ \begin{aligned} \partial _t {\widetilde{b}}^\varepsilon - \mathcal {R}_1 {\widetilde{\Omega }}^\varepsilon&= (\mathcal {R}_2 {\widetilde{\Omega }}^\varepsilon ) \partial _x {\widetilde{b}}^\varepsilon - (\mathcal {R}_1 {\widetilde{\Omega }}^\varepsilon ) \partial _y {\widetilde{b}}^\varepsilon , \\ \varepsilon ^2\partial _t {\widetilde{\Omega }}^\varepsilon - \mathcal {R}_1 {\widetilde{b}}^\varepsilon + {\widetilde{\Omega }}^\varepsilon&=\varepsilon ^2\Lambda ^{-1} [(\mathcal {R}_2 \widetilde{\Omega }^\varepsilon , - \mathcal {R}_1 \widetilde{\Omega }^\varepsilon ) \cdot (\nabla \Lambda \widetilde{\Omega }^\varepsilon )]. \end{aligned} \right. \end{aligned}$$

(10)

$\bullet$ Our first result is a systematic and improved proof of the global-in-time well-posedness, with uniform estimates in the relaxation parameter $\varepsilon \leqq \frac{1}{2}$, of the above system with initial data $(b_{\textrm{in}}, \Omega _{\textrm{in}}) \in \dot{H}^{1-\tau } \cap \dot{H}^s$ for any $0< \tau < 1$ and $s \geqq 3+\tau$. The approach is based on the use of the anisotropic Littlewood-Paley decomposition that allows to capture the (anisotropic) nature of the equation in a nearly optimal way. For references on the use of anisotropic Besov spaces in the analysis of incompressible flows we refer to [21] and we mention [17] for a study of the effect of anisotropy (the Riesz transform) in low regularity.

Sending $\varepsilon \rightarrow 0$ in system (10), one formally obtains that ${{\widetilde{\Omega }}}^\varepsilon \rightarrow \Omega$ and ${{\widetilde{b}}}^\varepsilon \rightarrow \rho$, where $\rho$ satisfies the Incompressible Porous Media equation (IPM-diss) and the Darcy law

$$\begin{aligned} {\Omega -\mathcal {R}_1 \rho =0.} \end{aligned}$$

(11)

We improperly call diffusive scaling the change of coordinates (9). Of course it is not the usual diffusive scaling $(t/\varepsilon ^2, x/\varepsilon , y/\varepsilon )$ under which the heat equation is invariant, but inserting (11) into the linear part of the equation for $\widetilde{b}^\varepsilon$ in (10) yields

$$\begin{aligned} \partial _t \rho - \mathcal {R}_1^2 \rho =0, \end{aligned}$$

where $\mathcal {R}_1^2=(-\Delta )^{-1} \partial _{xx}$ is a partially dissipative operator.

$\bullet$ Our next result is a mathematical proof of the relaxation limit of (10) towards (IPM-diss) as $\varepsilon \rightarrow 0$. To the best of our knowledge, the relaxation approximation of (IPM-diss) provided by (2D-B ) is new as well as its rigorous justification.

On the limit system (IPM-diss), the existence of global-in-time smooth solutions to (IPM-diss) with small data (or, equivalently, the asymptotic stability of (1) around the stratified steady state) has been first proved by Elgindi in [16], both in the full space $\mathbb {R}^2$ and in $\mathbb {T}^2$. In particular, Elgindi shows that the stratified steady state $\overline{\rho }_{\textrm{eq}}(y)=\rho _0-y$ is asymptotically stable in ${ H}^s(\mathbb {R}^2)$, for $s\geqq 20$, with the additional integrability assumption that initial perturbations $\rho _{\text {in}}(x,y)$ belong to the $L^1$-based Sobolev space $W^{4,1}$ (this is also a key hypothesis to use dispersion effects in [18]). Although groundbreaking, the result in [16] requires very high regularity of solutions.

$\bullet$ As a byproduct of the relaxation limit, in this article we provide a new proof of existence of global-in-time smooth solutions to (IPM-diss), only assuming that the initial datum $\rho _{\text {in}}(x,y) \in \dot{H}^{1-\tau } \cap \dot{H}^s$ for any $0< \tau < 1$ and $s \geqq \tau +3$.

RB thanks Ángel Castro for several useful discussions on the (IPM) equation, Klaus Widmayer for pointing out the use of anisotropic Besov spaces in fluid-dynamics problems. She also expresses her gratitude to Tarek M. Elgindi for insightful conversations and valuable suggestions.

2 Main Results

Recalling the notation $\Lambda =(-\Delta )^\frac{1}{2}$, we introduce the homogeneous Sobolev space for $s \in \mathbb {R}$

$$\begin{aligned} \Vert f\Vert _{\dot{H}^s}= \Vert \Lambda ^s f\Vert _{L^2(\mathbb {R}^2)}=\left( \int _{\mathbb {R}^2} |\xi |^{2s} |{\widehat{f}}(\xi )|^2 \, \textrm{d}\xi \right) ^\frac{1}{2}. \end{aligned}$$

(12)

We also use the notation, for any $s, s' \in \mathbb {R}$,

$$\begin{aligned} \Vert f\Vert _{\dot{H}^s\cap \dot{H}^{s'}} = \Vert f\Vert _{\dot{H}^s}+\Vert f\Vert _{\dot{H}^{s'}}. \end{aligned}$$

(13)

Our main results hold in Sobolev spaces, however we will rely on the properties of anisotropic Besov spaces to obtain some estimates that play a crucial role in our proof. Such anisotropic spaces allow to perfectly capture the anisotropic nature of the 2D Boussinesq system. Similar approaches have been applied to the MHD system by Lin and Zhang in [29, 30] and to the incompressible Navier–Stokes equations by Chemin and Zhang [8] and Paicu [33]. More recently, an approach based on anisotropic Besov spaces has been developed for 3D rotating incompressible fluids by Guo, Pausader and Widmayer [21].

Our first result concerns the uniform global well-posedness of system (2D-bouss). For any $r>0$, we define the following functional

$$\begin{aligned} \mathcal {M}_r (T)&:=\Vert (b,\Omega ,z)\Vert _{L^\infty _T({\dot{H}}^r)}+\sqrt{\varepsilon }\Vert \mathcal {R}_1b\Vert _{L^2_T({\dot{H}}^r)}\nonumber \\&\quad + \frac{1}{\sqrt{\varepsilon }}\Vert \Omega \Vert _{L^2_T({\dot{H}}^r)}+\frac{1}{\sqrt{\varepsilon }} \Vert z\Vert _{L^2_T({\dot{H}}^r)}. \end{aligned}$$

(14)

We obtain the following.

Theorem 2.1

(Global existence for (2D-B )) For any $0<\varepsilon \leqq 1/2$ and any $0< \tau < 1$, let $s\geqq 3+\tau$. For any couple of initial data $(b_{\textrm{in}},\Omega _{\textrm{in}})\in {\dot{H}}^{1-\tau }(\mathbb {R}^2) \cap {\dot{H}}^s(\mathbb {R}^2)$, there exists a constant value $0<\delta _0 \ll 1$ such that, under the following assumption that

$$\begin{aligned} \mathcal {M}(0)=\Vert (b_{\textrm{in}},\Omega _{\textrm{in}})\Vert _{\dot{H}^{1-\tau }\cap {\dot{H}}^s}\leqq \delta _0, \end{aligned}$$

(15)

there exists a unique global-in-time smooth solution $(b,\Omega )$ to system (2D-B ) satisfying the following inequality for all times $T>0$

$$\begin{aligned} X(T):=\mathcal {M}(T)+\Vert \nabla u_2\Vert _{L^1_T(L^\infty )}+\Vert \Lambda u_2\Vert _{L^1_T(L^\infty )} \leqq \mathcal {M}(0), \end{aligned}$$

(16)

where

$$\begin{aligned} u_2=(-\Delta )^{-1/2} \partial _x \Omega =\mathcal {R}_1 \Omega , \end{aligned}$$

(17)

and

$$\begin{aligned} \mathcal {M}(T):=\mathcal {M}_{1-\tau }(T)+\mathcal {M}_{s}(T). \end{aligned}$$

(18)

Remark 2.1

(On the expression of $\mathcal {M}(t)$) The functional $\mathcal {M}(t)$ in (18) is the sum of two terms, i.e. $\mathcal {M}_{1-\tau }(t)$ and $\mathcal {M}_{s}(t)$, defined in (14), which are both crucial for the embedding into anisotropic Besov spaces ${\dot{H}}^s \cap {\dot{H}}^{1-\tau }\subset B^{s_1, s_2}$ in Lemma 3.2. This is a key point: in fact, although the core of our analysis will be developed in anisotropic Besov spaces $B^{s_1, s_2}$ (introduced in Section 3.1), however the final result is stated in Sobolev regularity precisely thanks to the embedding ${\dot{H}}^s \cap {\dot{H}}^{1-\tau } \subset B^{s_1, s_2}$.

Remark 2.2

(Our setting and comparison with the result of Wan [35]) The global well-posedness of the 2D Boussinesq system with damping (4) in $\mathbb {R}^2$ was first established by Wan in [35]. Besides providing a more systematic and shorter proof (of Theorem 2.1) that exploits the anisotropic nature of the system, the present work improves several points.

• The regularity assumptions are lowered: [35] requires $b \in \dot{H}^{-1} \cap \dot{H}^{s_0}, \, \omega \in \dot{H}^{-2} \cap \dot{H}^{s_0-1}$ with $s_0 \geqq 6$, while here we only need $b \in \dot{H}^{1-\tau } \cap \dot{H}^{s}, \, \omega \in \dot{H}^{-\tau } \cap \dot{H}^{s-1}$ for any $0<\tau < 1$ and $s \geqq 3+\tau$.

• While [35] only bounds $\Vert u_2\Vert _{L^\frac{4}{3}_T L^\infty }$ relying on spectral analysis and sophisticated estimates, here we provide a control of $\Vert u_2\Vert _{L^1_T L^\infty }$ thanks to the anisotropic Littlewood-Paley approach; notice that the $L^1_T$ control of $\Vert u_2\Vert _{L^\infty }$ with $u_2=\mathcal {R}_1 \Omega$ is natural as $\Vert \mathcal {R}_1 \Omega \Vert _{L^\infty }$ is expected to decay at integrable rate like $\nabla u_2$ (see Proposition 3.4 in [16, 3], and Remark 4.2 in [35]).

• In stark contrast with [35] where $b \in \dot{H}^{-1}$ (see Remark 4.1 in [35]), we stress that here we do not need any unnatural assumption on b, which simply belongs to $\dot{H}^{1-\tau } \cap \dot{H}^s, \, s \geqq 3+\tau , \; 0< \tau < 1$.

• Finally, it is interesting to point out that in our framework $b \in \dot{H}^{1-\tau } \cap \dot{H}^{s}$ (homogeneous spaces with positive indexes) is not required to be square integrable, which is natural as both the Boussinesq equations (2D-bouss) and the incompressible porous media equation (IPM-diss) are invariant by the transformation $b\rightarrow b+C$, for any constant $C \in \mathbb {R}$.

Remark 2.3

(On the $\dot{H}^{1-\tau }$ estimate) As just remarked before, we only take $(b_\text {in}, \Omega _\text {in}) \in \dot{H}^{1-\tau } \cap \dot{H}^s$ without involving any negative Sobolev space (in [35] $b_\text {in} \in \dot{H}^{-1} \cap \dot{H}^{s_0}$ and $\omega _{\text {in}} \in \dot{H}^{-2} \cap H^{s_0-1}$ with $s_0 \geqq 6$). However, we point out that there is no need of $\omega \in \dot{H}^{-\tau }$ (i.e. $\Omega \in \dot{H}^{1-\tau }$) in the first part, namely the proof of Proposition 4.1. The assumption $\Omega _{\textrm{in}} \in \dot{H}^{1-\tau }$ is only used in the anisotropic Besov part (more precisely in the control of the Y(t) functional in (35)) to bound $\Vert \Omega \Vert _{L^\infty _T(B^{\frac{1}{2}, \frac{1}{2}})}$ by means of the embedding (Lemma 3.2) ${\dot{H}}^s \cap {\dot{H}}^{1-\tau } \subset B^{\frac{1}{2},\frac{1}{2}}$, and therefore to control $\mathcal {M}_{1-\tau }$.

The result below concerns the justification of the relaxation limit.

Theorem 2.2

(Relaxation limit) Let the hypotheses of Theorem 2.1 be fulfilled and let $(\widetilde{b}^\varepsilon ,\widetilde{\Omega }^\varepsilon )$ be the unique solution, scaled with (9), associated to the initial data $(b_{\textrm{in}},\Omega _{\text {in}})$ as in Theorem 2.1.

Then, for any $0<s'<s$ and $0<\tau<\tau '<1$, one has the limit as $\varepsilon \rightarrow 0$,

$$\begin{aligned} \widetilde{b}^\varepsilon \rightarrow \rho \text { strongly in }C([0,T],{\dot{H}}^{1-\tau '}_{loc}\cap {\dot{H}}^{s-s'}_{loc}), \end{aligned}$$

where $\rho$ is the unique solution of (IPM-diss) associated to the initial data $b_\text {in}$. Moreover, it holds that

$$\begin{aligned} \Vert \rho (\cdot ,t)\Vert _{\dot{H}^{1-\tau }\cap \dot{H}^s}\leqq C\Vert (b_{\textrm{in}},\Omega _{\textrm{in}})\Vert _{\dot{H}^{1-\tau }\cap \dot{H}^s}, \end{aligned}$$

where the constant C is independent of $\varepsilon$.

Finally, we recover the Darcy law in the following sense:

$$\begin{aligned} \Vert \widetilde{\Omega }^\varepsilon -\mathcal {R}_1\widetilde{b}^\varepsilon \Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})}\leqq \varepsilon \mathcal {M}(0). \end{aligned}$$

(19)

Remark 2.4

(On the Darcy law) Note that applying the operator $\nabla ^\perp \cdot$ to the velocity ${\textbf {u}}$ in (1) with $\kappa =g=1$ (and replacing the notation $\eta$ by $\rho$) yields

$$\begin{aligned} \omega =\nabla ^\perp \cdot {\textbf {u}}=\partial _x \rho , \end{aligned}$$

which in terms of the variables $(\Omega , \rho )$ reads exactly $\Omega =\mathcal {R}_1 \rho$.

Our analysis also provides a new proof of existence of global-in-time smooth solutions to the incompressible porous media equation (IPM-diss) for small data.

Theorem 2.3

(Existence for (IPM-diss)) For any $0< \tau < 1$, let $s\geqq 3+\tau$. For any initial datum $\rho _{\textrm{in}}\in {\dot{H}}^{1-\tau }(\mathbb {R}^2) \cap {\dot{H}}^s(\mathbb {R}^2)$, there exists a constant value $0<\delta _0 \ll 1$ such that, under the assumption

$$\begin{aligned} \Vert \rho _{\textrm{in}}\Vert _{\dot{H}^{1-\tau }\cap {\dot{H}}^s}\leqq \delta _0, \end{aligned}$$

there exists a unique global-in-time smooth solution $\rho$ to system (IPM-diss) satisfying the following inequality for all times $t>0$

$$\begin{aligned} X(t)&:=\Vert \rho \Vert _{L^\infty _T(\dot{H}^{1-\tau }\cap {\dot{H}}^s)}+\Vert \mathcal {R}_1\rho \Vert _{L^2_T(\dot{H}^{1-\tau }\cap {\dot{H}}^s)}\\&\quad +\Vert (\nabla \mathcal {R}_1^2 \rho ,\Lambda \mathcal {R}_1^2 \rho )\Vert _{L^1_T(L^\infty )}\lesssim \Vert \rho _{\textrm{in}}\Vert _{\dot{H}^{1-\tau }\cap {\dot{H}}^s}. \end{aligned}$$

Remark 2.5

(Comparison with the result of Elgindi [16]) The global well-posedness of (IPM-diss) for small data, namely the asymptotic stability of (1) around the stratified steady state $\overline{\rho }_{\textrm{eq}}(y)=\rho _0-y$, was first proved by Elgindi in [16] taking $b \in W^{4,1} \cap H^{s_0}, \, s_0\geqq 20$.

