Abstract
A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance, we capture the Navier-Stokes equations of
satisfying the no-slip boundary condition, as an intermediary approximation of the Euler equations through a new Hilbert-type expansion of the Boltzmann equation with the diffuse reflection boundary condition. Aiming to justify the approximation we establish a novel quantitative \(L^p\)-\(L^\infty \) estimate of the Boltzmann perturbation around a local Maxwellian of such viscous approximation, along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces; we also establish direct estimates of the Navier-Stokes equations in higher regularity with the aid of the initial-boundary and boundary layer weights using a recent Green’s function approach. The incompressible Euler limit follows as a byproduct of our framework.
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1 Introduction
An important and active research direction in mathematical physics/PDE is on the so-called Hilbert’s sixth problem [29] seeking a unified theory of the gas dynamics including different levels of descriptions from a mathematical standpoint by connecting the behavior of solutions to equations from kinetic theory to solutions of other systems that arise in formal limits, such as the N-body problem, the Euler equations, the Navier-Stokes equations, etc. In particular, the hydrodynamic limit of the Boltzmann equation has received a great deal of attention and enthusiasm in the mathematics and physics communities since the pioneering work [30] by Hilbert, which was the first example of his sixth problem. Remarkably, all the basic fluid equations of compressible, incompressible, inviscid, or viscous fluid dynamics can be derived from the Boltzmann equation of rarefied gas dynamics upon the choice of appropriate scalings in a small mean free path limit. Though formal derivations are rather well-understood, as far as mathematical justifications go, despite great progress over the decades (for example see [1,2,3, 13, 19, 23, 50] and the references therein), full understanding of the hydrodynamic limit incorporating important physical applications such as boundary effects or physical phenomena is still far from being complete. The goal of this paper is to make a rigorous connection between the Boltzmann equation and the incompressible Euler equations in the presence of the boundary by bypassing the inviscid limit of the incompressible Navier-Stokes equations.
The dimensionless Boltzmann equation with the Strouhal number 
and the Knudsen number 
takes the form of
Here the distribution function of the gas is denoted by \(F(t,x,v) \ge 0\) with the time variable \(t \in {\mathbb {R}}_+: = \{ t \ge 0\}\), the space variable \(x = (x_1,x_2,x_3)\in \Omega \subset {\mathbb {R}}^3\), and the velocity variable \(v = (v_1,v_2,v_3) \in {\mathbb {R}}^3\). The Boltzmann collision operator \(Q(\cdot , \cdot )\) of the hard sphere takes the form of
where \(v^\prime := v- ((v-v_*) \cdot {\mathfrak {u}} ) {\mathfrak {u}} \) and \(v_*^\prime := v_*+ ((v-v_*) \cdot {\mathfrak {u}} ) {\mathfrak {u}} \). This operator satisfies the so-called collision invariance property: for any \(F(v)\text { and } G(v) \) decaying sufficiently fast as \(|v|\rightarrow 0\),
which represents the local conservation laws of mass, momentum and energy. The celebrated Boltzmann’s H-theorem reveals the entropy dissipation:
for any \(F(v)>0\) decaying sufficiently fast as \(|v|\rightarrow \infty \). An intrinsic equilibrium, satisfying \(Q(\cdot , \cdot )=0\), is given by a local Maxwellian associated with the density \(R>0\), the macroscopic velocity \(U\in {\mathbb {R}}^3\) and the temperature \(T>0\)
which is known as the only configuration attaining the equality in (1.4).
In addition to the Strouhal number and Knudsen number we introduce the Mach number 
. By passing 
and 
to zero, one can formally derive PDEs of hydrodynamic variables for the fluctuations around the reference state (1, 0, 1), which are determined as
The famous Reynolds number appears as a ratio between the Knudsen number and Mach number through the von Karman relation:
For instance, the incompressible Navier-Stokes equations with 
, namely the viscosity of order one, can be derived by setting 
as \( \varepsilon \downarrow 0\). In this paper we are particularly interested in a scale of large Reynolds number as follows:
through which we will derive the incompressible Euler equations with the no-penetration boundary condition in the limit
with \(\partial _t \theta + u _E \cdot \nabla _x \theta =0\) and \( \nabla _x \theta (t,x) +\nabla _x \rho (t,x)= 0\). Here \(n=n(x)\) denotes an outward normal at x on the boundary \(\partial \Omega \).
For the sake of simplicity we set an initial datum \(\theta _{0}(x)= 0=\rho _{0}(x)\) so that
In many important physical applications such as a turbulence theory, it would be relevant to take into account the physical boundary in the hydrodynamic limit. A boundary condition of the Boltzmann equation is determined by the interaction law of the gas with the boundary surface. One of the physical conditions is the so-called diffuse reflection boundary condition, which takes into account an instantaneous thermal equilibration of reflecting gas particle (see [9, 12]): for \( (x,v) \in \{ \partial \Omega \times {\mathbb {R}}^3: n(x) \cdot v<0\},\)
where we have taken an isothermal boundary with a rescaled temperature 1 for the sake of simplicity. Here, the normalization constant \(c_\mu := 1/\big (\int _{n(x) \cdot {\mathfrak {v}}>0} M_{1,0,1} ({\mathfrak {v}}) (n(x) \cdot {\mathfrak {v}}) \mathrm {d}{\mathfrak {v}}\big )\) leads to the null flux condition \( \int _{{\mathbb {R}}^3} F(t,x,v) (n(x) \cdot v) \mathrm {d}v=0 \text { on }x \in \partial \Omega \). In particular, it is well-known that the diffuse boundary condition (1.12) is a kinetic boundary condition featuring a mismatch with the no-penetration boundary condition (1.10) of the the Euler flow under (1.8), without any small parameter with respect to 
or 
. One can readily see this by expanding F around a local Maxwellian \(M_{1,\varepsilon u_E, 1}(v)\) associated with a flow of the no-penetration boundary condition (1.10) directly. Unfortunately, this local Maxwellian does not honor the diffuse reflection boundary condition when a flow satisfies the no-penetration boundary condition (1.10). In fact a size of the boundary mismatch could be an order of the tangential component of the Euler flow \(u_E\) at the boundary. Therefore a uniform bound to verify the limit (1.6) in a scale of large Reynolds number (1.8) is not expected even at the formal level. This poses a major obstacle in the Euler limit from the Boltzmann equation with the diffuse reflection boundary. It is worth noting that such a mismatch does not appear at least at the formal level when the specular reflection boundary condition is imposed: \(F(t,x,v)= F(t,x,R_x v)\) on \(x \in \partial \Omega \) where \(R_x v= v- 2n(x) (n(x) \cdot v)\); while the mismatch can possess a small factor for the so-called Maxwell boundary condition, a convex combination of the specular reflection and the diffuse reflection boundary conditions, by choosing the coefficient for diffuse reflection known as the accommodation constant to vanish as 
.