Our Theorem 2.3 provides a new proof of Elgindi’s result, which allows to substantially lower the regularity assumption of [16], taking only $b \in \dot{H}^{1-\tau } \cap \dot{H}^s$ with $0< \tau < 1$ and $s \geqq 3+\tau$. Once again, we take advantage of the anisotropic Littlewood-Paley decomposition and anisotropic Besov spaces that capture the time-integrability of the solution without relying on Green function estimates of the linearized problem. We also mention the asymptotic stability result by Castro-Córdoba-Lear [6] of (1) around the stratified steady state $\overline{\rho }_{\text {eq}}=\rho _0-y$ in the domain $\mathbb {T} \times [-\pi , \pi ].$

Remark 2.6

(Asymptotic stability on the torus ${\mathbb {T}}^2$ and near more general density profiles) Besides the whole space $\mathbb {R}^2$, Elgindi [16] provides asymptotic stability of (IPM-diss) on the 2D torus ${\mathbb {T}}^2$ and near more general density profiles. We will discuss the adaptations of our proof to these cases in Section 7.

Remark 2.7

(Instability results from Kiselev-Yao) A consequence of the recent result [26, Theorem 1.5] by Kiselev and Yao is that there exists an initial perturbation $\rho _\text {in}(x,y)$ satisfying $\Vert \rho _\text {in}\Vert _{H^{2-\gamma }(\mathbb {T}\times [-\pi , \pi ])}\ll 1$ for any $\gamma >0$, such that the solution $\rho (t, x, y)$ to (IPM-diss) (provided it remains smooth for all times) displays the time growth

$$\begin{aligned} \lim \sup \limits _{t\rightarrow \infty } t^{-\frac{s}{2} }\Vert \rho (t)\Vert _{\dot{H}^{s+1}(\mathbb {T}\times [-\pi , \pi ])}=+\infty \end{aligned}$$

for all $s>0$. In [26, Remark 1.6], the authors ask whether $\rho _{\text {in}}(x,y)$ can be made small in higher Sobolev spaces, while $\rho (t, x, y)$ still displays time growth. Even though in the present work we study the case of the full space $\mathbb {R}^2$ rather than the bounded periodic channel, we underline that our Theorem 2.3 states that all perturbations $\rho _{\text {in}}$ that are small in $H^{s}(\mathbb {R}^2)$ with $s>3$ generate solutions $\rho (t, x, y)$ that remain small for all times. Thus, if a blow-up in finite time or a time-growth happens for solutions in the whole space $\mathbb {R}^2$, the initial perturbation must have less regularity than $H^s(\mathbb {R}^2)$, $s>3$.

Remark 2.8

(On the regularity assumption) As mentioned previously, our regularity setting is $b \in \dot{H}^{1-\tau } \cap \dot{H}^{s}, \, \omega \in \dot{H}^{-\tau } \cap \dot{H}^{s-1}$ for any $0<\tau < 1$ and $s \geqq 3+\tau$. While our results are significantly sharper than the previous ones, an interesting problem that remains open is the proof of asymptotic stability of stratified solutions to the IPM equation for $s>2$ (i.e. $H^2(\mathbb {R}^2)$ is a critical space for the IPM equation). The primary obstacle preventing us from reaching critical regularity lies in controlling the nonlinearity within anisotropic Besov spaces. For instance, the first term of the right-hand side of the estimate (51) requires a control of $\Vert \partial _y b\Vert _{B^{\frac{3}{2}, \frac{1}{2}}} \lesssim \Vert \partial _y b\Vert _{\dot{H}^{2-\tau } \cap \dot{H}^{2+\tau }} \lesssim \Vert b\Vert _{\dot{H}^{3-\tau } \cap \dot{H}^{3+\tau }}$. To overcome this difficulty, the use of product rules in the proof of Proposition 4.2 should be replaced by suitable commutator estimates. However, as far as our knowledge extends, commutator estimates within anisotropic Besov spaces have yet to be presented in existing literature. Providing such estimates would entail a considerable amount of technical effort and would stand as an independent achievement.

3 Anisotropic Besov Spaces

3.1 Anisotropic Littlewood–Paley Decomposition

We introduce the following anisotropic Littlewood-Paley decompositions: for $j,q,k\in \mathbb {Z}$, we denote that

• ${\dot{\Delta }}_j$ are the blocks associated to the Littlewood-Paley decomposition in $|\xi |$;

• ${\dot{\Delta }}_q^h$ are the blocks associated to the Littlewood-Paley decomposition in the direction $\xi _1$,

• ${\dot{\Delta }}_k^v$ are the blocks associated to the Littlewood-Paley decomposition in the direction $\xi _2$,

such that, denoting by $\mathcal {F}$ the Fourier transform,

$$\begin{aligned} {\dot{\Delta }}_ju=\mathcal {F}^{-1}(\varphi (2^{-j}|\xi |){\widehat{u}}) {\dot{\Delta }}_q^h u=\mathcal {F}^{-1}(\varphi (2^{-q}\xi _1){\widehat{u}}) \quad \!\text{ and }\!\quad {\dot{\Delta }}_k^v u=\mathcal {F}^{-1}(\varphi (2^{-k}\xi _2){\widehat{u}}), \end{aligned}$$

where $\varphi (\xi ) = \phi (\xi /2)-\phi (\xi )$ and $\phi \in C_c^\infty$ is such that $\phi =1$ for $|\xi | \leqq 1/2$ and $\phi (\xi ) = 0$ for $|\xi | \geqq 1$. We define the following homogeneous and anisotropic Besov semi-norms:

$$\begin{aligned} \Vert f\Vert _{\dot{B}^{s}_{p,r}}&\triangleq \bigl \Vert 2^{js}\Vert {\dot{\Delta }}_jf\Vert _{L^p(\mathbb {R}^d)}\bigr \Vert _{\ell ^r(j\in \mathbb {Z})},\\ \Vert f\Vert _{\dot{B}^{s_1,s_2}_{p,r}}&\triangleq \bigl \Vert 2^{js_1}2^{qs_2}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h f\Vert _{L^p(\mathbb {R}^d)}\bigr \Vert _{\ell ^r(j\in \mathbb {Z},k\in \mathbb {Z})}. \end{aligned}$$

Hereafter we will omit the dot (standing for homogeneous spaces) and the second and third Besov indexes will be dropped as well for lightening the notation

$$\begin{aligned} \Vert f\Vert _{B^{s_1,s_2}}\triangleq \Vert f\Vert _{\dot{B}^{s_1,s_2}_{2,1}}. \end{aligned}$$

3.2 Technical Results in Anisotropic Besov Spaces

We now state an anisotropic version of Bernstein’s lemma, the proof of which can be found in [30, 33].

Lemma 3.1

(Bernstein-type inequalities) For $x=(x_1,x_2)\in \mathbb {R}^2$, let $B_1$ be a ball of $\mathbb {R}_{x_1}$, $B_2$ be a ball of $\mathbb {R}_{x_2}$, $C_1$ be an annulus of $\mathbb {R}_{x_1}$ and $C_2$ an annulus of $\mathbb {R}_{x_2}$. Let $1\leqq p_1 \leqq p_2 \leqq \infty$ and $1\leqq q_1 \leqq q_2 \leqq \infty$. Then, we have that

• If the support of $\widehat{a}$ is included in $2^q B_1$, then

  $$\begin{aligned} \Vert \partial _{x_1}^s a\Vert _{L^{p_2}_{x_1}(L^{q_1}_{x_2})} \lesssim 2^{q(|s|+(\frac{1}{p_1}-\frac{1}{p_2}))}\Vert a\Vert _{L^{p_1}_{x_1}(L^{q_1}_{x_2})}. \end{aligned}$$

• If the support of $\widehat{a}$ is included in $2^k B_2$, then

  $$\begin{aligned} \Vert \partial _{x_2}^s a\Vert _{L^{p_1}_{x_1}(L^{q_2}_{x_2})} \lesssim 2^{k(|s|+(\frac{1}{q_1}-\frac{1}{q_2}))}\Vert a\Vert _{L^{p_1}_{x_1}(L^{q_1}_{x_2})}. \end{aligned}$$

• If the support of $\widehat{a}$ is included in $2^q C_1$, then

  $$\begin{aligned} \Vert a\Vert _{L^{p_1}_{x_1}(L^{q_1}_{x_2})} \lesssim 2^{-q |s|}\Vert \partial _{x_1}^s a\Vert _{L^{p_1}_{x_1}(L^{q_1}_{x_2})}. \end{aligned}$$

• If the support of $\widehat{a}$ is included in $2^k C_2$, then

  $$\begin{aligned} \Vert a\Vert _{L^{p_1}_{x_1}(L^{q_1}_{x_2})} \lesssim 2^{-k |s|}\Vert \partial _{x_2}^s a\Vert _{L^{p_1}_{x_1}(L^{q_1}_{x_2})}. \end{aligned}$$

Embeddings of Sobolev spaces into anisotropic Besov spaces are provided by the result below.

Lemma 3.2

(Embedding in Sobolev space, [29, Lemma 3.2]) Let $s_1,s_2,\tau _1,\tau _2\in \mathbb {R}$ such that $\tau _1<s_1+s_2<\tau _2$ and $s_2>0$. If $a\in {\dot{H}}^{\tau _1}(\mathbb {R}^2)\cap {\dot{H}}^{\tau _2}(\mathbb {R}^2)$ and $a\in B^{s_1,s_2}$, then

$$\begin{aligned} \Vert a\Vert _{B^{s_1,s_2}}\lesssim \Vert a\Vert _{B^{s_1+s_2}}\lesssim \Vert a\Vert _{\dot{H}^{\tau _1}}+\Vert a\Vert _{\dot{H}^{\tau _2}}. \end{aligned}$$

We will rely on the embeddings $B^{\frac{3}{2},\frac{1}{2}} \hookrightarrow \text {Lip},$ $B^{\frac{1}{2},\frac{3}{2}}\hookrightarrow \text {Lip}(\mathcal {R}_1 \cdot )$ and $B^{-\frac{1}{2},\frac{5}{2}} \hookrightarrow \text {Lip}(\mathcal {R}_1^2 \cdot )$, where for $n=1,2$, the notation $\text {Lip}(\mathcal {R}_1^n\cdot )$ denotes the space of functions whose Riesz transform of order n is Lipschitz, cf. the left-hand sides of (20) for the associated norms.

Lemma 3.3

(Embeddings in $\text {Lip}$) Let $a\in B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2}, \frac{1}{2}} \cap B^{\frac{1}{2}, \frac{3}{2}}\cap B^{-\frac{1}{2}, \frac{5}{2}}$. The following inequalities hold:

$$\begin{aligned}&\Vert a\Vert _{L^\infty }\lesssim \Vert a\Vert _{B^{\frac{1}{2},\frac{1}{2}}}, \quad \Vert \nabla a \Vert _{L^\infty } \lesssim \Vert a\Vert _{B^{\frac{3}{2},\frac{1}{2}}} \quad \!\text{ and }\!\quad \Vert \Lambda a\Vert _{L^\infty }\lesssim \Vert a\Vert _{B^{\frac{3}{2}, \frac{1}{2}}}. \end{aligned}$$

When the Riesz operator in the direction x is involved, one has

$$\begin{aligned} \Vert \nabla \mathcal {R}_1 a \Vert _{L^\infty }\lesssim \Vert a\Vert _{B^{\frac{1}{2},\frac{3}{2}}},\quad \Vert \Lambda \mathcal {R}_1 a\Vert _{L^\infty } \lesssim \Vert a\Vert _{B^{\frac{1}{2}, \frac{3}{2}}} \quad \!\text{ and }\!\quad \Vert \nabla \mathcal {R}_1^2 a \Vert _{L^\infty }\lesssim \Vert a\Vert _{B^{-\frac{1}{2},\frac{5}{2}}}. \end{aligned}$$

(20)

Proof

Using the anisotropic Bernstein Lemma 3.1, one obtains

$$\begin{aligned} \Vert \nabla a \Vert _{L^\infty }\lesssim \sum _{j,q\in \mathbb {Z}^2}2^{j}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h a \Vert _{L^\infty }&\lesssim \sum _{j,q\in \mathbb {Z}^2,j\geqq k}2^{j}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h{\dot{\Delta }}_k^v a \Vert _{L^\infty }\\&\lesssim \sum _{j,q,k\in \mathbb {Z}^3,j\geqq k}2^{j}2^{\frac{q}{2}}2^{\frac{k}{2}}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h{\dot{\Delta }}_k^v a \Vert _{L^2}\\&\lesssim \sum _{j,q\in \mathbb {Z}^2}2^{\frac{3j}{2}}2^{\frac{q}{2}}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h a \Vert _{L^2}\\&\lesssim \Vert a\Vert _{B^{\frac{3}{2},\frac{1}{2}}_{2,1}}. \end{aligned}$$

When replacing the operator $\nabla$ by $\Lambda$, the proof follows exactly the same lines. The estimates involving the Riesz operator $\mathcal {R}_1$ can be obtained in a similar manner noticing that, for $s\in \{1,2\}$

$$\begin{aligned} \Vert \nabla \mathcal {R}_1^s a \Vert _{L^\infty }\lesssim \sum _{j,q\in \mathbb {Z}^2}2^{j}2^{-s j}2^{s q}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h a \Vert _{L^\infty }. \end{aligned}$$

$\square$

4 Proof of Theorem 2.1

The proof of Theorem 2.1 is divided in two main steps. We first provide the estimates in Sobolev spaces, namely we control the functional $\mathcal {M}(t)$ in (18). The outcome of those estimates in Proposition 4.1 involves the quantities $\Vert \nabla u_2\Vert _{L^1_T(L^\infty )}, \, \Vert \Lambda u_2\Vert _{L^1_T(L^\infty )}$, whose control is not provided by the energy functional $\mathcal {M}(t)$. Thus, in a second step in Section 4.2, we rely on anisotropic Besov spaces to estimate the aforementioned quantities and to conclude the proof.

4.1 I. Control of $\mathcal {M}(t)$

The estimates in homogeneous Sobolev spaces are provided by the result below.

Proposition 4.1

Let $(b,\Omega )$ be a smooth solution to system (2D-B ). Then the following holds

$$\begin{aligned} \mathcal {M}(T) \lesssim X(0)+X^3(T). \end{aligned}$$

Proof

The estimate of $\mathcal {M}(t)$ is divided in two parts.

i) Control of $\mathcal {M}_s(t)$, with $s \geqq 3+\tau$. We consider the equations for the unknowns (b, z) in system (8),

applying the operator $\Lambda ^s=(-\Delta )^\frac{s}{2}$ to each term with the notation $(\dot{b}, \dot{z}, {\dot{\Omega }})=(\Lambda ^s b, \Lambda ^s z, \Lambda ^s \Omega )$.

Noticing that $\Lambda ^s\mathcal {R}_1=\Lambda ^{s-1}\partial _x$, the quasi-linearized system reads as

$$\begin{aligned} \left\{ \begin{aligned} \partial _t \dot{b} - \varepsilon \mathcal {R}_1^2 \dot{b}&= \mathcal {R}_1 \dot{z} + (\mathcal {R}_2 \Omega , - \mathcal {R}_1\Omega ) \cdot \nabla \dot{b} + [\Lambda ^s, \mathcal {R}_2 \Omega ] \partial _x b-[\Lambda ^s, \mathcal {R}_1 \Omega ] \partial _y b \\&:=\mathcal {R}_1 \dot{z}+\mathcal {I}_1+\mathcal {C}_{1,1}-\mathcal {C}_{1,2}, \\ \partial _t \dot{z} + \frac{\dot{z}}{\varepsilon }&=-\varepsilon \mathcal {R}_1^2 \dot{\Omega }- \varepsilon ((\mathcal {R}_2 \Omega , - \mathcal {R}_1\Omega ) \cdot \nabla \mathcal {R}_1 \dot{b}) + (\mathcal {R}_2 \Omega , - \mathcal {R}_1\Omega ) \cdot \nabla \dot{\Omega }\\&\quad + [\Lambda ^{s-1}, \mathcal {R}_2 \Omega ] \partial _x \Lambda \Omega -[\Lambda ^{s-1}, \mathcal {R}_1 \Omega ] \partial _y \Lambda \Omega \\&\quad - \varepsilon [\Lambda ^{s-1}\partial _x, \mathcal {R}_2 \Omega ] \partial _x b+\varepsilon [\Lambda ^{s-1}\partial _x, \mathcal {R}_1 \Omega ] \partial _y b\\&=:-\varepsilon \mathcal {R}_1^2 {\dot{\Omega }}- \mathcal {I}_2+\mathcal {I}_3 + \mathcal {C}_{2,1}-\mathcal {C}_{2,2}-\mathcal {C}_{3,1}+\mathcal {C}_{3,2}. \end{aligned} \right. \end{aligned}$$

(21)

Similarly, the quasi-linearized equation for $\Omega$ (from (2D-B )) reads as

$$\begin{aligned} \partial _t {\dot{\Omega }} + \frac{{\dot{\Omega }}}{\varepsilon }&=\mathcal {R}_1 \dot{b} + ((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot (\nabla {\dot{\Omega }}))\nonumber \\&\quad + [\Lambda ^{s-1}, \mathcal {R}_2 \Omega ] \partial _x \Lambda \Omega -[\Lambda ^{s-1}, \mathcal {R}_1 \Omega ] \partial _y \Lambda \Omega . \end{aligned}$$

(22)

Let us provide the desired estimate.

The linear evolution

We first look at the linear terms, neglecting the nonlinearity. Using the skew-symmetry of the Riesz transform $(\mathcal {R}_1 \dot{z}, \dot{b})_{L^2}=-(\dot{z}, \mathcal {R}_1 \dot{b})_{L^2}$ and $(\mathcal {R}_1^2 {\dot{\Omega }}, \dot{z})_{L^2}=-(\mathcal {R}_1 {\dot{\Omega }}, \mathcal {R}_1 \dot{z})_{L^2}$,

$$\begin{aligned}&\frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} (\Vert b\Vert ^2_{\dot{H}^s}+\Vert \Omega \Vert ^2_{\dot{H}^s}+\Vert z\Vert ^2_{\dot{H}^s}) + \varepsilon \Vert \mathcal {R}_1 b\Vert ^2_{\dot{H}^s}&+ \frac{1}{\varepsilon }\Vert z\Vert ^2_{\dot{H}^s}+\frac{1}{\varepsilon }\Vert \Omega \Vert ^2_{\dot{H}^s}\\&\quad =-(\mathcal {R}_1 \dot{b}, \dot{z})_{L^2}+\varepsilon (\mathcal {R}_1 {\dot{\Omega }}, \mathcal {R}_1 \dot{z})_{L^2}. \end{aligned}$$

Using the Cauchy-Schwarz inequality

$$\begin{aligned} |(\mathcal {R}_1 \dot{b}, \dot{z})_{L^2}|&\leqq \frac{\varepsilon }{2}\Vert \mathcal {R}_1 b\Vert _{\dot{H}^s}^2 + \frac{1}{2\varepsilon }\Vert z\Vert _{\dot{H}^s}^2,\\ \varepsilon |(\mathcal {R}_1 {\dot{\Omega }}, \mathcal {R}_1 \dot{z})_{L^2}|&\leqq \frac{\varepsilon ^2}{2}\Vert \mathcal {R}_1 b\Vert _{\dot{H}^s}^2 + \frac{1}{2} \Vert \mathcal {R}_1 z\Vert _{\dot{H}^s}^2 \end{aligned}$$

and the continuity of the Riesz transform $\Vert \mathcal {R}_j a\Vert _{L^2} \leqq \Vert a\Vert _{L^2}$ for any $a \in L^2$, the last two terms are absorbed by the linear dissipation, yielding the inequality

$$\begin{aligned} \frac{1}{2} \frac{\textrm{d}}{\textrm{d}t} (\Vert b\Vert ^2_{\dot{H}^s}+\Vert \Omega \Vert ^2_{\dot{H}^s}+\Vert z\Vert ^2_{\dot{H}^s}) + \frac{\varepsilon }{4} \Vert \mathcal {R}_1 b\Vert ^2_{\dot{H}^s}&+ \frac{1}{4\varepsilon }\Vert z\Vert ^2_{\dot{H}^s}+\frac{1}{\varepsilon }\Vert \Omega \Vert ^2_{\dot{H}^s} \leqq 0. \end{aligned}$$

Now we deal with the nonlinearity of system (21).

Estimates of the nonlinear term Consider the quasi-linearized terms $\mathcal {I}_j$ for $j\in \{1,2,3\}$. The divergence-free condition and integration by parts yield

$$\begin{aligned} (\mathcal {I}_1, \dot{b})_{L^2} = ((\mathcal {R}_2 \Omega , - \mathcal {R}_1\Omega ) \cdot \nabla \dot{b}, \dot{b})_{L^2}=0. \end{aligned}$$

Furthermore, it is easy to see that

$$\begin{aligned} -\mathcal {I}_2+\mathcal {I}_3=(\mathcal {R}_2 \Omega , - \mathcal {R}_1\Omega ) \cdot \nabla \dot{z}, \end{aligned}$$

readily implying, using again the divergence free condition, that $(-\mathcal {I}_2+\mathcal {I}_3, \dot{z})=0.$ Next, let us focus on the commutators.

Commutator estimates We begin with the terms of type $(\mathcal {C}_{2, j}, \dot{z})_{L^2}, (\mathcal {C}_{3, j}, \dot{z})_{L^2}$ for $j \in \{1,2\}$, which are easier being quadratic in (some norm of) the dissipative variable z. Notice indeed that as the products $(\mathcal {C}_{1, j}, \dot{b})_{L^2}$ are quadratic in b, they require a more careful treatment since the $L^2_T$ control of the (spatial norm of the) variable b is not provided by the energy functional (one only controls $\mathcal {R}_1 b$, see the definition of $\mathcal {M}_\tau (T)$ in (14)).

Let us first look at $\mathcal {C}_{2,1}=[\Lambda ^{s-1}, \mathcal {R}_2 \Omega ] \partial _x \Lambda \Omega$. By applying the commutator estimates in Lemma B.3 (in the “Appendix”) with $s\geqq 3+\tau >\frac{d}{2}+2$, one has

$$\begin{aligned} \Vert \mathcal {C}_{2,1}\Vert _{L^2}\ {}&\lesssim \Vert \nabla \mathcal {R}_2 \Omega \Vert _{L^\infty }\Vert \mathcal {R}_1\Omega \Vert _{{\dot{H}}^s}+\Vert \partial _x \Lambda \Omega \Vert _{L^\infty } \Vert \mathcal {R}_2 \Omega \Vert _{{\dot{H}}^{s-1}}. \end{aligned}$$

Now, applying first Lemma 3.3 and after Lemma 3.2 with $1-\tau =\tau _1< 2 < s=\tau _2$, we have

$$\begin{aligned} \Vert \nabla \mathcal {R}_2 \Omega \Vert _{L^\infty } \lesssim \Vert \mathcal {R}_2 \Omega \Vert _{B^{\frac{3}{2}, \frac{1}{2}}} \lesssim \Vert \mathcal {R}_2 \Omega \Vert _{\dot{H}^{1-\tau } \cap \dot{H}^{s}}. \end{aligned}$$

¹Similarly, using again Lemma 3.3 and Lemma 3.2

$$\begin{aligned} \Vert \partial _x \Lambda \Omega \Vert _{L^\infty }&\lesssim \Vert \Lambda ^2 \mathcal {R}_1 \Omega \Vert _{L^\infty } \lesssim \Vert \Lambda \Omega \Vert _{B^{\frac{1}{2}, \frac{3}{2}}} \lesssim \Vert \Lambda \Omega \Vert _{\dot{H}^{2-\tau } \cap \dot{H}^{2+\tau }}\\&\lesssim \Vert \Omega \Vert _{\dot{H}^{3-\tau } \cap \dot{H}^{s}}, \quad (s\geqq 3+\tau ), \end{aligned}$$

where now, using the interpolation Lemma B.4

$$\begin{aligned} \Vert \Omega \Vert _{\dot{H}^{3-\tau }} \lesssim \Vert \Omega \Vert _{\dot{H}^{1-\tau }}^\theta \Vert \Omega \Vert _{\dot{H}^s}^{1-\theta }, \quad \text {with} \quad \theta =\frac{s+\tau -3}{s+\tau -1}. \end{aligned}$$

Finally, the Young inequality $ab \leqq \frac{a^p}{p}+\frac{b^q}{q}$ with $p=\frac{1}{\theta }, q=\frac{1}{1-\theta }$ gives

$$\begin{aligned} \Vert \Omega \Vert _{\dot{H}^{3-\tau }} \lesssim \ \Vert \Omega \Vert _{\dot{H}^{1-\tau }} + \Vert \Omega \Vert _{\dot{H}^{s}}. \end{aligned}$$

In the same way, using that $\Vert \mathcal {R}_1 a\Vert _{\dot{H}^s} \leqq \Vert a\Vert _{\dot{H}^s}$ for any $s\geqq 0$ and $a \in \dot{H}^s$, by interpolation

$$\begin{aligned} \Vert \mathcal {R}_2 \Omega \Vert _{\dot{H}^{s-1}}&\lesssim \Vert \Omega \Vert _{\dot{H}^{s-1}} \lesssim \Vert \Omega \Vert _{\dot{H}^{1-\tau }}^{{\tilde{\theta }}} \Vert \Omega \Vert _{\dot{H}^{s}}^{{\tilde{\theta }}}\\&\lesssim \Vert \Omega \Vert _{\dot{H}^{1-\tau }} + \Vert \Omega \Vert _{\dot{H}^{s}}, \quad \text {with} \quad {\tilde{\theta }}=\frac{1}{s+\tau -1}. \end{aligned}$$

Altogether, this it yields that

$$\begin{aligned} \int _0^T |(\mathcal {C}_{2,1}, \dot{z})_{L^2}|\, \textrm{d}t&\leqq \int _0^T \Vert \mathcal {C}_{2,1}\Vert _{L^2}\Vert z\Vert _{{\dot{H}}^s} \, \textrm{d}t \\&\lesssim \int _0^T \Vert \Omega \Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \Vert \Omega \Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \Vert z\Vert _{\dot{H}^s} \, \textrm{d}t\\&\lesssim (\Vert \Omega \Vert _{L^\infty _T({\dot{H}}^{1-\tau })}+\Vert \Omega \Vert _{L^\infty _T(\dot{H}^s)})(\Vert \Omega \Vert _{L^2_T({\dot{H}}^{1-\tau })}\\&\quad +\Vert \Omega \Vert _{L^2_T(\dot{H}^s)}) \Vert z\Vert _{L^2_T({\dot{H}}^s)} \\&\lesssim \mathcal {M}^3(T). \end{aligned}$$

The commutator $\mathcal {C}_{2,2}$ is completely analogous, we omit it. Let us consider $\mathcal {C}_{3,1}=\varepsilon [\Lambda ^{s-1}\partial _x, \mathcal {R}_2 \Omega ] \partial _x b$, which gives

$$\begin{aligned} \Vert \mathcal {C}_{3,1}\Vert _{L^2}&\lesssim \varepsilon (\Vert \nabla \mathcal {R}_2 \Omega \Vert _{L^\infty }\Vert \mathcal {R}_1 b\Vert _{{\dot{H}}^s}+\Vert \partial _x b\Vert _{L^\infty } \Vert \mathcal {R}_1 \mathcal {R}_2 \Omega \Vert _{{\dot{H}}^s}). \end{aligned}$$

Now note that, applying the same reasoning as before (Lemmas 3.3, 3.2),

$$\begin{aligned} \Vert \partial _x b\Vert _{L^\infty } \lesssim \Vert \Lambda \mathcal {R}_1 b\Vert _{B^{\frac{1}{2}, \frac{1}{2}}} \lesssim \Vert \Lambda \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau } \cap \dot{H}^{s-1}} \lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{2-\tau } \cap \dot{H}^{s}}, \end{aligned}$$

where, again,

$$\begin{aligned} \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{2-\tau }} \lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau }}^{\theta } \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{s}}^{1-\theta } \lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau }} + \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{s}}, \quad \theta =\frac{s+\tau -2}{s+\tau -1}. \end{aligned}$$

This way,

$$\begin{aligned} \int _0^T |(\mathcal {C}_{3,1}, {\dot{\Omega }})_{L^2}| \, \textrm{d}t&\lesssim \varepsilon \int _0^T (\Vert \nabla \mathcal {R}_2 \Omega \Vert _{L^\infty }\Vert \mathcal {R}_1 b\Vert _{{\dot{H}}^s}+\Vert \partial _x b\Vert _{L^\infty } \Vert \mathcal {R}_2 \Omega \Vert _{{\dot{H}}^s})\Vert \Omega \Vert _{\dot{H}^s} \, \textrm{d}t \\&\lesssim \varepsilon \int _0^T \Vert \Omega \Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s}^2 \Vert b\Vert _{\dot{H}^s} + \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \Vert \Omega \Vert _{\dot{H}^s}^2 \, \textrm{d}t \\&\lesssim \varepsilon ^2 \Vert b\Vert _{L^\infty _T (\dot{H}^{1-\tau } \cap \dot{H}^s)}\times \frac{\Vert \Omega \Vert ^2_{L^2_T (\dot{H}^{1-\tau } \cap \dot{H}^s)}}{\varepsilon } \\&\lesssim \mathcal {M}^3(T). \end{aligned}$$

The commutator $\mathcal {C}_{3,2}$ is bounded similarly.

Now we deal with the more delicate estimates involving the commutators $\mathcal {C}_{1,j}$ for $j \in \{1,2\}$ in the equation of $\dot{b}$. Let us start with the term with $\mathcal {C}_{1,1}=[\Lambda ^s, \mathcal {R}_2\Omega ] \partial _x b$, where it is easy to reconstruct the term $\mathcal {R}_1 \dot{b}$, which is controlled in $L^2_T$ by the energy functional $\mathcal {M}(T)$. We have, applying the above reasoning, that

$$\begin{aligned} \Vert \mathcal {C}_{1,1}\Vert _{L^2}&\lesssim \Vert \nabla \mathcal {R}_2 \Omega \Vert _{L^\infty } \Vert \partial _x b\Vert _{\dot{H}^{s-1}} + \Vert \mathcal {R}_2 \Omega \Vert _{\dot{H}^s} \Vert \partial _x b\Vert _{L^\infty } \\&\lesssim \Vert \Omega \Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \Vert \mathcal {R}_1 b\Vert _{\dot{H}^s} + \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \Vert \Omega \Vert _{\dot{H}^s}, \end{aligned}$$

so that

$$\begin{aligned} \int _0^T |(\mathcal {C}_{1,1}, \dot{b})_{L^2}| \, \textrm{d}t&\leqq \int _0^T \Vert \mathcal {C}_{1,1}\Vert _{L^2} \Vert b\Vert _{{\dot{H}}^s} \, d t \\&\lesssim \int _0^T \frac{1}{\sqrt{\varepsilon }} \Vert \Omega \Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau } \cap \dot{H}^s} \sqrt{\varepsilon }\Vert b\Vert _{{\dot{H}}^s}\, \textrm{d}t \\&\lesssim \frac{1}{\sqrt{\varepsilon }}\Vert \Omega \Vert _{L_T^2 ({\dot{H}}^{1-\tau } \cap \dot{H}^s)}\Vert b\Vert _{L^\infty _T( {\dot{H}}^s)} \sqrt{\varepsilon }\Vert \mathcal {R}_1 b\Vert _{L^2_T ({\dot{H}}^{1-\tau }\cap \dot{H}^s)}\\&\lesssim \mathcal {M}^3(T). \end{aligned}$$

The next commutator $\mathcal {C}_{1,2}=[\Lambda ^s, \mathcal {R}_1 \Omega ] \partial _y b$ requires a more careful treatment. We rely on the fractional Leibniz rule, which is an extension, due to Li [27] (see also D’Ancona [14]) of the Kenig–Ponce–Vega inequality to the case $s \geqq 1$.