Remarkably, an analogous, better-known boundary mismatch phenomenon exists in the realm of mathematical fluid dynamics, specifically in the inviscid limit problem of the Navier-Stokes equations that addresses the validity of the Euler solutions as the leading order approximation of the Navier-Stokes solutions in the vanishing viscosity limit. The inviscid limit for the no-slip boundary condition features a boundary mismatch between two different boundary conditions for the Navier-Stokes and Euler flows. In fact, whether the solution to the Navier-Stokes equations with a \( \kappa \eta _0 \)-viscosity (a physical constant \(\eta _0\) can be computed explicitly from the Boltzmann theory as in (1.37)) satisfying the no-slip boundary condition
converges to the solution of the Euler equations satisfying the no-penetration boundary condition (1.9)-(1.10) in 
is an outstanding problem, which is arguably the most relevant and challenging because of the mismatch of two boundary conditions between (1.15) and (1.10) resulting in the formation of boundary layers such as Prandtl layer and unbounded vorticity near the boundary. While the verification of the inviscid limit is still largely open, it holds under certain symmetry assumption on the domain and data or under the flat boundary and strong regularity such as analyticity at least near the boundary [45]. A classical way to tackle the inviscid limit problem is to study the Prandtl expansion [44, 48, 49]: \(u(t,x_1,x_2,x_3)= u_E(t,x_1,x_2,x_3) + u_P (t,x_1,x_2,\frac{x_3}{\sqrt{\kappa }})+ O(\sqrt{\kappa })\). Recently, different frameworks that avoid the boundary layer expansion have become available [38, 47, 54].
The incompressible Euler limit from the Boltzmann equation turns out to be intimately tied to the inviscid limit of the incompressible Navier-Stokes equations, which accounts for the similarity of two boundary mismatches. A beautiful connection stems from the Navier-Stokes solutions of (1.13)-(1.15) in large Reynolds numbers: at least formally, not only they are approximated by the Euler equations (1.9)-(1.10) but also they approximate the Boltzmann equation (1.1) under (1.8) with (1.12), in fact better than the Euler equations (1.9)-(1.10) at each Mach number \(\varepsilon >0\), because the Navier-Stokes equations contain a high order correction term \(\kappa \eta _0 \Delta u\) that captures the dissipative nature of the Boltzmann collision operator (as we will see in Section 1.1). And importantly, a local Maxwellian \(M_{1,\varepsilon u, 1}(v)\) associated with u satisfying the no-slip boundary condition (1.15), satisfies the diffuse reflection boundary condition (1.12) without singular terms. In other words, the Navier-Stokes solutions are compatible with the diffuse reflection boundary condition. Therefore, under the scale (1.8) the Navier-Stokes solution of (1.13)-(1.15) stands in between the Boltzmann solution of (1.1), (1.12) and the Euler solution (1.9)-(1.10).
In this paper, inspired by these observations, we propose to study the Euler limit from the Boltzmann equation through the Navier-Stokes solutions that hold both features of the Euler and the Boltzmann under (1.8) at each Mach number \(\varepsilon >0\). To this end, we expand the Boltzmann solution F around a local Maxwellian associated with a Navier-Stokes flow u to (1.13)-(1.15):
as
and analyze (1.17) via a new Hilbert expansion presented in Section 1.1. Although the notations \(F^\varepsilon \) and \(f^\varepsilon \) may be more precise for the equation depending on \(\varepsilon \), we will abuse the notations by dropping the superscript \(\varepsilon \) for the sake of simplicity. The next order correction \(f_2\) can be entirely determined by the Navier-Stokes flow and it turns out that its contribution is always smaller than \(f_R\)’s one in our choice of \(\varepsilon \) and \(\kappa \). A choice of the range of the Mach number with respect to the Reynolds number: 
in \(\varepsilon \downarrow 0\) plays an important role in our analysis and the formal expansion. We will discuss the relation and its role in Section 1.2. With such a choice of the scale, uniform-in-\(\varepsilon \) estimates of the Boltzmann remainder \(f_R\) are achieved by a novel quantitative \(L^p\)-\(L^\infty \) estimate in a setting of the local Maxwellian of the Navier-Stokes approximation (1.16), along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces.
In order to establish the Euler limit by using the Navier-Stokes solutions of (1.13)-(1.15) as a reference state as \( \varepsilon \downarrow 0\) in a scale of large Reynolds number (1.8), it is imperative to show the uniform-in-\(\kappa \) convergence of the Navier-Stokes solutions to the Euler solutions of (1.9)-(1.10), where the inviscid limit comes into play. In this paper, we take the spatial domain to be the upper-half space with periodic boundary conditions in the horizontal components and analytic data for the Navier-Stokes solutions of (1.13)-(1.15) and obtain uniform-in-\(\kappa \) estimates built upon a recent development on the inviscid limit problem in the half-space based on the Green’s function approach using the boundary vorticity formulation [38, 44, 47, 54].
Our main result concerns a rigorous justification of the passage from the solutions to the dimensionless Boltzmann equation (1.1) of the scale (1.8) with the diffuse reflection boundary condition (1.12) to the solution of the incompressible Euler equation (1.9) with the no-penetration boundary condition (1.10), without introducing any boundary expansion of the Boltzmann equation:
Theorem 1
(Informal statement) We consider a half space in 3D
For some choice of \(\varepsilon \) and \(\kappa (\varepsilon )\), there exists a large set of initial data \(u_{in}\), \(f_{2,in}\) and \(f_{R,in}\) such that a unique solution F(t, x, v) of the form (1.17) to (1.1) and (1.12) with (1.8) exists on [0, T] for some \(T>0\) and satisfies
and
while u and \(u_E\) denote solutions of the Navier-Stokes (1.13)-(1.15) and Euler equations (1.9)-(1.10), respectively.
The precise statement of Theorem 1 is given in Theorem 4 and Corollary 5 in Section 2.3.
Remark 1
To the best of our knowledge our result of this paper appears to be the first rigorous incompressible Euler limit result from the Boltzmann solutions with the sole diffuse reflection (therefore the accommodation constant \(\sim \)1) in the boundary condition! Moreover, our framework captures the inviscid limit of mathematical fluid dynamics from the Boltzmann theory.