We introduce the notation $\alpha =(\alpha _1, \alpha _2) \in \mathbb {N}^2 \, (\beta =(\beta _1, \beta _2) \in \mathbb {N}^2)$ and $\nabla ^\alpha =(\partial _x^{\alpha _1}, \partial _y^{\alpha _2})$, while the operator $\Lambda ^{s, \alpha }$ is defined via Fourier transform as

$$\begin{aligned} \widehat{\Lambda ^{s, \alpha } f}(\xi ) = \widehat{\Lambda ^{s, \alpha }}(\xi ) {\widehat{f}}(\xi ), \qquad \widehat{\Lambda ^{s, \alpha }}(\xi )=i^{-|\alpha |}\partial _\xi ^\alpha (|\xi |^s), \end{aligned}$$

(23)

where $|\alpha |=\alpha _1+\alpha _2$ (resp. $|\beta |=\beta _1+\beta _2$). Notice that $\Lambda ^{s,\alpha }$ is a pseudo-differential operator of order $s-|\alpha |.$ Now, we apply Lemma B.6, with $s_1=1, s_2=s-1$, which gives

$$\begin{aligned}&\left\| \mathcal {C}_{1,2} -\sum _{|\alpha |=1} \frac{1}{\alpha !} \nabla ^\alpha \mathcal {R}_1 \Omega \Lambda ^{s, \alpha } \partial _y b - \sum _{|\beta | \leqq s-2}\frac{1}{\beta !} \nabla ^\beta \partial _y b \Lambda ^{s, \beta } \mathcal {R}_1 \Omega \right\| _{L^2}\nonumber \\&\quad \lesssim \Vert \Lambda \mathcal {R}_1 \Omega \Vert _{\text {BMO}} \Vert \partial _y b\Vert _{\dot{H}^{s-1}}. \end{aligned}$$

(24)

Then we write the scalar product adding and subtracting the above right-hand side as

$$\begin{aligned} (\mathcal {C}_{1,2}, \dot{b})_{L^2}&= \left( \mathcal {C}_{1,2} -\sum _{|\alpha |=1} \frac{1}{\alpha !} \nabla ^\alpha \mathcal {R}_1 \Omega \Lambda ^{s, \alpha } \partial _y b - \sum _{|\beta | \leqq s-2}\frac{1}{\beta !} \nabla ^\beta \partial _y b \Lambda ^{s, \beta } \mathcal {R}_1 \Omega , \dot{b}\right) _{L^2} \\&\quad +\sum _{|\alpha |=1} \frac{1}{\alpha !} (\nabla ^\alpha \mathcal {R}_1 \Omega \Lambda ^{s, \alpha } \partial _y b, \dot{b})_{L^2} + \sum _{|\beta | \leqq s-2}\frac{1}{\beta !} (\nabla ^\beta \partial _y b \Lambda ^{s, \beta } \mathcal {R}_1 \Omega , \dot{b})_{L^2}, \end{aligned}$$

yielding, thanks to (24),

$$\begin{aligned} |(\mathcal {C}_{1,2}, \dot{b})_{L^2}|&\lesssim \Vert \Lambda \mathcal {R}_1 \Omega \Vert _{\text {BMO}} \Vert \partial _y b\Vert _{\dot{H}^{s-1}}\Vert b\Vert _{\dot{H}^s}+|\mathcal {J}_1|+|\mathcal {J}_2|, \end{aligned}$$

(25)

where we denote that

$$\begin{aligned} \mathcal {J}_1:&= \sum _{|\alpha |=1} \frac{1}{\alpha !}(\nabla ^\alpha \mathcal {R}_1 \Omega \Lambda ^{s, \alpha } \partial _y b, \dot{b})_{L^2} \quad \!\text{ and }\!\quad \mathcal {J}_2\\&:= \sum _{|\beta | \leqq s-2} \frac{1}{\beta !} (\nabla ^\beta \partial _y b \Lambda ^{s, \beta } \mathcal {R}_1 \Omega , \dot{b})_{L^2}. \end{aligned}$$

Now note that $\mathcal {J}_1$ can be rewritten as

$$\begin{aligned} \sum _{|\alpha |=1} \nabla ^\alpha \mathcal {R}_1 \Omega \Lambda ^{s, \alpha } \partial _y b&=- s(\nabla \mathcal {R}_1 \Omega )( \Lambda ^{s-2} \nabla \partial _y b) = - s(\nabla \mathcal {R}_1 \Omega )( \Lambda ^{s-1} \nabla \mathcal {R}_2 b), \end{aligned}$$

so that using the continuity of $\mathcal {R}_2$ in $L^2$,

$$\begin{aligned} |\mathcal {J}_1|&\lesssim \Vert \nabla \mathcal {R}_1 \Omega \Vert _{L^\infty } \Vert \Lambda ^{s} \mathcal {R}_2 b\Vert _{L^2} \Vert b\Vert _{\dot{H}^s} \lesssim \Vert \nabla \mathcal {R}_1 \Omega \Vert _{L^\infty }\Vert b\Vert _{\dot{H}^s}^2, \end{aligned}$$

(26)

and integrating in time, recalling that $u_2=-\mathcal {R}_1 \Omega$,

$$\begin{aligned} \int _0^T |\mathcal {J}_1| \, \textrm{d}t \lesssim \Vert \nabla \mathcal {R}_1 \Omega \Vert _{L^1_T (L^\infty )}\Vert b\Vert _{L^\infty _T( \dot{H}^s)}^2 \lesssim \Vert \nabla u_2 \Vert _{L^1_T( L^\infty )}\mathcal {M}_s(T) \lesssim X(T)^3. \end{aligned}$$

Now let us deal with $\mathcal {J}_2$. As $\mathcal {R}_1= \Lambda ^{-1}{\partial _x}$, we integrate by parts in the horizontal direction

$$\begin{aligned} \mathcal {J}_2=\sum _{|\beta |\leqq s-2} ((\nabla ^\beta \partial _y b) \Lambda ^{s-1,\beta } \partial _x\Omega , \dot{b})_{L^2}&=-\sum _{|\beta |\leqq s-2} ((\nabla ^\beta \partial ^2_{xy} b) \Lambda ^{s-1,\beta } \Omega , \dot{b})_{L^2}\nonumber \\&\quad -\sum _{|\beta |\leqq s-2} ((\nabla ^\beta \partial _y b) \Lambda ^{s-1,\beta }\Omega , \partial _x \dot{b})_{L^2}. \end{aligned}$$

(27)

Now, we deal with the last term in (27). Using that $\partial _x \dot{b}=\Lambda (\partial _x \Lambda ^{s-1} b)$ and the symmetry of $\Lambda$, it can be written as

$$\begin{aligned} \sum _{|\beta | \leqq s-2} ((\nabla ^\beta \partial _y b) \Lambda ^{s-1,\beta }\Omega , \partial _x \dot{b})_{L^2}&= \sum _{|\beta | \leqq s-2} (\Lambda (\nabla ^\beta \partial _y b \Lambda ^{s-1, \beta } \Omega ), \partial _x \Lambda ^{s-1} b)_{L^2}\\&=\sum _{|\beta | \leqq s-2}(\Lambda (\nabla ^\beta \partial _y b \Lambda ^{s-1, \beta } \Omega ), \mathcal {R}_1 \dot{b})_{L^2}. \end{aligned}$$

Now, we deal with the last term in (27). Using that $\partial _x \dot{b}=\Lambda (\partial _x \Lambda ^{s-1} b)$ and the symmetry of $\Lambda$, it can be written as

$$\begin{aligned} \sum _{|\beta |\leqq s-2 } ((\nabla ^\beta \partial _y b) \Lambda ^{s-1,\beta }\Omega , \partial _x \dot{b})_{L^2}&= \sum _{|\beta | \leqq s-2} (\Lambda (\nabla ^\beta \partial _y b \Lambda ^{s-1, \beta } \Omega ), \partial _x \Lambda ^{s-1} b)_{L^2}\\&=\sum _{|\beta | \leqq s-2}(\Lambda (\nabla ^\beta \partial _y b \Lambda ^{s-1, \beta } \Omega ), \mathcal {R}_1 \dot{b})_{L^2}. \end{aligned}$$

We use the following decomposition:

$$\begin{aligned}&\sum _{|\beta | \leqq s-2}(\Lambda (\nabla ^\beta \partial _y b \Lambda ^{s-1, \beta } \Omega ), \mathcal {R}_1 \dot{b})_{L^2}\\&\quad =(\Lambda ( \partial _y b \Lambda ^{s-1} \Omega ), \mathcal {R}_1 \dot{b})_{L^2} +\sum _{|\beta | =1 \, \text {and} \, \beta =(1,1)}(\Lambda (\nabla ^\beta \partial _y b \Lambda ^{s-1, \beta } \Omega ), \mathcal {R}_1 \dot{b})_{L^2} \\&\qquad + \sum _{2 \leqq |\beta |\leqq s-2 \, \text {and} \, \beta \ne (1,1)}(\Lambda (\nabla ^\beta \partial _y b \Lambda ^{s-1, \beta } \Omega ), \mathcal {R}_1 \dot{b})_{L^2}\\&\quad =\mathcal {J}_{2}^a + \mathcal {J}_{2}^b+\mathcal {J}_{2}^c. \end{aligned}$$

By virtue of the product Lemma B.2, one obtains

$$\begin{aligned} \mathcal {J}_{2}^a&\lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^s} (\Vert \Lambda \partial _y b\Vert _{L^\infty } \Vert \Lambda ^{s-1}\Omega \Vert _{L^2} + \Vert \partial _y b\Vert _{L^\infty } \Vert \Lambda ^{s}\Omega \Vert _{L^2} ) \qquad (\beta =(0,0))\\ \mathcal {J}_2^b&\lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^s} (\Vert \Lambda \nabla \partial _y b\Vert _{L^2} \Vert \Lambda ^{s-3}\nabla \Omega \Vert _{L^\infty }\\&\quad + \Vert \nabla \partial _y b\Vert _{L^\infty } \Vert \Lambda ^{s-2} \nabla \Omega \Vert _{L^2}) \qquad (|\beta |=1 \text {and} \beta =(1,1))\\ \mathcal {J}_2^c&\lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^s} \sum _{2 \leqq |\beta | \leqq s-2} (\Vert \Lambda \nabla ^\beta \partial _y b\Vert _{L^2} \Vert \Lambda ^{s-1, \beta } \Omega \Vert _{L^\infty }\\&\quad + \Vert \nabla ^\beta \partial _y b\Vert _{L^2} \Vert \Lambda ^{s, \beta } \Omega \Vert _{L^\infty }) \qquad (|\beta |\geqq 2). \end{aligned}$$

Concerning the first term of $\mathcal {J}_2$ in (27), one has

$$\begin{aligned} \sum _{|\beta |\leqq s-2} |((\nabla ^\beta \partial ^2_{xy} b) \Lambda ^{s-1,\beta } \Omega , \dot{b})_{L^2}|&\lesssim \sum _{|\beta |\leqq s-2}\Vert (\nabla ^\beta \partial ^2_{xy} b) \Lambda ^{s-1,\beta } \Omega \Vert _{L^2} \Vert b\Vert _{\dot{H}^s}. \end{aligned}$$

For $\beta =(0,0)$, we have

$$\begin{aligned} \Vert (\partial _{xy}^2 b) \Lambda ^{s-1} \Omega \Vert _{L^2}&\lesssim \Vert \partial _{xy} b\Vert _{L^\infty } \Vert \Omega \Vert _{\dot{H}^{s-1}}\lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{3-\tau } \cap \dot{H}^{3+\tau }} \Vert \Omega \Vert _{\dot{H}^{s-1}}. \end{aligned}$$

The remaining terms yield

$$\begin{aligned}&\sum _{|\beta |=1 \, \text {and} \, \beta =(1,1)} |((\nabla ^\beta \partial ^2_{xy} b) \Lambda ^{s-1,\beta } \Omega , \dot{b})_{L^2}|\\&\quad \lesssim \sum _{|\beta |=1 \, \text {and} \, \beta =(1,1)}\Vert \nabla ^\beta \partial _{xy}^2 b\Vert _{L^2} \Vert \Lambda ^{s-1, \beta } \Omega \Vert _{L^\infty }\Vert b\Vert _{\dot{H}^s}\\&\quad \lesssim \Vert \Omega \Vert _{\dot{H}^{3-\tau } \cap \dot{H}^{3+\tau }}\Vert b\Vert ^2_{\dot{H}^s} \end{aligned}$$

and

$$\begin{aligned}&\sum _{2\leqq |\beta |\leqq s-2} |((\nabla ^\beta \partial ^2_{xy} b) \Lambda ^{s-1,\beta } \Omega , \dot{b})_{L^2}|\\&\quad \lesssim \sum _{2\leqq |\beta |\leqq s-2} \Vert \nabla ^\beta \partial ^2_{xy} b\Vert _{L^2} \Vert \Lambda ^{s-1, \beta } \Omega \Vert _{L^\infty }\Vert b\Vert _{\dot{H}^s}. \end{aligned}$$

Altogether, appealing to the embeddings of Lemma 3.3 and Lemma 3.2 ($s \geqq 3+\tau$), one obtains

$$\begin{aligned} \mathcal {J}_{2}&\lesssim (\Vert \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau }}+\Vert \mathcal {R}_1 b\Vert _{\dot{H}^s}) ( \Vert b\Vert _{\dot{H}^{1-\tau }}+ \Vert b\Vert _{\dot{H}^{s}}) ( \Vert \Omega \Vert _{\dot{H}^{1-\tau }}+ \Vert \Omega \Vert _{\dot{H}^{s}}), \end{aligned}$$

so that

$$\begin{aligned} \int _0^T |\mathcal {J}_2| \, \textrm{d}t&\lesssim (\Vert \mathcal {R}_1 b\Vert _{L^2_T (\dot{H}^{1-\tau })}+\Vert \mathcal {R}_1 b\Vert _{L^2_T (\dot{H}^s)}) ( \Vert b\Vert _{L^\infty _T (\dot{H}^{1-\tau })}+ \Vert b\Vert _{L^\infty _T (\dot{H}^{s})})\\&\quad \times ( \Vert \Omega \Vert _{L^2_T (\dot{H}^{1-\tau })}+ \Vert \Omega \Vert _{L^2_T (\dot{H}^{s})}) \\&\lesssim \mathcal {M}(T)^3. \end{aligned}$$

Inserting the latter in (25) together with (26) and using the embedding $L^\infty \hookrightarrow \text {BMO}$ yields

$$\begin{aligned} \int _0^T |(\mathcal {C}_{1,2}, \dot{b})_{L^2}|\, \textrm{d}t&\lesssim (\Vert \Lambda \mathcal {R}_1 \Omega \Vert _{L^1_T (L^\infty )} + \Vert \nabla \mathcal {R}_1 \Omega \Vert _{L^1_T( L^\infty )})\mathcal {M}^2(T)+\mathcal {M}^3(T). \end{aligned}$$

To control the terms $\Vert \nabla \mathcal {R}_1\Omega \Vert _{L^1_T (L^\infty )}, \Vert \Lambda \mathcal {R}_1 \Omega \Vert _{L^1_T (L^\infty )}$, we shall rely on anisotropic Besov spaces in Step II (Section 4.2).

ii) Control of $\mathcal {M}_{1-\tau }(t)$, $0<\tau < 1$. We apply $\Lambda ^{1-\tau }$ to system (8), yielding

$$\begin{aligned} \left\{ \begin{aligned} \partial _t \dot{b} -\varepsilon \mathcal {R}_1^2 \dot{b}&= \mathcal {R}_1 \dot{z}+ \Lambda ^{1-\tau }((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot \nabla b), \\ \partial _t \dot{z} + \frac{\dot{z}}{\varepsilon }&=-\varepsilon \mathcal {R}_1^2 {\dot{\Omega }} - \varepsilon \Lambda ^{1-\tau }\mathcal {R}_1((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot \nabla b)\\&\quad +\Lambda ^{-\tau } ((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot (\nabla \Lambda \Omega )). \end{aligned} \right. \end{aligned}$$

(28)

Moreover, the equation for ${\dot{\Omega }}$ reads as

$$\begin{aligned} \partial _t {\dot{\Omega }} + \frac{{\dot{\Omega }}}{\varepsilon }&=\Lambda ^{-\tau } ((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot (\nabla \Lambda \Omega )). \end{aligned}$$

The linear terms work exactly the same way as before, thus we focus on the nonlinearities. We begin with

$$\begin{aligned} (\Lambda ^{1-\tau }((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot \nabla b), \dot{b})_{L^2}&=: \mathcal {I}_1+\mathcal {I}_2. \end{aligned}$$

Let us start with $\mathcal {I}_1$. Appealing to the product estimate (Lemma B.2) with $p_1=r_1=\infty$ yields

$$\begin{aligned} \mathcal {I}_1&=(\Lambda ^{1-\tau }(\mathcal {R}_2\Omega \partial _x b), \dot{b})_{L^2}\\&\lesssim ({\Vert \mathcal {R}_2 \Omega \Vert _{L^\infty } \Vert \partial _x b\Vert _{\dot{H}^{1-\tau }}+\Vert \Lambda ^{1-\tau }\mathcal {R}_2 \Omega \Vert _{L^2} \Vert \partial _x b\Vert _{L^\infty })\Vert b\Vert _{\dot{H}^{1-\tau }}}, \end{aligned}$$

where interpolation (Lemma B.4) and Young inequality give

$$\begin{aligned} \Vert \partial _x b\Vert _{\dot{H}^{1-\tau }} \leqq \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{2-\tau }} \lesssim \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{1-\tau }}^\theta \Vert \mathcal {R}_1 b\Vert _{\dot{H}^{s}}^{1-\theta }, \quad \theta =\frac{s+\tau -2}{s+\tau -1}. \end{aligned}$$

This yields

$$\begin{aligned} \int _0^T \mathcal {I}_1 \, \textrm{d}t&\lesssim \sqrt{\varepsilon }(\Vert \mathcal {R}_1 b\Vert _{L^2_T( \dot{H}^{1-\tau })} +\Vert \mathcal {R}_1 b\Vert _{L^2_T( \dot{H}^{s})}) \Vert b\Vert _{L^\infty _T (\dot{H}^{1-\tau })} \frac{1}{\sqrt{\varepsilon }} (\Vert \Omega \Vert _{L^2_T( \dot{H}^{1-\tau })}\\&\quad +\Vert \Omega \Vert _{L^2_T (\dot{H}^s)})\\&\lesssim \mathcal {M}^3(T). \end{aligned}$$

We cannot use the same trick for the next term, which gives

$$\begin{aligned} \int _0^T \mathcal {I}_2 \, \textrm{d}t&\lesssim \Vert b\Vert _{L^\infty _T (\dot{H}^{1-\tau })} (\Vert b\Vert _{L^\infty _T (\dot{H}^s)}+ \Vert b\Vert _{L^\infty _T (\dot{H}^{1-\tau })}) \Vert \mathcal {R}_1 \Omega \Vert _{L^1_T( W^{1,\infty })}\\ {}&\lesssim \Vert u_2 \Vert _{L^1_T( W^{1,\infty })}\mathcal {M}^2(T) \lesssim X^3(T). \end{aligned}$$

Next, it is easy to see that $\varepsilon \mathcal {R}_1 (\mathcal {R}_2\Omega , -\mathcal {R}_1 \Omega ) \cdot \nabla b$ is very similar to the terms treated before, then it is omitted.

It remains to deal with the last term in the equation of $\dot{z}$ (which, in the energy estimate, is multiplied both by $\dot{z}$ and by ${\dot{\Omega }}$, but the computations are identical, so that we only detail one case). Using the symmetry of the multiplier $\Lambda ^{-\tau }$, one obtains

$$\begin{aligned} \mathcal {I}_3:&=(\Lambda ^{-\tau } ((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot (\nabla \Lambda \Omega )), \dot{z})_{L^2} \\&= ((\mathcal {R}_2\Omega , -\mathcal {R}_1\Omega )\cdot (\nabla \Lambda \Omega ), \Lambda ^{1-2\tau } z)_{L^2}. \end{aligned}$$

Now, we integrate by parts in x the first addend and in y the second one. As the term $\mathcal {R}_2\partial _x \Omega - \mathcal {R}_1 \partial _y \Omega =0$ (by definition of the Riesz Transform), it remains

$$\begin{aligned} \mathcal {I}_3&=- (\mathcal {R}_2 \Omega \, \Lambda \Omega , \Lambda ^{1-2\tau } \partial _x z)_{L^2} +(\mathcal {R}_1 \Omega \, \Lambda \Omega , \Lambda ^{1-2\tau } \partial _y z)_{L^2}\\&= - (\mathcal {R}_2 \Omega \, \Lambda \Omega , \Lambda ^{2(1-\tau )} \mathcal {R}_1 z)_{L^2} +(\mathcal {R}_1 \Omega \, \Lambda \Omega , \Lambda ^{2(1-\tau )} \mathcal {R}_2 z)_{L^2}\\&= - (\Lambda ^{1-\tau }(\mathcal {R}_2 \Omega \, \Lambda \Omega ) , \Lambda ^{1-\tau } \mathcal {R}_1 z)_{L^2} +(\Lambda ^{1-\tau }(\mathcal {R}_1 \Omega \, \Lambda \Omega ), \Lambda ^{1-\tau } \mathcal {R}_2 z)_{L^2}\\&=:\mathcal {I}_3^a+\mathcal {I}_3^b. \end{aligned}$$

This way

$$\begin{aligned} |\mathcal {I}_3^a|&\leqq \Vert \Lambda ^{1-\tau }(\mathcal {R}_2 \Omega \, \Lambda \Omega ) \Vert _{L^2} \Vert z\Vert _{\dot{H}^{1-\tau }}, \end{aligned}$$

and using the product Lemma B.2, Lemma 3.3 and Lemma 3.2,

$$\begin{aligned} \Vert \Lambda ^{1-\tau }(\mathcal {R}_2 \Omega \, \Lambda \Omega ) \Vert _{L^2}&\lesssim \Vert \mathcal {R}_2 \Omega \Vert _{\dot{H}^{1-\tau }} \Vert \Lambda \Omega \Vert _{L^\infty }+ \Vert \mathcal {R}_2 \Omega \Vert _{L^\infty }\Vert \Lambda \Omega \Vert _{\dot{H}^{1-\tau }}\\&\lesssim (\Vert \Omega \Vert _{\dot{H}^{1-\tau }}+\Vert \Omega \Vert _{\dot{H}^{s}})^2. \end{aligned}$$

The estimate of $\mathcal {I}_3^b$ is identical. Finally,

$$\begin{aligned} \int _0^T \mathcal {I}_3 \, \textrm{d}t&\lesssim \varepsilon (\Vert \Omega \Vert _{L^\infty _T (\dot{H}^{1-\tau })}+ \Vert \Omega \Vert _{L^\infty _T( \dot{H}^{s})}) \frac{1}{\sqrt{\varepsilon }} (\Vert \Omega \Vert _{L^2_T (\dot{H}^{1-\tau })}\\&\quad +\Vert \Omega \Vert _{L^2_T (\dot{H}^{s})}) \frac{1}{\sqrt{\varepsilon }} \Vert z\Vert _{L^2_T (\dot{H}^{1-\tau })} \\&\lesssim \varepsilon \mathcal {M}^3(T). \end{aligned}$$

The proof of Proposition 4.1 is concluded. $\quad \square$

4.2 II. $L^\infty$ and Lipschitz bounds for $u_2=\mathcal {R}_1 \Omega$

The purpose of this section is to prove the following proposition:

Proposition 4.2

For $\varepsilon >0$, let (b, z) be a smooth solution of (2D-B ). One has

$$\begin{aligned} \Vert (\nabla \mathcal {R}_1\Omega ,\Lambda \mathcal {R}_1\Omega )\Vert _{L^1_T(L^\infty )}+\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )}\lesssim X(0)+ X(t)^2. \end{aligned}$$

4.3 Linear a Priori Estimates in Anisotropic Spaces

Applying ${\dot{\Delta }}_j{\dot{\Delta }}_q^h$ to the linear part of (8), we obtain

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t b_{j,q} - \varepsilon \mathcal {R}_1^2 b_{j,q} = \mathcal {R}_1z_{j,q} +h_{j,q}, \\&\partial _t z_{j,q} + \frac{z_{j,q}}{\varepsilon }=-\varepsilon \mathcal {R}_1^2\Omega _{j,q} +g_{j,q}, \end{aligned} \right. \end{aligned}$$

(29)

where $h=(\mathcal {R}_2 \Omega , - \mathcal {R}_1 \Omega ) \cdot \nabla b$ and $g=-\varepsilon \mathcal {R}_1 (\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot \nabla b + \Lambda ^{-1} ((\mathcal {R}_2 \Omega , -\mathcal {R}_2\Omega ) \cdot (\nabla \Lambda \Omega ))$. Performing standard energy estimates, one obtains

$$\begin{aligned} \frac{1}{2} \dfrac{\textrm{d}}{\textrm{d}t}\Vert b_{j,q}\Vert _{L^2}^2+\varepsilon \Vert \mathcal {R}_1 b_{j,q}\Vert _{L^2}^2\leqq \Vert \mathcal {R}_1 z_{j,q}\Vert _{L^2}\Vert b_{j,q}\Vert _{L^2}+ \Vert h_{j,q}\Vert _{L^2}\Vert b_{j,q}\Vert _{L^2}. \end{aligned}$$

(30)

Using Fourier-Plancherel theorem and the anisotropic Bernstein inequality in Lemma 3.1 yields

$$\begin{aligned} \frac{1}{2} \dfrac{\textrm{d}}{\textrm{d}t}\Vert b_{j,q}\Vert _{L^2}^2+2^{-2j}2^{2q}\varepsilon \Vert b_{j,q}\Vert _{L^2}^2\leqq \Vert \mathcal {R}_1z_{j,q}\Vert _{L^2}\Vert b_{j,q}\Vert _{L^2}+\Vert h_{j,q}\Vert _{L^2}\Vert b_{j,q}\Vert _{L^2}. \end{aligned}$$

(31)

Now, applying Lemma B.1, multiplying by $2^{js_1}2^{qs_2}$ and summing on $j,k\in \mathbb {Z}$ give, that

$$\begin{aligned} \Vert b\Vert _{L^\infty _T(B^{s_1,s_2})}+ \varepsilon \Vert b\Vert _{L^1_T(B^{s_1-2,s_2+2})} \lesssim \Vert b_0\Vert _{B^{s_1,s_2}}+\Vert z\Vert _{B^{s_1-1,s_2+1}} +\Vert h\Vert _{L^1_T(B^{s_1,s_2})} \end{aligned}$$

(32)

Following a similar procedure for z, one infers that

$$\begin{aligned} \Vert z\Vert _{L^\infty _T(B^{s_{1'},s_{2'}})}+ \frac{1}{\varepsilon } \Vert z\Vert _{L^1_T(B^{s_1',s_2'})} \lesssim \Vert z_0\Vert _{B^{s_1',s_2'}}+\varepsilon \Vert \Omega \Vert _{B^{s_1-2,s_2+2}} +\Vert g\Vert _{L^1_T(B^{s_{1'},s_{2'}})}. \end{aligned}$$

(33)

Notice that the linear term $\Vert z\Vert _{B^{s_1-1,s_2+1}}$ in (32) can be absorbed by the left-hand side of (33) if $s_1'\geqq s_1$ and $s_2'\leqq s_2$ via Lemma B.5 with $s=1$. The linear term $\Vert \Omega \Vert _{B^{s_1-2,s_2+2}}$ will be absorbed in a similar fashion once the estimates for $\Omega$ are obtained.

Since $\Omega =z+\varepsilon \mathcal {R}_1b$, this linear analysis together with Lemma 3.3 suggests us to choose that

$$\begin{aligned}&s_1=s_1'=\frac{3}{2}\quad \!\text{ and }\!\quad s_2=s_2'=\frac{1}{2}, \text { to ensure that one controls} \Vert \nabla \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )},\\&s_1=s_1'=\frac{1}{2} \quad \!\text{ and }\!\quad s_2=s_2'=\frac{1}{2},\text { to ensure that one controls } \Vert \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )}. \end{aligned}$$

Remark 4.1

The term $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )}$ is not needed to close the a priori estimates in Sobolev spaces (proof of Proposition 4.1), but it is actually crucial to deal with some nonlinear terms appearing in this anisotropic analysis. The control of $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(W^{1,\infty })}$ requires a priori estimates in two different regularity settings (as just remarked above).

With these regularity indexes, one obtains

$$\begin{aligned}&\Vert b\Vert _{L^\infty _T(B^{\frac{3}{2},\frac{1}{2}})}+ \varepsilon \Vert b\Vert _{L^1_T(B^{-\frac{1}{2},\frac{5}{2}})}+\sqrt{\varepsilon }\Vert b\Vert _{L^2_T(B^{\frac{1}{2},\frac{3}{2}})} \lesssim \Vert b_0\Vert _{B^{\frac{3}{2},\frac{1}{2}}} +\Vert h\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})}, \\&\Vert b\Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}+ \varepsilon \Vert b\Vert _{L^1_T(B^{-\frac{3}{2},\frac{5}{2}})}+\sqrt{\varepsilon }\Vert b\Vert _{L^2_T(B^{-\frac{1}{2},\frac{3}{2}})} \lesssim \Vert b_0\Vert _{B^{\frac{1}{2},\frac{1}{2}}} +\Vert h\Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}, \end{aligned}$$

and

$$\begin{aligned}&\Vert z\Vert _{L^\infty _T(B^{\frac{3}{2},\frac{1}{2}})}+ \frac{1}{\varepsilon }\Vert z\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})} +\frac{1}{\sqrt{\varepsilon }}\Vert z\Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})} \lesssim \Vert z_0\Vert _{B^{\frac{3}{2},\frac{1}{2}}} +\Vert g\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})}, \\ {}&\Vert z\Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}+ \frac{1}{\varepsilon }\Vert z\Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}+\frac{1}{\sqrt{\varepsilon }}\Vert z\Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})} \lesssim \Vert z_0\Vert _{B^{\frac{1}{2},\frac{1}{2}}} +\Vert g\Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}, \end{aligned}$$

where we used interpolation inequalities to recover the $L^2_T$ terms. Since $\Omega =z-\varepsilon \mathcal {R}_1b$, one derives the following bounds

$$\begin{aligned}&\Vert \Omega \Vert _{L^\infty _T(B^{\frac{3}{2},\frac{1}{2}})}+\Vert \Omega \Vert _{L^1_T(B^{\frac{1}{2},\frac{3}{2}})} +\Vert \Omega \Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})}\\&\quad \lesssim \Vert (\Omega _0,b_0)\Vert _{B^{\frac{3}{2},\frac{1}{2}}} +\Vert (g,h)\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})}, \\ {}&\Vert \Omega \Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}+\Vert \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{3}{2}})} +\Vert \Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\\&\quad \lesssim \Vert (\Omega _0,b_0)\Vert _{B^{\frac{1}{2},\frac{1}{2}}} +\Vert (g,h)\Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}. \end{aligned}$$

As expected, via Lemma 3.3, one has that

$$\begin{aligned} \Vert \nabla \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )} \lesssim \Vert \Omega \Vert _{L^1_T(B^{\frac{1}{2},\frac{3}{2}})}\quad \!\text{ and }\!\quad \Vert \mathcal {R}_1 \Omega \Vert _{L^1_T(L^\infty )}\lesssim \Vert \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{3}{2}})}. \end{aligned}$$

(34)

Accordingly, we define the functional

$$\begin{aligned} Y(t)&= \Vert (b,z,\Omega )\Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}}\cap B^{\frac{3}{2},\frac{1}{2}})}+\varepsilon \Vert b\Vert _{L^1_T(B^{-\frac{3}{2},\frac{5}{2}}\cap B^{-\frac{1}{2},\frac{5}{2}})}\nonumber \\&\quad +\sqrt{\varepsilon }\Vert b\Vert _{L^2_T(B^{-\frac{1}{2},\frac{3}{2}}\cap B^{\frac{1}{2},\frac{3}{2}})} \nonumber \\ {}&\quad +\frac{1}{\varepsilon }\Vert z\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})}+\frac{1}{\sqrt{\varepsilon }}\Vert z\Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})}\nonumber \\&\quad +\Vert \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{3}{2}}\cap B^{\frac{1}{2},\frac{3}{2}})}+\Vert \Omega \Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})}, \end{aligned}$$

(35)

which will be useful to close the estimates.

5 Estimates of the Nonlinearities

The term $\Vert (h,g)\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})}$ is bounded by the following lemma:

Lemma 5.1

Let $(b,\Omega )$ be a smooth solution of (2D-B ), then one has the following estimate:

$$\begin{aligned} \Vert (h,g)\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})} \lesssim Y(t)^2+Y(t)X(t)+X(t)^2 \end{aligned}$$

The proof of Lemma 5.1 is postponed to Section A and it is based on product laws in anisotropic Besov spaces (again, postponed to Section A).

Proof of Proposition 4.2

It is a direct consequence of Lemma 5.1. $\quad \square$

5.1 Conclusion of the Proof of Theorem 2.1

Gathering the estimates from Proposition 4.1 and 4.2, we obtain

$$\begin{aligned} X(t)\lesssim X(0)+X(t)^3+X(t)^2+Y(t)X(t)+Y(t)^2. \end{aligned}$$

(36)

Then, by Lemma 3.2, one has

$$\begin{aligned} Y(t)\lesssim X(t), \end{aligned}$$

which yields

$$\begin{aligned} X(t)\lesssim X(0)+X(t)^2+X(t)^3. \end{aligned}$$

(37)

From there, a standard bootstrap argument leads to the existence of global-in-time solutions of (2D-B ). Then the uniqueness follows from stability estimate below and Theorem 2.1 is proven.