Remark 2
Another natural choice of the scale in the study of the Euler limit might be \(\varepsilon ^{{\mathfrak {q}}}=\kappa \) with an integer \({\mathfrak {q}}\ge 1\). Then the second correction \(\frac{1}{\kappa }Lf_2\) is shifted to the next hierarchy (see (1.27)) and as a consequence the Euler equations become the leading approximation with loss of \(\kappa \eta _ 0 \Delta u\). Without the boundary, a higher order expansion \(F=\mu _E+ [\varepsilon f_1 + \varepsilon ^2 f_2 + \varepsilon ^3 f_3 + \cdots + \varepsilon ^r f_R] \sqrt{\mu _E}\) for \(\mu _E=M_{1 , \varepsilon u_E , 1 }\) has been established in [10, 56]. In the presence of the boundary, on the other hand, such an expansion features a boundary mismatch. The usual approach is then drawn on a boundary layer expansion, correcting an interior Hilbert-like expansion at the boundary to satisfy the boundary conditions (for example, see [27, 55]). Our approach is based on an interior expansion up to the second correction \(f_2\) that avoids the boundary layer expansion under our choice of scale \(\varepsilon \ll \kappa \) (see (2.11)).
Before discussing the essence of the methodology and novelty of our result, we shall briefly overview some relevant literatures on the hydrodynamic limit of the Boltzmann equation. One of the first mathematical studies of the limits at the formal level may go back to a work [30] of Hilbert, in which he introduced so-called the Hilbert expansion. Based on the truncated Hilbert expansion rigorous justifications of fluid limits have been shown as long as the solutions of corresponding fluids are bounded in some suitable spaces, for example, in the compressible fluid limits in [7, 53], incompressible fluid limits in [5, 10, 23], diffusive limits from the Vlasov-Maxwell-Boltzmann system in [32], and relativistic fluid limits in [52]. All the derivations mentioned above did not take into account the boundary, while one of the main obstacles to study the Boltzmann solutions with the boundary is its boundary singularity (see [20, 21, 35]). In [22], an \(L^p\)-\(L^\infty \) framework has been developed to construct a unique global solution of the Boltzmann equation with physical boundary conditions. Such a framework has been developed successfully in various problems of the Boltzmann theory (for example [8, 14, 24,25,26,27, 36, 37, 55]). In particular, in [12, 13], the authors have constructed a solution of the Boltzmann equation satisfying the diffuse reflection boundary condition and proved the validity of the hydrodynamic limit toward the incompressible Navier-Stokes-Fourier system in both steady and unsteady settings, based on a novel \(L^6\)-bound of the hydrodynamic part.
Rigorous passage from the renormalized solutions of [11] ([46] with the physical boundary) of the Boltzmann equation toward (weak) solutions of fluid equations has been also extensively explored (see [17, 33, 50] for the references in this direction). In particular, the program of the incompressible Navier-Stokes limit to the Leray-Hopf weak solutions has been developed successfully in [2, 3, 19, 40, 41] without the physical boundary and with the boundary in [33, 42]. As for the incompressible Euler limit in terms of the entropy production, based on the relative entropy method, a dissipative solution of the incompressible Euler equations in [39] has been studied in [40, 41, 51] without the boundary. Notably the results have been extended to the domain with the boundary for the specular reflection boundary condition in [50], and for the Maxwell boundary condition in [4], assuming to set that the accommodation constant (a factor of diffuse reflection) vanishes as \(\varepsilon \downarrow 0\).
For the rest of this section, we present the strategy and key ideas developed in the proof of our result starting with a new (formal) Hilbert expansion followed by the control of the Boltzmann remainder \(f_R\) and higher regularity of Navier-Stokes flows, for the rigorous justification of the formal expansion.
1.1 Hilbert Expansion in a Scale of Large Reynolds Number
Through a new formal Hilbert-type expansion of Boltzmann equation with the diffuse reflection boundary condition we aim to capture the Navier-Stokes equations of vanishing viscosity proportional to 
and satisfying the no-slip boundary condition.
It is worth pointing out that although more convenient choice of an expansion of F is seemingly the one around the global Maxwellian \(\mu _0:= M_{1,0,1}\) such as \(F=\mu _0 + \varepsilon (u \cdot v) \mu _0 + \varepsilon ^2 {\tilde{f}}_2 \sqrt{\mu _0} + \varepsilon \delta {\tilde{f}}_R \sqrt{\mu _0}\), unfortunately this choice will produce, in the Hilbert expansion (1.26)-(1.30), an unbounded term \(\frac{2}{\kappa \varepsilon } \frac{1}{\sqrt{\mu _0 }}Q(u \cdot v \mu _0 ,f_R \sqrt{\mu _0 })\) even compared to the strongest control in hand, namely a dissipation term (see (1.31))! To achieve a sharper estimate, which provides weaker restriction on \(\kappa \) and \(\varepsilon \), and hence weaker restriction on the initial data, we work on an expansion around the local Maxwellian \(\mu \).
It is conceptually convenient in our analysis to introduce an auxiliary parameter \(\delta =\delta (\varepsilon )\downarrow 0\) as \(\varepsilon \downarrow 0\), which indicates a size of the fluctuation \((F-\mu )/ {\varepsilon }\):
In (1.17) we have chosen \(\delta =\sqrt{\varepsilon }\) and in Section 2.3 we will have the same choice such as (2.11), however in Section 2.1, Section 3 and Section 4, \(\delta \) will be regarded as a free parameter and will be chosen at the last step of closing our argument (as (2.11)!).
Interior Expansion. We investigate an expansion (1.19) of the Boltzmann equation (1.1) at the local Maxwellian \(\mu \) in (1.16). Let
The operators L and \(\Gamma \) can be read as
where the precise form of \({\mathbf {k}}\) is delayed to be presented in (3.19). We will demonstrate basic properties of operators L and \(\Gamma \) in Section 3.1. From (1.3) the null space of L, denoted by \({\mathcal {N}}\), is a subspace of \(L^2({\mathbb {R}}^3)\) spanned by orthonormal bases \(\{\varphi _i \sqrt{\mu }\}_{i=0}^4\) with
We define a hydrodynamic projection \({\mathbf {P}}\) as an \(L_v^2\)-projection on \({\mathcal {N}}\) such as
where \(\langle \cdot , \cdot \rangle \) stands for an \(L^2_v\)-inner product. It is well-known that the operators enjoy \({\mathbf {P}}L=L {\mathbf {P}}={\mathbf {P}} \Gamma =0\). Importantly the linear operator L enjoys a coercivity away from the kernel \({\mathcal {N}}\): for \(\nu (v)\ge 0\) defined in (1.21)
Now we plug the expansion (1.19) into the rescaled equation (1.1) with the scale (1.8). It turns out that by relating \(f_2\) with the flow and locating it carefully in the hierarchy we can exhibit the dissipative nature of the Boltzmann collision operator at the leading order of the fluid approximation. In particular we locate \((v-\varepsilon u) \cdot \nabla _x ({\mathbf {I}} - {\mathbf {P}})f_2\) in \(\frac{1}{\delta }\)-order hierarchy to capture \(\kappa \)-order viscosity in the fluid equation (1.13):
We can readily see an \(L^2\)-energy structure of \(f_R\) with a strong dissipation
which inherits its lower bound from the coercivity (1.25).