5.2 Uniqueness: Stability Estimate

Let $(\rho _1,\rho _2)$ be two solutions of (IPM-diss) associated to the same initial data. We Defining that $w=\rho _1-\rho _2$, thus satisfies that

$$\begin{aligned} \partial _tw-\mathcal {R}^2_1w=(\mathcal {R}_2 \mathcal {R}_1 \rho _1 )\partial _x \rho _1 - (\mathcal {R}_1^2 \rho _1) \partial _y \rho _1-(\mathcal {R}_2 \mathcal {R}_1 \rho _2) \partial _x \rho _2 + (\mathcal {R}_1^2 \rho _2) \partial _y \rho _2.\nonumber \\ \end{aligned}$$

(38)

The right-hand side terms may be rewritten as follows:

$$\begin{aligned} \partial _tw-\mathcal {R}^2_1w=(\mathcal {R}_2 \mathcal {R}_1 w,-\mathcal {R}_1^2w)\cdot \nabla \rho _1+(\mathcal {R}_2 \mathcal {R}_1 \rho _2,-\mathcal {R}_1^2\rho _2)\cdot \nabla w. \end{aligned}$$

(39)

Applying $\Lambda ^s=(-\Delta )^\frac{s}{2}$ to (39) and using the notation $\dot{w}=\Lambda ^s w$, one infers that

$$\begin{aligned} \partial _t{\dot{w}}-\mathcal {R}^2_1{\dot{w}}=\Lambda ^{s}((\mathcal {R}_2 \mathcal {R}_1 w,-\mathcal {R}_1^2w)\cdot \nabla \rho _1)+\Lambda ^s((\mathcal {R}_2 \mathcal {R}_1 \rho _2,-\mathcal {R}_1^2\rho _2)\cdot \nabla w).\nonumber \\ \end{aligned}$$

(40)

Combining the use of commutator estimates for the nonlinear terms (as in the previous section) with the bounds on $(\rho _1,\rho _2)$ from Theorem 2.1 and the Gronwall inequality easily gives $w=0$, from which the uniqueness of smooth solutions for (IPM-diss) follows. Analogous arguments lead to the uniqueness of smooth solutions to system (2D-B ).

6 Proof of the Relaxation Limit Theorem 2.2

Recall the scaled variables

$$\begin{aligned} ({\widetilde{b}}^\varepsilon ,{\widetilde{\Omega }}^\varepsilon )(\tau ,x) \triangleq (b,\frac{\Omega }{\varepsilon })(t,x) \quad \!\text{ with }\!\quad \tau =\varepsilon t. \end{aligned}$$

The system reads as

$$\begin{aligned} \left\{ \begin{aligned} \partial _t {\widetilde{b}}^\varepsilon - \mathcal {R}_1 {\widetilde{\Omega }}^\varepsilon&= (\mathcal {R}_2 {\widetilde{\Omega }}^\varepsilon ) \partial _x {\widetilde{b}}^\varepsilon - (\mathcal {R}_1 {\widetilde{\Omega }}^\varepsilon ) \partial _y {\widetilde{b}}^\varepsilon , \\ \varepsilon ^2\partial _t {\widetilde{\Omega }}^\varepsilon - \mathcal {R}_1 {\widetilde{b}}^\varepsilon + {\widetilde{\Omega }}^\varepsilon&=\varepsilon ^2\Lambda ^{-1} [(\mathcal {R}_2 \widetilde{\Omega }^\varepsilon , - \mathcal {R}_1 \widetilde{\Omega }^\varepsilon ) \cdot (\nabla \Lambda \widetilde{\Omega }^\varepsilon )]. \end{aligned} \right. \end{aligned}$$

(41)

Let $(\widetilde{b}^\varepsilon ,\widetilde{\Omega }^\varepsilon )$ be the unique solution of (41) from Theorem 2.1. Under such scaling, that satisfies the following estimate:

$$\begin{aligned}&\Vert \widetilde{b}^\varepsilon \Vert _{L^\infty _T({\dot{H}^{1-\tau }\cap \dot{H}}^s)}+\varepsilon \Vert \widetilde{\Omega }^\varepsilon \Vert _{L^\infty _T(\dot{H}^{1-\tau }\cap {\dot{H}}^s)}+ \Vert \mathcal {R}_1 \widetilde{b}^{{\varepsilon }}\Vert _{L^2_T(\dot{H}^{1-\tau }\cap {\dot{H}}^s)}\nonumber \\&\qquad +\Vert \widetilde{\Omega }^{{\varepsilon }}\Vert _{L^2_T(\dot{H}^{1-\tau }\cap {\dot{H}}^s)} + \Vert \widetilde{\Omega }^\varepsilon -\mathcal {R}_1\widetilde{b}^\varepsilon \Vert _{L^2_T(\dot{H}^{1-\tau }\cap {\dot{H}}^s)}\leqq \widetilde{\mathcal {M}}_0. \end{aligned}$$

(42)

there $\widetilde{\mathcal {M}}_0=\Vert \widetilde{b}_0^\varepsilon \Vert _{\dot{H}^{1-\tau }\cap \dot{H}^s}+\Vert \widetilde{\Omega }_0^\varepsilon \Vert _{\dot{H}^{1-\tau }\cap {\dot{H}}^s}$. Owing to (42), $\varepsilon \widetilde{\Omega }^\varepsilon$ and $\widetilde{\Omega }^\varepsilon$ (and therefore $\mathcal {R}_1\Omega$ and $\mathcal {R}_2 \Omega$) are uniformly bounded in the spaces $L^\infty (\mathbb {R}^+;\dot{H}^{1-\tau }\cap \dot{H}^s)$ and $L^2(\mathbb {R}^+;\dot{H}^{1-\tau }\cap \dot{H}^s),$ respectively. This implies that

$$\begin{aligned} \varepsilon ^2\Lambda ^{-1} [(\mathcal {R}_2 \widetilde{\Omega }^\varepsilon , - \mathcal {R}_1 {\widetilde{\Omega }}^\varepsilon ) \cdot (\nabla \Lambda {\widetilde{\Omega }}^\varepsilon )]\rightarrow 0 \quad \text{ in } \ L^2(\mathbb {R}^+;{\dot{H}}^{s-1}). \end{aligned}$$

Therefore $\varepsilon ^2\partial _t{\widetilde{\Omega }}^\varepsilon$ goes to 0 in the sense of distributions. Putting this information into the second equation of (41), one infers that

$$\begin{aligned} {\widetilde{\Omega }}^\varepsilon -{\widetilde{\mathcal {R}}}_1b^\varepsilon \rightharpoonup 0 \ \text { in }\ \mathcal {D}'(\mathbb {R}^+\times \mathbb {R}^d). \end{aligned}$$

(43)

Concerning the other unknown, ${\widetilde{b}}^\varepsilon$ is uniformly bounded in $L^\infty (\mathbb {R}^+;\dot{H}^{1-\tau }\cap \dot{H}^{s})$ and $\partial _t \widetilde{b}^\varepsilon$ is bounded in $L^2_T \dot{H}^{s-1}$. Therefore, there exists $\rho \in L^\infty (\mathbb {R}^+;\dot{H}^{1-\tau }\cap \dot{H}^{s})$ such that, up to subsequence,

$$\begin{aligned} {\widetilde{b}}^\varepsilon \overset{*}{\rightharpoonup }\ \rho \ \text { in}\ L^\infty (\mathbb {R}^+;{\dot{H}}^{1-\tau }\cap {\dot{H}}^{s})\cap H^1(\mathbb {R}_+; {\dot{H}}^{s-1}). \end{aligned}$$

(44)

Then, as the bounds from (42) easily ensure bounds for the time-derivative of the solution $(\partial _t {\widetilde{b}}^\varepsilon ,\partial _t {\widetilde{\Omega }}^\varepsilon )$, a standard procedure involving compactness argument and Aubin-Lions lemma leads to ${\widetilde{b}}^\varepsilon \rightarrow \rho$ strongly² in $C([0,T], {\dot{H}}^{1-\tau '}_{loc}\cap {\dot{H}}^{s-s'}_{loc})$ for $0<\tau<\tau '<1$ and $0<s'<s$. For more details, we refer to Coulombel and Lin in [28] or Xu and Wang in [36] for the relaxation limit of the compressible Euler system with damping in the inhomogeneous Sobolev and Besov settings respectively. Now, defining

$$\begin{aligned} {\widetilde{Z}}^\varepsilon :={\widetilde{\Omega }}^\varepsilon - \mathcal {R}_1 {\widetilde{b}}^\varepsilon \end{aligned}$$

(45)

the first equation of (41) may be rewritten as

$$\begin{aligned} \partial _t{\widetilde{b}}^\varepsilon -\mathcal {R}_1^2{\widetilde{b}}^\varepsilon = {\widetilde{S}}^\varepsilon \quad \!\text{ with }\!\quad {\widetilde{S}}^\varepsilon = \mathcal {R}_1 {\widetilde{Z}}^\varepsilon + (\mathcal {R}_2 {\widetilde{\Omega }}^\varepsilon , - \mathcal {R}_1:={\widetilde{\Omega }}^\varepsilon ) \cdot \nabla {\widetilde{b}}^\varepsilon . \end{aligned}$$

(46)

Hence, combining (42), (43), (44) and (45), one can deduce that $\rho$ satisfies

$$\begin{aligned} \partial _t\rho -\mathcal {R}_1^2\rho =-(\mathcal {R}_2 \mathcal {R}_1 \rho , - \mathcal {R}_1^2 \rho ) \cdot \nabla \rho .\end{aligned}$$

(47)

The convergence rate of $\widetilde{Z}^\varepsilon$ stated in (19) comes directly from the control of Y(t) (35), which is ensured by Theorem 2.1.

7 Extensions

In this section, we discuss the adaptations of our proof to the case of the periodic domain ${\mathbb {T}}^2$ and slightly more general density profiles.

7.1 Asymptotic Stability on the Torus ${\mathbb {T}}^2$

from the whole space $\mathbb {R}^2$, Elgindi [16] provides asymptotic stability of (IPM-diss) on the 2D torus ${\mathbb {T}}^2$. His proof is completely different from the analogous result in the whole space. In particular, in the periodic domain ${\mathbb {T}}^2$ the density $\rho$ does not decay in $L^2({\mathbb {T}}^2)$ and proving decay requires a loss of derivatives. It is natural to inquire whether our approach is also applicable to the torus $\mathbb {T}^2$. We observe that the anisotropic Littlewood-Paley decomposition (anisotropic Besov spaces) and anisotropic Bernstein inequalities indeed extend to the torus $\mathbb {T}^2$. Our approach of exploiting the anisotropic dissipation through a suitable (anisotropic) functional space works on ${\mathbb {T}}^2$ as well. The only additional difficulty lies in the control of the horizontal average of the density . To overcome this problem, we can rely on Elgindi’s idea (see also [34]) of decomposing $\rho (x,y)={\bar{\rho }} (x,y)+ {\tilde{\rho }} (y)$ and writing the two equations

$$\begin{aligned} \partial _t {\bar{\rho }} - \mathcal {R}_1^2 {\bar{\rho }} (1- \partial _y {\tilde{\rho }})&=- \overline{{\varvec{u}} \cdot \nabla {\bar{\rho }}}, \end{aligned}$$

(48)

$$\begin{aligned} \partial _t {\tilde{\rho }} + \widetilde{\partial _y (u_2 {\bar{\rho }})}&=0 \quad \Leftrightarrow \quad \partial _t {\tilde{\rho }} + \partial _y ({\bar{\rho }}\mathcal {R}_1^2 {\tilde{\rho }})=0. \end{aligned}$$

(49)

As showed by Elgindi in the proof of [16, Proposition 4.1], the horizontal average ${\tilde{\rho }}$ is preserved by the linear evolution, i.e. neglecting the nonlinear terms yields

$$\begin{aligned} \partial _t {\tilde{\rho }}&= \partial _t \int _{-\pi }^\pi \rho (t, x, y) \, \textrm{d}x = \int _{-\pi }^\pi \mathcal {R}_1^2 \rho (1 - \partial _y {\tilde{\rho }}) \, \textrm{d}x\\&= \int _{-\pi }^\pi \partial _x (-\Delta ) s^{-1} \partial _x \rho (1- \partial _y {\tilde{\rho }}) \, \textrm{d}x =0, \end{aligned}$$

as well as any vertical derivative $\frac{\textrm{d}}{\textrm{d}t}\partial ^i_y \tilde{\rho }=0$, i.e. $\partial _y^i {\tilde{\rho }} (t)=\partial _y^i \tilde{\rho }_0$ for all $i \in \mathbb {N}$. On the other hand, the linear evolution of ${\bar{\rho }}$ displays anisotropic dissipation and decay. Towards a proof of our result on ${\mathbb {T}}^2$, taking horizontal average at the initial time $\Vert {\tilde{\rho }}_0\Vert _{\dot{H}^{1-\tau }\cap \dot{H}^s} \leqq \delta _0$ with $\delta _0$ small enough (and $s>3$) we should achieve the control of our functional X(t) in (16) (adjusted for ${\bar{\rho }}$) with very similar computations. More precisely, the Sobolev estimates for controlling $\mathcal {M}(t)$ would resemble the computations in Section 4.1, further capitalizing on the smallness of ${\tilde{\rho }}$. The primary challenge lies in the (linear part of the) Besov estimate in Section 4.2. More precisely, using the same ideas requires controlling the term $\Vert {\dot{\Delta }}_j \dot{\Delta }_q^h(\partial _y {\tilde{\rho }} \mathcal {R}_1^2 \bar{\rho })\Vert _{L^2}$. Being $\partial _y {\tilde{\rho }}$ a function of y only, it commutes with ${\dot{\Delta }}_q^h$ and therefore that term can be written as $\Vert {\dot{\Delta }}_j (\partial _y {\tilde{\rho }} \mathcal {R}_1^2 {\dot{\Delta }}_q^h{\bar{\rho }})\Vert _{L^2}$. In turn, the latter can be bounded using the usual (isotropic) product laws in Besov spaces, yielding

$$\begin{aligned} \Vert 2^{\frac{3}{2} j}2^{\frac{q}{2}}\Vert {\dot{\Delta }}_j (\partial _y \tilde{\rho }\mathcal {R}_1^2 {\dot{\Delta }}_q^h\bar{\rho })\Vert _{L^2(\mathbb {R}^2)}\Vert _{\ell ^1}&\lesssim \Vert \partial _y \tilde{\rho }\Vert _{L^\infty } \Vert \mathcal {R}_1^2 {\bar{\rho }}\Vert _{\dot{B}^{\frac{3}{2}, \frac{1}{2}}}\\&\quad +\Vert \partial _y {\tilde{\rho }}\Vert _{\dot{B}^{\frac{3}{2}, \frac{1}{2}}}\Vert \mathcal {R}_1^2 {\bar{\rho }}\Vert _{L^\infty } \\&\lesssim \Vert \partial _y {\tilde{\rho }}\Vert _{\dot{B}^{1}} \Vert \bar{\rho }\Vert _{\dot{B}^{-\frac{1}{2}, \frac{5}{2}}}+\Vert \partial _y \tilde{\rho }\Vert _{\dot{B}^{\frac{3}{2}}}\Vert {\bar{\rho }}\Vert _{\dot{B}^{-\frac{3}{2}, \frac{5}{2}}}. \end{aligned}$$

Integrating in time,

$$\begin{aligned}&\int _0^t \Vert 2^{\frac{3}{2} j}2^{\frac{q}{2}}\Vert {\dot{\Delta }}_j (\partial _y {\tilde{\rho }} \mathcal {R}_1^2 {\dot{\Delta }}_q^h\bar{\rho })\Vert _{L^2(\mathbb {R}^2)}\Vert _{\ell ^1} \, \textrm{d}t\\&\quad \lesssim \Vert \partial _y {\tilde{\rho }}\Vert _{L_T^\infty \dot{B}^{1}} \Vert {\bar{\rho }}\Vert _{L_T^1\dot{B}^{-\frac{1}{2}, \frac{5}{2}}}+\Vert \partial _y {\tilde{\rho }}\Vert _{L_T^\infty \dot{B}^{\frac{3}{2}}}\Vert {\bar{\rho }}\Vert _{L_T^1 \dot{B}^{-\frac{3}{2}, \frac{5}{2}}}, \end{aligned}$$

where $\Vert {\bar{\rho }}\Vert _{L_T^1\dot{B}^{-\frac{1}{2}, \frac{5}{2}}}$ and $\Vert {\bar{\rho }}\Vert _{L_T^1\dot{B}^{-\frac{3}{2}, \frac{5}{2}}}$ would be controlled by (the adaptation to ${\bar{\rho }}$ of) the functional Y(t) in (35). Of course some extra work would be needed to manage (and ensure the smallness of) the additional term $\partial _y {\tilde{\rho }}$ in $\dot{B}^{1} \cap \dot{B}^{\frac{3}{2}}$. In summary, we believe that our results for $\mathbb {R}^2$ can be extended to the torus $\mathbb {T}^2$ under the same regularity setting.

7.2 Nearly Linear Density Profiles

Besides the linear density profile ${\bar{\rho }}_{\text {eq}}=\rho _0-y$, we believe that our proof could be extended to the slightly more general case of nearly linear density profile $\bar{\rho }_{\text {eq}}=\rho _0+G(y)$, with $G'(y)<0$ and $\Vert 1+G'(y)\Vert _{H^r}\leqq \delta _0$ with $\delta _0$ sufficiently small and r big enough. Linearizing the IPM equation around it gives

$$\begin{aligned} \partial _t \rho +G'(y) \mathcal {R}_1^2 \rho =\text {NL} \quad \Leftrightarrow \quad \partial _t \rho - \mathcal {R}_1^2 \rho +(1+G'(y))\mathcal {R}_1^2 \rho =\text {NL}, \end{aligned}$$

where $\text {NL}$ stands for the nonlinear term. While the Sobolev estimates in Section 4.1 should work similarly exploiting the smallness of $\Vert 1+G'(y)\Vert _{H^{r}}$, the delicate point would be the (linear part of the) Besov estimate in Section 4.2. Similarly to the previous case, this point requires controlling $\Vert 2^{\frac{3}{2} j}2^{\frac{q}{2}}\Vert {\dot{\Delta }}_j ((1+G'(y))\mathcal {R}_1^2 {\dot{\Delta }}_q^h \rho )\Vert _{L^2(\mathbb {R}^2)}\Vert _{\ell ^1}$, where, once again ${\dot{\Delta }}^h_q$ and $1+G'(y)$ commute.

This leads to

$$\begin{aligned} \int _0^t \Vert 2^{\frac{3}{2} j}2^{\frac{q}{2}}\Vert {\dot{\Delta }}_j ((1+G') \mathcal {R}_1^2 {\dot{\Delta }}_q^h \rho )\Vert _{L^2(\mathbb {R}^2)}\Vert _{\ell ^1} \, \textrm{d}t&\lesssim \Vert 1+G'\Vert _{ \dot{B}^{1}} \Vert \rho \Vert _{L_T^1\dot{B}^{-\frac{1}{2}, \frac{5}{2}}}\\&\quad +\Vert 1+G'\Vert _{ \dot{B}^{\frac{3}{2}}}\Vert \rho \Vert _{L_T^1 \dot{B}^{-\frac{3}{2}, \frac{5}{2}}}, \end{aligned}$$

where $\Vert \rho \Vert _{L_T^1\dot{B}^{-\frac{1}{2}, \frac{5}{2}}}$ and $\Vert \rho \Vert _{L_T^1\dot{B}^{-\frac{3}{2}, \frac{5}{2}}}$ are controlled by the functional Y(t) in (35) and $1+G'$ is small enough in a sufficiently regular Sobolev space.

Appendices

Appendix A

Lemma A.1

(Anisotropic product laws) Let $-\frac{1}{2}\leqq s_1 \leqq \frac{5}{2}$ and $-\frac{1}{2} \leqq s_2 \leqq \frac{1}{2}$. Let $\delta _1,\delta _2,\delta _3,\delta _4\geqq 0$ and f, g be two smooth functions, one has

$$\begin{aligned} \Vert fg\Vert _{B^{s_1,s_2}}&\lesssim \Vert f\Vert _{L^\infty }\Vert g\Vert _{B^{s_1,s_2}}+\Vert f\Vert _{B^{s_1,s_2}}\Vert g\Vert _{L^\infty } \\ {}&\quad +\Vert f\Vert _{B^{s_1+\frac{1}{2}+\delta _1,-\delta _2}}\Vert g\Vert _{B^{-\delta _1,s_2+\frac{1}{2}+\delta _2}}+ \Vert f\Vert _{B^{-\delta _3,s_2+\frac{1}{2}+\delta _4}} \Vert g\Vert _{B^{s_1+\frac{1}{2}+\delta _3,-\delta _4}}. \end{aligned}$$

Moreover,

$$\begin{aligned} \Vert fg\Vert _{B^{-\frac{1}{2},\frac{1}{2}}}&\lesssim \Vert f\Vert _{L^\infty }\Vert g\Vert _{B^{-\frac{1}{2},\frac{1}{2}}}+\Vert f\Vert _{B^{\frac{1}{2},\frac{1}{2}}}\Vert \Lambda ^{-1} g\Vert _{L^\infty } \\&\quad +\Vert f\Vert _{B^{\delta _1,-\delta _2}}\Vert g\Vert _{B^{-\delta _1,1+\delta _2}}+ \Vert f\Vert _{B^{-\delta _3,1+\delta _4}} \Vert g\Vert _{B^{\delta _3,-\delta _4}}. \end{aligned}$$

1.1 A.1 Proof of Lemma A.1

Proof of Lemma A.1

Recalling the definitions of the Littlewood-Paley blocks $\dot{\Delta }_j$ and ${\dot{\Delta }}_q^h$ in Section 3.1, we introduce the isotropic and anisotropic paradifferential decomposition due to Bony in [5]. Let $f,g\in \mathcal {S}'(\mathbb {R}^d)$,

$$\begin{aligned} fg=T(f,g)+R(f,g) \quad \!\text{ and }\!\quad fg=T^h(f,g)+R^h(f,g) \end{aligned}$$

where

$$\begin{aligned} T(f,g)= \sum _{j\in \mathbb {Z}}S_{j-1}f{\dot{\Delta }}_jg \quad \!\text{ and }\!\quad R(f,g)=\sum _{j\in \mathbb {Z}}{\dot{\Delta }}_jf S_{j+2}g, \end{aligned}$$

and in the horizontal direction

$$\begin{aligned} T^h(f,g)= \sum _{q\in \mathbb {Z}}S_{q-1}f{\dot{\Delta }}_q^h g \quad \!\text{ and }\!\quad R^h(f,g)=\sum _{q\in \mathbb {Z}}{\dot{\Delta }}_q^h f S_{q+2}g. \end{aligned}$$

Applying the double paraproduct decomposition to fg, one has

$$\begin{aligned} fg=TT^h(f,g)+TR^h(f,g)+RT^h(f,g)+RR^h(f,g). \end{aligned}$$

(50)

Let us estimate each of these terms separately. For the first one, we have

$$\begin{aligned} \Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h TT^1(f,g)\Vert _{L^2}&\lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} \Vert S_{j'-1}S_{q'-1}f\dot{\Delta }_{j'}\dot{\Delta }_{q'}\partial _yg\Vert _{L^2}\\&\lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4}\Vert S_{j'-1}S_{q'-1}f\Vert _{L^\infty }\Vert \dot{\Delta }_{j'}\dot{\Delta }_{q'}\partial _yg\Vert _{L^2}\\&\lesssim c_{j,q}2^{-js_1}2^{-qs_2}\Vert f\Vert _{L^\infty }\Vert g\Vert _{B^{s_1,s_2}}. \end{aligned}$$

Concerning the fourth term, one has

$$\begin{aligned} \Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h RR^1(f,g)\Vert _{L^2}&\lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} \Vert \dot{\Delta }_{j'}\dot{\Delta }_{q'}fS_{j'+2}S_{q'+2}g\Vert _{L^2}\\ {}&\lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} \Vert \dot{\Delta }_{j'}\dot{\Delta }_{q'}f\Vert _{L^2}\Vert S_{j'+2}S_{q'+2}g\Vert _{L^\infty }\\&\lesssim c_{j,q}2^{-js_1}2^{-qs_2}\Vert f\Vert _{B^{s_1,s_2}}\Vert g\Vert _{L^\infty }. \end{aligned}$$

The following computations in the special case $s_1=-\frac{1}{2}$ and $s_2=\frac{1}{2}$ will be needed in the proof of Lemma A.1:

$$\begin{aligned} \Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h RR^1(f,g)\Vert _{L^2}&\lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} \Vert \dot{\Delta }_{j'}\dot{\Delta }_{q'}fS_{j'+2}S_{q'+2}g\Vert _{L^2}\\&\lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} \Vert \dot{\Delta }_{j'}\dot{\Delta }_{q'}f\Vert _{2}\Vert S_{j'+2}S_{q'+2}g\Vert _{L^\infty }\\&\lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4}2^{j}\Vert \dot{\Delta }_{j'}\dot{\Delta }_{q'}f\Vert _{L^2}2^{-j}\Vert S_{j'+2}S_{q'+2}g\Vert _{L^2}\\&\lesssim c_{j,q}2^{\frac{j}{2}}2^{-\frac{q}{2}}\Vert f\Vert _{B^{\frac{1}{2},\frac{1}{2}}}\Vert \Lambda ^{-1} g\Vert _{L^\infty }. \end{aligned}$$

For the third term, using the anisotropic Bernstein inequality, one has

$$\begin{aligned}&\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h TR^1(f,g)\Vert _{L^2} \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} \Vert \dot{\Delta }_{j'}S_{q'+2}fS_{j'+2}\dot{\Delta }_{q'}g\Vert _{L^2}\\&\quad \lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} \Vert \dot{\Delta }_{j'}S_{q'+2}f\Vert _{L^2_x L^\infty _y}\Vert S_{j'+2}\dot{\Delta }_{q'}g\Vert _{L^\infty _x L^2_y }\\&\quad \lesssim \sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} 2^{\frac{j'}{2}} \Vert \dot{\Delta }_{j'}S_{q'+2}f\Vert _{L^2}2^{\frac{q'}{2}}\Vert S_{j'+2}\dot{\Delta }_{q'}g\Vert _{L^2}\\&\quad \lesssim 2^{-js_1}2^{-qs_2}\sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} 2^{j'(s_1+\frac{1}{2})} \Vert \dot{\Delta }_{j'}S_{q'+2}f\Vert _{L^2}2^{q'(s_2+\frac{1}{2})}\Vert S_{j'+2}\dot{\Delta }_{q'}g\Vert _{L^2}\\&\quad \lesssim 2^{-js_1}2^{-qs_2}\sum _{|j'-j|\leqq 4, |q'-q|\leqq 4} 2^{j'(s_1+\frac{1}{2}+\delta _1)}2^{-q\delta _2} \Vert \dot{\Delta }_{j'}S_{q'+2}f\Vert _{L^2}2^{q'(s_2+\frac{1}{2}+\delta _2)} 2^{-j'\delta _1}\Vert S_{j'+2}\dot{\Delta }_{q'}g\Vert _{L^2}\\&\quad \lesssim 2^{-js_1}2^{-qs_2} c_{j,q} \left\{ \begin{aligned}&\Vert f\Vert _{B^{s_1+\frac{1}{2}+\delta _1,-\delta _2}}\Vert g\Vert _{B^{-\delta _1,s_2+\frac{1}{2}+\delta _2}} \quad&\text {if } \delta _1,\delta _2>0 \\&\Vert f\Vert _{B^{s_1+\frac{1}{2}}}\Vert g\Vert _{B^{s_2+\frac{1}{2}}_h} \quad&\text {if } \delta _1,\delta _2=0, \end{aligned} \right. \end{aligned}$$

where $B_h^s$ refers to the Besov spaces with horizontal localisation ${\dot{\Delta }}_q$. Since

$$\begin{aligned} \Vert f\Vert _{B^{s_1+\frac{1}{2}}}\Vert g\Vert _{B^{s_2+\frac{1}{2}}_h}\lesssim \Vert f\Vert _{B^{s_1+\frac{1}{2},0}}\Vert g\Vert _{B^{0,s_2+\frac{1}{2}}}, \end{aligned}$$

we obtain the desired estimate. Similar computations lead to

$$\begin{aligned} \Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h&RT^1(f,g)\Vert _{L^2} \lesssim 2^{-js_1}2^{-qs_2} c_{j,q} \Vert f\Vert _{B^{-\delta _3,s_2+\frac{1}{2}+\delta _4}} \Vert g\Vert _{B^{s_1+\frac{1}{2}+\delta _3,-\delta _4}}. \end{aligned}$$

Multiplying the above estimates by $2^{js_1}2^{qs_2}$ and summing on $j,q\in \mathbb {Z}$, the desired result follows. $\quad \square$

1.2 A.2 Proof of Lemma 5.1

Proof of Lemma 5.1

I) Estimates for $h=(\mathcal {R}_2 \Omega , - \mathcal {R}_1 \Omega ) \cdot \nabla b$. i) Applying Lemma A.1 with $f=\mathcal {R}_1 \Omega$, $g=\partial _yb$, $s_1=\frac{3}{2}$, $s_2=\frac{1}{2}$, $\delta _1=\delta _2=\frac{1}{2}$ and $\delta _3=\delta _4=0$, one has

$$\begin{aligned} \Vert \mathcal {R}_1\Omega \partial _yb\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})}&\lesssim \Vert \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )}\Vert \partial _yb\Vert _{L^\infty _T(B^{\frac{3}{2},\frac{1}{2}})}\nonumber \\&\quad +\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})}\Vert \partial _yb\Vert _{L^\infty _T(L^\infty )}\nonumber \\&\quad + \Vert \mathcal {R}_1\Omega \Vert _{L^2_T(B^{\frac{5}{2},-\frac{1}{2}})}\Vert \partial _yb\Vert _{L^2_T(B^{-\frac{1}{2},\frac{3}{2}})}\nonumber \\&\quad +\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{0,1})}\Vert \partial _yb\Vert _{L^\infty _T(B^{2,0})}. \end{aligned}$$

(51)

Now, using Lemma 3.2 and B.5 we have

• $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )}\Vert \partial _yb\Vert _{L^\infty _T(B^{\frac{3}{2},\frac{1}{2}})} \lesssim X(t)\Vert b\Vert _{L^\infty ({\dot{H}}^s\cap H^{3-\varepsilon })} \lesssim X(t)^2$,

• $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})}\Vert \partial _yb\Vert _{L^\infty _T(L^\infty )} \lesssim Y(t)\Vert b\Vert _{L^\infty (H^{2+\varepsilon })} \lesssim Y(t)X(t)$,

• $\Vert \mathcal {R}_1\Omega \Vert _{L^2_T(B^{\frac{5}{2},-\frac{1}{2}})}\Vert \partial _yb\Vert _{L^2_T(B^{-\frac{1}{2},\frac{3}{2}})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})}\Vert b\Vert _{L^2_T(B^{\frac{1}{2},\frac{3}{2}})} \lesssim Y(t)^2$,

• $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{0,1})}\Vert \partial _yb\Vert _{L^\infty _T(B^{2,0})}\lesssim \Vert \Omega \Vert _{L^1_T(B^{-1,2})}\Vert b\Vert _{L^\infty _T(B^{3,0})}\lesssim \Vert \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{3}{2}})} \Vert b\Vert _{L^\infty _T(B^{3,0})} \lesssim X(t)^2$.