Let us first consider an \(\frac{1}{\varepsilon \delta }\)-hierarchy (1.27). For any non-vanishing term of (1.27) would cause unpleasant unboundedness, we make the term vanish entirely by solving an equation (1.27)\(=0\). By the Fredholm alternative, an inverse map
is well-defined and hence the solvability condition is given by
This condition indeed implies the incompressible condition (1.14).
Once (1.14) holds, we have \(\sum _{\ell , m=1}^3 \partial _\ell u_m\varphi _\ell \varphi _m \sqrt{\mu }= \sum _{\ell , m=1}^3 \partial _\ell u_m (\varphi _\ell \varphi _m - \frac{|v-\varepsilon u|^2}{3} \delta _{\ell m} ) \sqrt{\mu }\). Now we solve (1.27)\(=0\) by setting
Then we move to an \(\frac{1}{\delta }\)-hierarchy (1.28). The hydrodynamic part of (1.28), unless it vanishes, would induce an unbounded term again. We expand \(\delta \times \) (1.28), using (1.16) and (1.34), as
The leading order term of the last term in (1.35) contributes the following to the hydrodynamic part of (1.35) as
where we have used the fact \(A_{\ell m } \in {\mathcal {N}}^\perp \) and \(\frac{|v-\varepsilon u|^2}{3} \sqrt{\mu } \in {\mathcal {N}}\) at the first step and the definition of \(A_{ik}\) at the last step. It is well-known (e.g. Lemma 4.4 in [3]) that for some constant \(\eta _0 >0\)
Therefore we deduce that (1.36) vanishes for \(i=0,4,\) and the \(\kappa \eta _0\)-viscosity term in (1.13) can be captured:
Here we have used the incompressible condition (1.14) at the last step. On the other hand, a leading order term of the hydrodynamic part of \((v- \varepsilon u) \cdot \nabla _x {\mathbf {P}} f_2\) contributes to the pressure term of (1.13) by choosing a special form of \({\mathbf {P}}f_2\) as in (3.1). Therefore the whole leading order terms of the hydrodynamic part in (1.28) do vanish by solving the Navier-Stokes equations (1.13) and (1.14)! For the sake of brevity we refer to Section 3 for the full expansion of (1.26)-(1.30).
Boundary Conditions. Now we consider a boundary condition of \(f_R\). Noticeably the local Maxwellian \(\mu \) becomes \(M_{1,0,1}\) on the boundary from the no-slip boundary condition (1.15), and hence \(\mu \) satisfies the diffuse reflection boundary condition (1.12). For the detailed study of the boundary condition of \(f_R\) we introduce the incoming and outgoing boundaries
Since \(\mu \) satisfies the diffuse reflection boundary condition (1.12) with a constant wall temperature \( =1\), by plugging (1.19) into the boundary condition, we arrive at
Letting \(P_{\gamma _+}\) be an \(L^2 (\{ v : n(x) \cdot v>0 \})\)-projection of \(\sqrt{c_\mu \mu }\), we derive that
Note that \(\int _{n(x) \cdot v>0} c_\mu \mu (v) (n(x) \cdot v) \mathrm {d}v=1\).
On the other hand, we emphasize that, with the no-penetrate boundary condition of (1.10), the associated local Maxwellian \(M_{1,\varepsilon u_E,1}\) does not satisfy the diffuse reflection boundary condition in general. Therefore the Boltzmann remainder \(f_R\) would have a singularity of an order of \(1/\sqrt{\varepsilon }\) in (1.17).
1.2 Uniform Controls of the Boltzmann Remainder \(f_R\)
For a rigorous justification of the Hilbert expansion (1.19), the major task is to establish uniform-in-\(\varepsilon \) estimates of the Boltzmann remainder \(f_R\) in \(L^2\). The equation of the Boltzmann remainder \(f_R\) in (1.26)-(1.30) with the boundary condition (1.39) features a discrepancy between the behavior of the hydrodynamic part \({\mathbf {P}}f_R\) and pure kinetic part \(({\mathbf {I}}-{\mathbf {P}})f_R\): schematically an \(L^2\)-energy estimate reads
A key difficulty arises from a growth of the hydrodynamic part at least as \( e^{\Vert \nabla _x u\Vert _{L^\infty } } \) which might behave as an exponential of the reciprocal of some power of the viscosity \(\kappa \) due to the unbounded vorticity formed near the boundary, while such strong singularity of the hydrodynamic part enters the nonlinear estimate in turn. In fact such trilinear estimate can be effectively handled only by a point-wise bound of the solutions. Unfortunately as the physical boundary conditions create singularities in general ([35]), the high Sobolev estimates would not be possible. In this paper we develop a quantitative \(L^p\)-\(L^\infty \) estimate solely in the setting of the local Maxwellian associated with the Navier-Stokes flow, in the presence of the diffuse reflection boundary.
Thanks to a strong control of the dissipation from the spectral gap of (1.25), the nonlinear term can be bounded as
Notably an integrability gain of the hydrodynamic part \(Pf_R\) should play a role; however a classical velocity average lemma \(Pf_R \in H^{1/2}_x\subset L^3_x\) fails to fulfill the need in 3D. We achieve such a higher integrability by developing a recent \(L^6\)-bound of hydrodynamic part of [13] in the setting of the local Maxwellian on the scale of large Reynolds number. We utilize the micro-macro decomposition and the equation to control \(\kappa ^{1/2}v\cdot \nabla _x {\mathbf {P}} f_R\) mainly by \(\frac{1}{\varepsilon \kappa ^{1/2}} L({\mathbf {I}} - {\mathbf {P}} )f_R\) and \( \varepsilon \kappa ^{1/2}\partial _t f_R\). We invert the operator \(v\cdot \nabla _x {\mathbf {P}}\), employing a recent test function method of [12] in the local Maxwellian setting, to establish a crucial \(L^6\)-bound of the hydrodynamic part, which is controlled by the dissipation plus the a priori \(L^2\)-bound of \(\partial _t f_R\):
In other words we can achieve the \(L^6\)-estimate of (1.41) as “one spatial derivative gain” through the dissipation provided a temporal derivative being controlled, while the temporal derivative preserves the boundary conditions. It is a critical point in which a temporal derivative gets involved in our analysis of Boltzmann and fluids as well!
New difficulties arise as commutator estimates of \(\frac{1}{\varepsilon ^2 \kappa } \{\partial _t Lf_R- L \partial _t f_R\}\) induce singularities even at the linear level, as well as \(\partial _t(\partial _t + \varepsilon ^{-1} v\cdot \nabla _x) \sqrt{\mu } f_R/\sqrt{\mu }\) and the source terms in the equation of \(\partial _t f_R\) possess higher temporal derivatives of the fluid with an initial layer. In fact after a careful analysis we realize such singular terms amount to
while \({\mathfrak {P}}\) depends on the singularity of derivatives of the Navier-Stokes flow in large Reynolds numbers.
We establish a unified \(L^\infty \)-estimate in the local Maxwellian setting, devising a special weight function \({\mathfrak {w}}_{\varrho , \ss }(x,v)\) in order to control an extra growth in |v| from \((\partial _t + \varepsilon ^{-1} v\cdot \nabla _x) \sqrt{\mu } f_R/\sqrt{\mu }\) and its temporal derivative. We control \(f_R\) in \(L^\infty _t L^\infty _x\) by the hydrodynamic part \(Pf_R\) in \(L^6\) and the dissipation, studying the particle-trajectory bouncing against the diffuse reflection boundary and geometric change of variables related to the bouncing trajectories. The temporal derivative \(\partial _t f_R\) needs some special attention since the source term of the equation of \(\partial _t f_R\) possesses \(\nabla _x \partial _t^2 u\), which turns out to have an initial-boundary layer. For that we measure \(\partial _t f_R\) using a different time-space norm, namely a weighted \(L^2_tL^\infty _x\), and control it by the hydrodynamic part of \(\partial _t f_R\) in \(L^2_tL^3_x\) (with more singular factor-in-\(\varepsilon \) than the counterpart for \(f_R\)) and the dissipation. Although our estimate of \(\partial _tf_R\) is singular than \(f_R\) due to our choice of different spaces, we are able to balance such extra singularity by the strong dissipation and careful trilinear estimates.
We establish \(L^2_tL^3_x-\)controls for \(Pf_R\) and \(P\partial _t f_R\) via the trajectory rather than the classical average lemma. In fact a direct application of such average lemma has some subtle issue since the source terms of \(f_R\) and \(\partial _t f_R\) equations are known to be bounded within a finite time interval only, while the \(L^2_tL^3_x\)-control enters the nonlinear estimates. In fact it is not clear whether our iteration of estimates would guarantee a nonempty finite time interval of validity. Instead we utilize the Duhamel formula along the trajectories and an extension of solutions in specially designed domains, and employ the \(TT^*\)-method developed in [14, 28, 31]. As a result we achieve \(L^2_tL^P_x\) estimates for \(f_R\) and \(\partial _t f_R\) uniformly for all \(p<3\), which gives us a sufficient bound in \(L^2_tL^3_x\) by interpolating with our \(L^\infty \)-estimates.
Finally upon combining all the estimates above together we are able to bound an energy by the Gronwall’s inequality. The resulting bound is not uniform but growing exponentially as \(e^{1/\kappa ^{{\mathfrak {P}}}}\), in which the power depends on the higher regularity of the fluid. Luckily we are able to find a range of \(\varepsilon \) with respect to \(\kappa \) in a scale of large Reynolds number to absorb the Gronwall growth, and achieve a uniform bound of the Boltzmann remainder, which ensures the rigorous justification of the Hilbert expansion in Section 1.1. The main theorem of the uniform controls of the Boltzmann remainder \(f_R\) is given in Theorem 2.
1.3 Higher Regularity of Navier-Stokes Equations in the Inviscid Limit
The inviscid limit of the Navier-Stokes equations (1.13)-(1.15) is at the heart of our approach. Furthermore, in order to control \(f_R\), as explained in the above, we need to derive quantitative higher regularity estimates of the Navier-Stokes solutions which are not directly available in the usual inviscid limit results. Before discussing new features of our analysis, we briefly discuss some prior works on the inviscid limit most relevant to our result. Due to the formation of boundary layers in the limit caused by the mismatch of boundary conditions (1.15) and (1.10), a classical way to tackle the inviscid limit problem is via the Prandtl expansion, of which rigorous justification was shown in [48, 49] for well-prepared data with analytic regularity and in [44] for the initial datum with Sobolev regularity when the initial vorticity is bounded away from the boundary. In particular, the author of [44] introduced the boundary vorticity formulation of (1.13)-(1.15) (see (2.16)-(2.18)) which prompted subsequent interesting works in the field. Among others, in a recent work [47], the authors proved the inviscid limit in 2D based on the Green’s function approach based on Maekawa’s vorticity formulation without having to construct Prandtl boundary layer corrections but by utilizing the boundary layer weights in the norm. In [38, 54], the inviscid limit was shown for initial data that is analytic only near the boundary and has finite Sobolev regularity in the complement in 2D and 3D respectively.
Our analysis of the Navier-Stokes solutions in the limit is based on the Green’s function approach for the Stokes problem using the vorticity formulation (2.16)-(2.18) in the same spirit of [47]. However, the existing methods [38, 47, 54] do not immediately fulfill the goal of our hydrodynamic limit because the analysis of our remainder \(f_R\) requires higher regularity of Navier-Stokes solutions, more specifically \(L^2\) and \(L^\infty \) bounds for higher order derivatives up to two temporal derivatives of \(\nabla _x u\) and p and two spatial derivatives of \(\partial _tu\), while the existing methods do not decipher any bounds for temporal derivatives and the boundedness of the conormal derivatives in their analytic norms does not rule out \(\frac{1}{x_3}\) singularity of the normal derivative of the vorticity in the boundary layer, which may cause the loss of \(L^2\) integrability. To get around these issues, we pursue new estimates of temporal derivatives of the vorticity \(\omega \) by demanding the compatibility conditions for the initial data. With such conditions, the initial layer is absent for \(\omega \) and \(\partial _t\omega \); we can derive an analogous integral representation formula for \(\partial _t \omega \) so that we may run the same fixed point argument for \(\partial _t\omega \) as in [47] without the initial layer. For the second temporal derivative, we handle the initial-boundary layer for the horizontal part with the initial-boundary weight function. These new features allow us to attain the derivative estimates of the vorticity in the normal direction without \(\frac{1}{x_3}\) singularity near the boundary at the expense of losing a power of \(\sqrt{\kappa }\), which is crucial for the control of \(f_R\). The velocity and pressure estimates are then recovered by utilizing elliptic regularity results and the Biot-Savart law in the analytic setting. The main results of Navier-Stokes solutions to (1.13)-(1.15) are given in Theorem 3.
2 Main Results
For the sake of the readers we present the precise statement of main theorems and their notations in this section. We first present the uniform controls of the Boltzmann remainder \(f_R\) of Theorem 2, and the higher regularity of the Navier-Stokes equations in the inviscid limit of Theorem 3. As a consequence of those two theorems we will show a rigorous justification of kinetic approximation of Navier-Stokes in high Reynolds numbers of Theorem 4. Then using the vorticity estimates in Theorem 3 and the famous Kato’s condition in the inviscid limit, we prove a hydrodynamic limit toward the incompressible Euler equations in Corollary 5.