Gathering the above estimates, one obtains

$$\begin{aligned} \Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h \mathcal {R}_1\Omega \partial _yb\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2})}}\lesssim X(t)^2. \end{aligned}$$

ii) Applying Lemma A.1 with $f=\mathcal {R}_1 \Omega$, $g=\partial _yb$, $s_1=\frac{1}{2}$, $s_2=\frac{1}{2}$, $\delta _1=\delta _2=\frac{1}{2}$ and $\delta _3=\delta _4=0$, we get

$$\begin{aligned} \Vert \mathcal {R}_1\Omega \partial _yb\Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}&\lesssim \Vert \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )}\Vert \partial _y b\Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}+\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \partial _yb\Vert _{L^\infty _T(L^\infty )}\\ {}&\quad +\Vert \mathcal {R}_1\Omega \Vert _{L^2_T(B^{\frac{3}{2},-\frac{1}{2}})}\Vert \partial _yb\Vert _{L^2_T(B^{-\frac{1}{2},\frac{3}{2}})}\\&\quad +\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{0,1})}\Vert \partial _yb\Vert _{L^\infty _T(B^{1,0})}. \end{aligned}$$

Using Lemma 3.2 and B.5, we have

• $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(L^\infty )}\Vert \partial _yb\Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})} \lesssim Y(t)\Vert b\Vert _{L^\infty _T({\dot{H}}^s\cap H^{2-\tau })} \lesssim X(t)^2$,

• $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \partial _yb\Vert _{L^\infty _T(L^\infty )} \lesssim Y(t)\Vert b\Vert _{L^\infty (H^{2+\tau })} \lesssim Y(t)X(t)$,

• $\Vert \mathcal {R}_1\Omega \Vert _{L^2_T(B^{\frac{3}{2},-\frac{1}{2}})}\Vert \partial _yb\Vert _{L^2_T(B^{-\frac{1}{2},\frac{3}{2}})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert b\Vert _{L^2_T(B^{\frac{1}{2},\frac{3}{2}})} \lesssim Y(t)^2$,

• $\Vert \mathcal {R}_1\Omega \Vert _{L^1_T(B^{0,1})}\Vert \partial _yb\Vert _{L^\infty _T(B^{1,0})}\lesssim \Vert \Omega \Vert _{L^1_T(B^{-1,2})}\Vert b\Vert _{L^\infty _T(B^{2,0})}\lesssim \Vert \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{3}{2}})}\Vert b\Vert _{L^\infty _T(B^{2,0})} \lesssim X(t)^2$.

iii) For the second addend of h, applying Lemma A.1 with $f=\mathcal {R}_2 \Omega$, $g=\partial _xb$, $s_1=\frac{3}{2}$, $s_2=\frac{1}{2}$, $\delta _1=\delta _2=\frac{1}{2}$ and $\delta _3=\delta _4=0$, one obtains

$$\begin{aligned} \Vert \mathcal {R}_2\Omega \partial _yb\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}})}&\lesssim \Vert \mathcal {R}_2\Omega \Vert _{L^2_T(L^\infty )}\Vert \partial _xb\Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})}\\&\quad +\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})}\Vert \partial _xb\Vert _{L^2_T(L^\infty )}\\&\quad + \Vert \mathcal {R}_2\Omega \Vert _{L^\infty _T(B^{\frac{5}{2},-\frac{1}{2}})}\Vert \partial _xb\Vert _{L^1_T(B^{-\frac{1}{2},\frac{3}{2}})}\\&\quad +\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{0,1})}\Vert \partial _xb\Vert _{L^2_T(B^{2,0})}. \end{aligned}$$

Let us deal with each r.h.s. term:

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(L^\infty )}\Vert \partial _xb\Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})}\lesssim \Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2})}}\Vert \mathcal {R}_1b\Vert _{L^2_T({\dot{H}}^s\cap H^{3-\tau )}} \lesssim Y(t)X(t),$

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})}\Vert \partial _xb\Vert _{L^2_T(L^\infty )}\lesssim Y(t)X(t),$

• $\Vert \mathcal {R}_2\Omega \Vert _{L^\infty _T(B^{\frac{5}{2},-\frac{1}{2}})}\Vert \partial _xb\Vert _{L^1_T(B^{-\frac{1}{2},\frac{3}{2}})}\lesssim \Vert \mathcal {R}_2\Omega \Vert _{L^\infty _T(B^{\frac{5}{2},-\frac{1}{2}})}\Vert b\Vert _{L^1_T(B^{-\frac{1}{2},\frac{5}{2}})}\lesssim X(t)Y(t),$

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{0,1})}\Vert \partial _xb\Vert _{L^2_T(B^{2,0})}\lesssim \Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \partial _xb\Vert _{L^2_T(B^{2,0})}\lesssim Y(t)X(t)$.

iv) Applying Lemma A.1 with $f=\mathcal {R}_2 \Omega$, $g=\partial _xb$, $s_1=\frac{1}{2}$, $s_2=\frac{1}{2}$, $\delta _1=\delta _2=0$ and $\delta _3=\delta _4=0$,

$$\begin{aligned} \Vert \mathcal {R}_2\Omega \partial _xb\Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2})}}&\lesssim \Vert \mathcal {R}_2\Omega \Vert _{L^2_T(L^\infty )}\Vert \partial _xb\Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\\&\quad +\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \partial _xb\Vert _{L^2_T(L^\infty )}\\&\quad +\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{1,0})}\Vert \partial _xb\Vert _{L^2_T(B^{0,1})}\\&\quad +\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{0,1})}\Vert \partial _xb\Vert _{L^2_T(B^{1,0})}. \end{aligned}$$

Again, we deal with each term separately:

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(L^\infty )}\Vert \partial _xb\Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert b\Vert _{L^2_T(B^{\frac{1}{2},\frac{3}{2}})}\leqq Y(t)^2$,

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \partial _xb\Vert _{L^2_T(L^\infty )} \lesssim \Vert \Omega \Vert _{L^2_T(B^{\frac{1}{2},\frac{1}{2}})}\Vert b\Vert _{L^2_T(B^{\frac{1}{2},\frac{3}{2}})}\lesssim Y(t)^2$,

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{1,0})}\Vert \partial _xb\Vert _{L^2_T(B^{0,1})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{1,0})}\Vert \mathcal {R}_1b\Vert _{L^2_T(B^{2,0})}\lesssim X(t)^2$,

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{0,1})}\Vert \partial _xb\Vert _{L^2_T(B^{1,0})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{1,0})}\Vert \mathcal {R}_1b\Vert _{L^2_T(B^{2,0})}\lesssim X(t)^2$.

Gathering the above estimates yields

$$\begin{aligned} \Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^h \mathcal {R}_1\Omega \partial _yb\Vert _{L^1_T(B^{\frac{1}{2},\frac{1}{2}})}&\lesssim X(t)^2. \end{aligned}$$

Adding the estimates from $i)-iv)$ and using that $Y(t)\lesssim X(t)$ thanks to Lemma 3.2, one obtains

$$\begin{aligned} \Vert h\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})}&\lesssim X(t)^2. \end{aligned}$$

(52)

II) Estimates for $g=\varepsilon \mathcal {R}_1 (\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot \nabla b + \Lambda ^{-1} ((\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot (\nabla \Lambda \Omega )).$

The estimates of the first term $\varepsilon \mathcal {R}_1 (\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot \nabla b$ can be obtained in the same way as the previous terms. Indeed, notice that by Lemma B.5 and thanks to the boundedness of the Riesz transform $\mathcal {R}_1: B^{1,0}\rightarrow B^{1,0}$, one has

$$\begin{aligned} \Vert \mathcal {R}_1 (\mathcal {R}_2 \Omega \partial _x b)\Vert _{L_T^1(B^{\frac{1}{2}, \frac{1}{2}})}&\lesssim \Vert \mathcal {R}_1 (\mathcal {R}_2 \Omega \partial _x b)\Vert _{L_T^1(B^{1, 0})}\lesssim \Vert \mathcal {R}_2 \Omega \partial _x b\Vert _{L_T^1(B^{1, 0})}, \end{aligned}$$

which is exactly the same term as the one we treated in $\varvec{I})$. A similar argument can be applied for the bound in $B^{\frac{3}{2},\frac{1}{2}}$. We then turn to the second addend of g.

i) First, observing that

$$\begin{aligned} \Vert \Lambda ^{-1} ((\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot (\nabla \Lambda \Omega ))\Vert _{B^{\frac{3}{2},\frac{1}{2}}}\leqq \Vert (\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot (\nabla \Lambda \Omega )\Vert _{B^{\frac{1}{2},\frac{1}{2}}} \end{aligned}$$

and since $\Omega$ has better decay properties than b and we control $\Omega$ in $L^2_T(H^s)$ for $s> 3$, we can directly deduce from our previous computations that

$$\begin{aligned} \Vert \Lambda ^{-1} ((\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot (\nabla \Lambda \Omega ))\Vert _{B^{\frac{3}{2},\frac{1}{2}}}\leqq X(t)^2. \end{aligned}$$

ii) For the second regularity setting it is a bit trickier as one has

$$\begin{aligned} \Vert \Lambda ^{-1} ((\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot (\nabla \Lambda \Omega ))\Vert _{B^{\frac{1}{2},\frac{1}{2}}}&\lesssim \Vert (\mathcal {R}_2 \Omega , -\mathcal {R}_1\Omega ) \cdot (\nabla \Lambda \Omega )\Vert _{B^{-\frac{1}{2},\frac{1}{2}}}. \end{aligned}$$

Applying the second inequality of Lemma A.1 with $f=\mathcal {R}_2 \Omega$, $g=\partial _x\Lambda \Omega$, $s_1=-\frac{1}{2}$, $s_2=\frac{1}{2}$, $\delta _1=1$, $\delta _2=0$ and $\delta _3=\delta _4=0$, we obtain

$$\begin{aligned} \Vert \mathcal {R}_2\Omega \partial _x\Lambda \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{1}{2}})}&\lesssim \Vert \mathcal {R}_2\Omega \Vert _{L^\infty _T(L^\infty )}\Vert \partial _x\Lambda \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{1}{2}})}\\&\quad +\Vert \mathcal {R}_2\Omega \Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \Lambda ^{-1} \partial _x\Lambda \Omega \Vert _{L^1_T(L^\infty )}\\&\quad + \Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{1,0})}\Vert \partial _x\Lambda \Omega \Vert _{L^2_T(B^{-1,1})}\\&\quad +\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{0,1})}\Vert \partial _x\Lambda \Omega \Vert _{L^2_T(B^{0,0})}. \end{aligned}$$

$\square$

Remark A.1

Alternatively, one could think to exploit the first inequality of Lemma A.1, which would require a control of $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{-\frac{1}{2},\frac{1}{2}})}\Vert \partial _x\Lambda \Omega \Vert _{L^2_T(L^\infty )}$. However, such bounds hold under additional low-regularity assumptions on the initial data, for instance $B^{0,0}$, that we want to avoid here.

We estimate the above terms as follows.

• $\Vert \mathcal {R}_2\Omega \Vert _{L^\infty _T(L^\infty )}\Vert \partial _x\Lambda \Omega \Vert _{L^1_T(B^{-\frac{1}{2},\frac{1}{2}})}\lesssim \Vert \Omega \Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \Omega \Vert _{L^1_T(B^{\frac{1}{2},\frac{3}{2}})}\lesssim Y(t)^2$,

• $\Vert \mathcal {R}_2\Omega \Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \Lambda ^{-1} \partial _x\Lambda \Omega \Vert _{L^1_T(L^\infty )} \lesssim \Vert \Omega \Vert _{L^\infty _T(B^{\frac{1}{2},\frac{1}{2}})}\Vert \Omega \Vert _{L^1_T(B^{\frac{1}{2},\frac{3}{2}})}\lesssim X(t)Y(t)$,

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{1,0})}\Vert \partial _x\Lambda \Omega \Vert _{L^2_T(B^{-1,1})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{1,0})}\Vert \Omega \Vert _{L^2_T(B^{0,2})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{1,0})}\Vert \Omega \Vert _{L^2_T(B^{2,0})}\lesssim X(t)^2$,

• $\Vert \mathcal {R}_2\Omega \Vert _{L^2_T(B^{0,1})}\Vert \partial _x\Lambda \Omega \Vert _{L^2_T(B^{0,0})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{1,0})}\Vert \Omega \Vert _{L^2_T(B^{1,1})}\lesssim \Vert \Omega \Vert _{L^2_T(B^{1,0})}\Vert \Omega \Vert _{L^2_T(B^{\frac{3}{2},\frac{1}{2}})}\lesssim X(t)Y(t)$.

The estimates for the last term $\mathcal {R}_1\Omega \partial _y\Lambda \Omega$ follow the exact same lines, then we omit them. We have therefore

$$\begin{aligned} \Vert g\Vert _{L^1_T(B^{\frac{3}{2},\frac{1}{2}}\cap B^{\frac{1}{2},\frac{1}{2}})}&\lesssim X(t)^2. \end{aligned}$$

(53)

Adding (52) and (53) concludes the proof of Lemma 5.1. $\quad \square$

Appendix B. Toolbox

We collect some technical lemmas that are used in the course of the proof the results of this article. In some case, we provide (short) proofs, while in other cases, appealing to the existing literature to which we refer explicitly, we omit the proofs. When no explicit reference is provided being the results classical, the reader can look for instance at [20].

Lemma B.1

Let $X: [0,T]\rightarrow \mathbb {R}_+$ be a continuous function such that $X^2$ is differentiable. Assume that there exists a constant $B\geqq 0$ and a measurable function $A: [0,T]\rightarrow \mathbb {R}_+$ such that

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}X^2+BX^2\leqq AX\quad \hbox {a.e. on }\ [0,T]. \end{aligned}$$

Then, for all $t\in [0,T],$ we have

$$\begin{aligned} X(t)+B\int _0^tX\leqq X_0+\int _0^tA. \end{aligned}$$

Lemma B.2

(Product estimates, [35, Lemma 2.1]) Let $s>0, 1 \leqq p,r \leqq \infty$, then

$$\begin{aligned} \Vert \Lambda ^s(fg)\Vert _{L^p(\mathbb {R}^2)} \lesssim \Vert f\Vert _{L^{p_1}(\mathbb {R}^2)}\Vert \Lambda ^{s}g\Vert _{L^{p_2}(\mathbb {R}^2)}+\Vert g\Vert _{L^{r_1}(\mathbb {R}^2)}\Vert \Lambda ^s f\Vert _{L^{r_2}(\mathbb {R}^2)}, \end{aligned}$$

where $1 \leqq p_1, r_1 \leqq \infty$ such that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{r_1}+\frac{1}{r_2}$.

Lemma B.3

(Commutator estimates, [35, Lemma 2.1]) Let $s>0$, $1 \leqq p_1, r_1 \leqq \infty$ and $1<p, p_1, r_1 <1$ such that $\frac{1}{p}=\frac{1}{p_1} + \frac{1}{p_2} = \frac{1}{r_1} + \frac{1}{r_2}$. Then

$$\begin{aligned} \Vert [\Lambda ^s, f]g\Vert _{L^p} \lesssim (\Vert \nabla f\Vert _{L^{p_1}}\Vert \Lambda ^{s-1}g\Vert _{L^{p_2}}+\Vert \Lambda ^s f\Vert _{L^{r_1}}\Vert g\Vert _{L^{r_2}}). \end{aligned}$$

Lemma B.4

(Interpolation) Let $s_0 \leqq s \leqq s_1$. Then, for $\theta \in (0,1)$, such that $s=\theta s_0+(1-\theta )s_1$, it holds

$$\begin{aligned} \Vert f\Vert _{\dot{H}^s} \lesssim \Vert f\Vert _{\dot{H}^{s_0}}^\theta \Vert f\Vert _{\dot{H}^{s_1}}^{1-\theta }. \end{aligned}$$

Lemma B.5

(Embedding in Besov spaces) For $s>0$, $s_1,s_2\in \mathbb {R}$, one has

$$\begin{aligned} B^{s_1+s,s_2-s} \subset B^{s_1,s_2} \end{aligned}$$

(54)

Proof

Let $f\in B^{s_1+s,s_2-s}\cap B^{s_1,s_2}$. By definition of the localisation ${\dot{\Delta }}_j$ and ${\dot{\Delta }}_q^h$, when applying ${\dot{\Delta }}_j{\dot{\Delta }}_q^h$ to a function, one can use that there exists $N_0$ such that $j\geqq q- N_0$. This implies that

$$\begin{aligned} \sum _{j,q\in \mathbb {Z}} 2^{js_1}2^{qs_2}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^hf\Vert _{L^2}&\lesssim \sum _{j,q\in \mathbb {Z}} 2^{js_1}2^{js}2^{-js}2^{qs_2}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^hf\Vert _{L^2}\\&\lesssim \sum _{j,q\in \mathbb {Z},j\geqq q-N_0} 2^{js_1}2^{js}2^{-qs}2^{qs_2}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^hf\Vert _{L^2}\\&\lesssim \sum _{j,q\in \mathbb {Z}} 2^{j(s_1+s)}2^{q(s_2-s)}\Vert {\dot{\Delta }}_j{\dot{\Delta }}_q^hf\Vert _{L^2}. \end{aligned}$$

$\square$

The next lemma provides a generalized version of the Kenig–Ponce–Vega inequality (the fractional version of the Leibniz rule) for all $s>0$, see [27] and [14].

Recall the notation $\alpha , \beta \in \mathbb {N}^2$ (multi-index) and $\nabla ^\alpha =(\partial _x^{\alpha _1}, \partial _y^{\alpha _2})$, while the operator $\Lambda ^{s, \alpha }$ is defined via Fourier transform as

$$\begin{aligned} \widehat{\Lambda ^{s, \alpha } f}(\xi ) = \widehat{\Lambda ^{s, \alpha }}(\xi ) {\widehat{f}}(\xi ), \qquad \widehat{\Lambda ^{s, \alpha }}(\xi )=i^{-|\alpha |}\partial _\xi ^\alpha (|\xi |^s). \end{aligned}$$

Lemma B.6

(Generalized Kenig–Ponce–Vega inequality [27, Theorem 5.1]) Let $s>0$ and $1<p<\infty$. Then, for any $s_1, s_2 \geqq 0$ such that $s_1+s_2=s$, and any $f, g \in \mathcal {S}(\mathbb {R}^d)$,

$$\begin{aligned}&\left\| \Lambda ^s (fg)- \sum _{|\alpha | \leqq s_1} \frac{1}{\alpha !} (\nabla ^\alpha f) (\Lambda ^{s,\alpha }g) - \sum _{|\beta | \leqq s_2-1} \frac{1}{\beta !} (\nabla ^\beta g) (\Lambda ^{s,\beta }f) \right\| _{L^p}\\&\quad \lesssim \Vert \Lambda ^{s_1} f\Vert _{\text {BMO}} \Vert \Lambda ^{s_2} g\Vert _{L^p}. \end{aligned}$$