2.1 Uniform Controls of the Boltzmann Remainder \(f_R\) (Theorem 2)
We recall the expansion of Boltzmann solution \(F = \mu + \varepsilon ^2 f_2 \sqrt{\mu } + \delta \varepsilon f_R\sqrt{\mu } \) in (1.19) around the local Maxwellian \(\mu (v) : = M_{1 , \varepsilon u , 1 }(v)\) for any given flow (u, p) solving the incompressible Navier-Stokes equation with the no-slip boundary condition (1.13)-(1.15).
Inspired by the energy structure of the PDE and the coercivity of the linear operator L in (1.25), we define an energy and a dissipation as
As explained in Section 1.2, the temporal derivative gets involved mainly in order to access the \(L^6\)-bound of the hydrodynamic part \({\mathbf {P}}f_R\), while we will control the following auxiliary norm to be used in order to handle the nonlinearity: for \(p<3\) and \(t>0\)
Here we have introduced weight functions, in order to control an extra quadratic growth in |v| from \((\partial _t + \varepsilon ^{-1} v\cdot \nabla _x) \sqrt{\mu } f_R/\sqrt{\mu }\)
where \({\mathfrak {z}}_{\ss }: {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+\) is defined as, for \(\ss >0\)
We have denoted \({\mathfrak {w}}_{\varrho ', \ss }(x,v) ={\mathfrak {w}}'\) for \(\varrho '<\varrho \). Also we have denoted the boundary norms and integral as
Next we discuss the initial data of the Boltzmann equation. We note that an initial datum of \(f_2\) is already determined by given flow (u, p). For given initial data \(f_{R,0}:=f_{R,in}\), inspired by the PDE, we define
where \(({\mathbf {I}}- {\mathbf {P}}){\mathfrak {R}}_1\) and \({\mathfrak {R}}_2\) are defined in (3.2) with \(\delta =\sqrt{\varepsilon }\) and \(\mu _{in}\), \(L_{in}\), \(\Gamma _{in}\) are induced by the initial Naiver-Stokes velocity \(u_{in}\). For the remainder \(f_R\) in (1.17), we will use the norms of the initial data:
Theorem 2
(Uniform controls of the Boltzmann remainder \(f_R\)) Suppose for \(T>0\) and \({\mathfrak {P}}\ge 1/2\)
We further assume that, for \(0 \le {\mathfrak {P}}^\prime < {\mathfrak {P}}\),
For given such \(T>0\), let us choose \(\varepsilon , \delta \) and \(\kappa \) as, for some \({\mathfrak {C}}\gg 1\),
Assume that an initial datum for the remainder \(f_{R,in }\) satisfies, for some \(p<3\) and \(|p-3|\ll 1\),
Then we construct a unique solution \(f_R(t,x,v)\) of the form of
which solves the Boltzmann equation (1.1) and the diffuse reflection boundary condition (1.12) with the scale of (1.8) and (2.11), and satisfies the initial condition \(F|_{t=0}= M_{1, \varepsilon u _{in} ,1 } + \varepsilon ^2 f_{2} \sqrt{M_{1, \varepsilon u ,1 } } |_{t=0} + \delta \varepsilon f_{R,in} \sqrt{M_{1, \varepsilon u ,1 } }|_{t=0}\), in a time interval \(t \in [0,T]\). Moreover, we have
Remark 3
The condition (2.11) in the theorem is indeed the largest \(\sqrt{\varepsilon }\) can be allowed. Any smaller \(\sqrt{\varepsilon }\) than \(\exp \Big ( \frac{ - {\mathfrak {C}} T}{ \kappa ^{1/2} }\Big )\) (which means \(\sqrt{\varepsilon }\) decaying faster than \(\exp \Big ( \frac{ - {\mathfrak {C}} T}{ \kappa ^{1/2} }\Big )\) as \(\kappa \downarrow 0\)) will produce the same result. In terms of (1.8) the relation (2.11) implies that the Knudsen number 
has to vanish only slightly faster than the Mach number 
:
The proof of Theorem 2 will be given in Section 4.
2.2 Higher Regularity of Navier-Stokes Equations in the Inviscid Limit (Theorem 3)
For the Navier-Stokes solutions to (1.13)-(1.15), we introduce real analytic norms and function spaces, adopted from [47] and [54] for the 3D counter part with slight modifications.
In this subsection and Section 5, we will use the following notations: \(x=(x_h,x_3)=(x_1,x_2,x_3)\in {\mathbb {T}}^2\times {\mathbb {R}}_+=\Omega \), \(\nabla _x=\nabla =(\nabla _h,\partial _3)=(\partial _{x_1},\partial _{x_2},\partial _{x_3})\); for a vector valued function \(g\in {\mathbb {R}}^3\), \(g=(g_h,g_3)=(g_1,g_2,g_3)\).
We denote the vorticity by
while the second identity is the famous Biot-Savart law. Here \((- \Delta )^{-1}\) denotes the inverse of \(-\Delta \) with the zero Dirichlet boundary condition on \(\partial \Omega \).
Our analysis of the Navier-Stokes solutions is based on the vorticity formulation in 3D ([43, 44]):
where \(\sqrt{- \Delta _h}=|\nabla _h|\) is defined as
Here, \(g_{\xi } (x_3)= \frac{1}{(2\pi )^2} \iint _{{\mathbb {T}}^2} e^{-i x_h \cdot \xi } g(x_h,x_3) \mathrm {d}x_h \in {\mathbb {C}} \text { with } \xi = (\xi _1, \xi _2) \in {\mathbb {Z}}^2\) denotes the Fourier transform in the horizontal variables, which satisfies \(g(x_1,x_2,x_3) = \sum _{\xi \in {\mathbb {Z}}^2} g_{\xi } (x_3) e^{i x_h \cdot \xi }.\) The Fourier transform can be regarded as a function \(g_{\xi } (z)\) where z is sitting in a pencil-like complex domain: for any \(\lambda > 0\),
We define analytic function spaces without the boundary layer, \({\mathfrak {L}}^{p,\lambda }\), for holomorphic functions with a finite norm, for \(p\ge 1\),
Next we introduce an \(L^\infty \)-based analytic boundary layer function space, for \(\lambda >0\) and \(\kappa \ge 0\), that consists of holomorphic functions in \({\mathcal {H}}_\lambda \) with a finite norm
where \(\Vert g_{\xi } \Vert _{{\mathcal {L}}^\infty _{\lambda ,0}} : = \Vert e^{{\bar{\alpha }} \text {Re}\,z} g_{\xi } (z) \Vert _{{\mathcal {L}}^\infty _\lambda } := \sup _{z \in {\mathcal {H}}_\lambda }e^{{\bar{\alpha }} \text {Re}\,z} g_{\xi } (z) \) and
Here, a boundary layer weight function is defined as
We define \({\mathfrak {B}}^{\lambda , \kappa }\) for holomorphic functions \(g= (g_1,g_2,g_3)\) with a finite norm
We note that \({\mathfrak {B}}^{\lambda , \kappa } \subset {\mathfrak {L}}^{1,\lambda }\), but \({\mathfrak {B}}^{\lambda , 0} \subsetneqq {\mathfrak {L}}^{\infty ,\lambda }\) if \({\bar{\alpha }}>0\).
Due to its singular nature of the Navier-Stokes flow in the inviscid limit, we introduce the conormal derivatives
With the multi-indices \(\beta =(\beta _h,\beta _3):=(\beta _1,\beta _2,\beta _3)\in {\mathbb {N}}_0^3\), the higher derivatives are denoted by \(D^\beta = \partial _1^{\beta _1} \partial _2 ^{\beta _2} D_3^{\beta _3}\) and \(D^\beta _\xi = (i \xi _1)^{\beta _1} (i \xi _2) ^{\beta _2} D_3^{\beta _3}\).
Now we define, for \(\lambda _0>0\), \(\gamma _0>0\), \(\alpha >0\), \(\kappa \ge 0\), and \(t \in (0, \frac{\lambda _0}{2 \gamma _0})\)
With an initial-boundary layer weight function as in [47]
we define an initial-boundary layer function space \({\mathfrak {B}}^{\lambda , \kappa t}\) for holomorphic functions \(g= (g_1,g_2,g_3)\) with a finite norm
where an \(L^\infty \)-based analytic norm with the initial-boundary layer is defined as
We finally define, for \(t \in (0, \frac{\lambda _0}{2\gamma _0})\),
In this subsection and Section 5, \(\alpha \), \({\bar{\alpha }}\) are given positive small constants, \(\lambda _0\) is a given positive constant, and \(\gamma _0\) is a sufficiently large constant to be determined in Theorem 3.
Next we discuss the initial data of the velocity \(u_{in}\) and the corresponding vorticity \(\omega _{in}= \nabla _x \times u_{in}\). Inspired by the PDEs, let
Theorem 3
Let \(\lambda _0>0\) and \(\omega _{in} \in {\mathfrak {B}}^{\lambda _0,\kappa }\) with (2.32) satisfy
Further assume that \(\omega _{in}=\omega _0\) and (2.32) satisfies the compatibility conditions on \(\partial \Omega \)
Then there exists a constant \(\gamma _0>0\) and a time \(T>0\) depending only on \(\lambda _0\) and the size of the initial data such that the solution \(\omega (t)\) to the vorticity formulation of the Navier-Stokes equations (2.16)–(2.18) exists in \(C^1([0,T]; {\mathfrak {B}}^{\lambda , \kappa })\) with \(\partial _t^2\omega \) in \(C(0,T; {\mathfrak {B}}^{\lambda , \kappa t})\) for \(0<\lambda <\lambda _0\) satisfying
Furthermore, for each \((t,x)\in [0,T]\times \Omega \),
-
(1)
(Bounds on the vorticity and its derivatives) \(\omega (t,x)\) enjoys the following bounds:
$$\begin{aligned} | \nabla _{h}^i \partial _t^\ell \omega _h (t,x) |&\lesssim e^{-{\bar{\alpha }} x_3} \left( 1 + \phi _\kappa (x_3) \right) , \ \ | \nabla _{h}^i \partial _t^\ell \omega _3 (t,x) | \lesssim e^{-{\bar{\alpha }} x_3} \text { for } i,\ell =0,1, \end{aligned}$$(2.36)$$\begin{aligned} | \partial _t^2 \omega _h (t,x) |&\lesssim e^{-{\bar{\alpha }} x_3} \left( 1 + \phi _\kappa (x_3) + \phi _{\kappa t} (x_3) \right) , \ \ | \partial _t^2 \omega _3 (t,x) | \lesssim e^{-{\bar{\alpha }} x_3}, \end{aligned}$$(2.37)$$\begin{aligned} | \partial _{x_3} \partial _t^\ell \omega _h (t,x) |&\lesssim {\kappa }^{-1} e^{-{\bar{\alpha }} x_3}, \ \ | \partial _{x_3} \partial _t^\ell \omega _3 (t,x) | \lesssim e^{-{\bar{\alpha }} x_3} \left( 1 + \phi _\kappa (x_3) \right) \text { for } \ell =0,1. \end{aligned}$$(2.38) -
(2)
(Bounds on the velocity and its derivatives) The corresponding velocity field u(t, x) satisfies the following:
$$\begin{aligned} |\partial _t^\ell u (t,x)|&\lesssim 1 \ \ \text {for} \ \ell =0,1,2, \end{aligned}$$(2.39)$$\begin{aligned} \sum _{1 \le |\beta | \le 2} |\nabla ^{\beta } \partial _t^\ell u(t,x )|&\lesssim \big (1+ \phi _\kappa (x_3) + (|\beta |-1) {\kappa }^{-1} \big ) e^{-\min (1, \frac{{\bar{\alpha }}}{2} )x_3} \ \ \text {for} \ \ell =0,1, \end{aligned}$$(2.40)$$\begin{aligned} \sum _{ |\beta | =1} |\nabla ^{\beta } \partial _t^2 u(t,x )|&\lesssim \big (1+ \phi _\kappa (x_3) + \phi _{\kappa t} (x_3)\big )e^{-\min (1, \frac{{\bar{\alpha }}}{2} )x_3} . \end{aligned}$$(2.41)Moreover, we have the decay estimate for \(\partial _t^\ell u\):
$$\begin{aligned} |\partial _t^\ell u|\lesssim \kappa ^{-\frac{1}{2}} e^{-\min (1,\frac{{{\bar{\alpha }}}}{2}) x_3} \ \ \text {for} \ \ell =1,2. \end{aligned}$$(2.42) -
(3)
(Bounds on the pressure and its derivatives) The pressure defined in (5.73) satisfies the following:
$$\begin{aligned} |\partial _t^\ell p (t,x)|&\lesssim 1 \ \ \text {for} \ \ell =0,1,2, \end{aligned}$$(2.43)$$\begin{aligned} \sum _{0 \le |\beta | \le 1} |\nabla ^{\beta } \partial _t^\ell p(t,x )|&\lesssim \kappa ^{-\frac{1}{2}} e^{-\min (1, \frac{{\bar{\alpha }}}{2} )x_3} \ \ \text {for} \ \ell =0,1, \end{aligned}$$(2.44)$$\begin{aligned} |\partial _t^2 p|&\lesssim ( \kappa ^{-\frac{1}{2}} +\phi _{\kappa t}(x_3) )e^{-\min (1, \frac{{\bar{\alpha }}}{2} )x_3} . \end{aligned}$$(2.45)
Remark 4
For simplicity of the presentation, we have taken the analytic data with the same analyticity radius in \(x_1\), \(x_2\) and \(x_3\) with the exponential decay for large \(x_3\). As shown in [38, 54], more general initial data requiring the analyticity only near the boundary can be taken.
Remark 5
The horizontal vorticity \(\omega _h\) and the vertical vorticity \(\omega _3\) obey different boundary conditions (2.18) which enforce different behaviors near the boundary. This is well-reflected in our \(L^\infty \) based norms in (2.24) and (2.29). As noted in [54], such incompatible behaviors of \(\omega _h\) and \(\omega _3\) in 3D are dealt with the \(L^1\) based norm (2.27) which contains one more tangential derivative \((1+|\nabla _h|)\), which is different from 2D analysis [38, 47].
Remark 6
We demand the compatibility conditions in (2.34) in order to avoid singular initial-boundary layers for the temporal derivatives of the vorticity. If the first two conditions in (2.34) were not satisfied, the initial-boundary layers would occur for the first temporal derivative of the vorticity. For the second temporal derivative, we handle the initial-boundary layer for the horizontal part with the initial-boundary layer weight, while for the vertical part we further demand \(\partial _t\omega _{0,3}|_{x_3=0}=0\) in order to rule out a singular initial-boundary layer caused by the Dirichlet boundary condition. This amounts to requiring the second order vanishing condition at the boundary for \(\omega _{0,3}\), which is satisfied by a large class of \(\omega _0\). We remark that the first condition of (2.34) is also satisfied by a large class of \(\omega _0\). In fact, if not, by the result of [47], we can obtain a short time solution \({\tilde{\omega }}(t)\) to (2.16)–(2.18) and may reset the initial data by \(\omega _0 = {\tilde{\omega }}(t=t_0)\) for sufficiently small \(t_0>0\).
The proof of Theorem 3 will be given in Section 5.
2.3 Main Theorem
Now we present the full statement of the main theorems of this paper:
Theorem 4
(Kinetic approximation of Navier-Stokes in large Reynolds numbers) We consider a half space \(\Omega \) in 3D as in (1.18). Suppose an initial datum of the Navier-Stokes flow \(u_{in}\) is divergence-free \(\nabla _x \cdot u_{in}=0\) in \(\Omega \) and the corresponding initial vorticity \(\omega _{in}= \nabla _x \times u_{in}\) belongs to the real analytic space \({\mathfrak {B}}^{\lambda _0,\kappa }\) of (2.24) for some \(\lambda _0>0\) such that (2.33) holds. Further we assume that \(\omega _{in}\) satisfies the compatibility conditions (2.34) on \(\partial \Omega \). Then there exists a unique real analytic solution \((u(t,x), \nabla _x p(t,x))\) to (1.13)–(1.15) in \([0,T] \times \Omega \), while \(T>0\) only depends on \(\lambda _0\) and the size of the initial data as in (2.33).
Choosing a pressure p(t, x) such that \(p(t,x)\rightarrow 0\) as \(x_3 \uparrow \infty \), we set the local Maxwellian and the second order correction \(f_2\) as
For given such \(T>0\), let us choose \(\varepsilon \) and \(\kappa \) in the relation of (2.11).
Assume that an initial datum for the remainder \(f_{R,in }\) satisfies (2.12) for some \(p<3\) and \(|p-3|\ll 1\).
Then we construct a unique solution \(f_R(t,x,v)\) of the form of
which solves the Boltzmann equation (1.1) and the diffuse reflection boundary condition (1.12) with the scale of (1.8) and (2.11), and satisfies the initial condition \(F|_{t=0}= M_{1, \varepsilon u _{in} ,1 } + \varepsilon ^2 f_{2} \sqrt{M_{1, \varepsilon u ,1 }} |_{t=0} + \varepsilon ^{3/2} f_{R,in} \sqrt{M_{1, \varepsilon u ,1 }}|_{t=0}\).
Moreover we derive that, for each \(\varepsilon \) and \(\kappa \) of (2.11),
Proof
The existence of the Navier-Stokes solutions follows from Theorem 3. For the remaining assertions, we note that all the estimates (2.39)–(2.42) of Theorem 3 ensure the conditions of Theorem 2 with \({\mathfrak {P}}=\frac{1}{2}\). Therefore the conclusion follows directly as a consequence of Theorem 2 and Theorem 3. \(\square \)
The incompressible Euler limit follows as a byproduct of the main theorem:
Corollary 5
(Hydrodynamic limit toward the incompressible Euler equation) Let \(u_E(t,x)\) be a (unique) solution of the incompressible Euler equations (1.9)-(1.10) with the initial condition \(u_E|_{t=0}= u_{in}\) in \(\Omega \). Then
Proof
Note that
The first term can be bounded as in (2.46). We bound the second term by an expansion:
Note that \(\Vert \varepsilon u \Vert _{L^\infty } \ll 1 \) and \(\Vert \varepsilon u_E\Vert _{L^\infty }\ll 1\) from Theorem 3. Then we conclude that the second term converges to 0 as \(\kappa \downarrow 0\) from Theorem 3 and the famous Kato’s condition for vanishing viscosity limit in [34].\(\square \)
3 Hilbert Expansion Around a Local Maxwellian and Source Terms
In this section we complete the Hilbert expansion along with the outline of the introduction. As a result we prove
Proposition 6
Suppose that F of (1.19), with a free parameter \(\delta \), solve (1.1) and (1.12) with (1.8) and that (u, p) solves (1.13)-(1.15). We choose a hydrodynamic part \(f_2\) as
with the pressure p of the Navier-Stokes flow in (1.13), and \(({\mathbf {I}}- {\mathbf {P}})f_2\) has been given in (1.34). Then \(f_{R}\) in (1.19) satisfies that
where the commutators \(L_t\), \({\mathbf {P}}_t\) and \(\Gamma _t\) are given in (3.34), while
At the boundary \(f_R\) and \(\partial _t f_R\) satisfy
In addition,
Remark 7
We note that due to the choice of (3.1) we remove a contribution of \(p^2\) in \(\frac{\varepsilon }{\delta \kappa } \Gamma (f_2,f_2)\). And also we remark that \({\mathfrak {R}}_4\) is quasi-linear for \(\partial _t^2p\) and \(\nabla _x \partial _t^2 u\).
3.1 Derivatives of \(A_{ij}\) and Commutators in the Local Maxwellian Setting
First we check properties of L and \(\Gamma \) defined in (1.20). Recall the notation of the global Maxwellian \(\mu _0:= M_{1,0,1}\). It is convenient to define
For a given \(\varepsilon u\), we define \({\tilde{f}}(\cdot ) := f(\cdot + \varepsilon u)\). Then we have
As in (1.23) a null space of \(L_0\), denoted by \({\mathcal {N}}_0\), is a subspace of \(L^2({\mathbb {R}}^3)\) spanned by orthonormal bases \(\{ {\tilde{\varphi }}_i \sqrt{\mu _0 }\}_{i=0}^4\) with
We denote a projection \(\tilde{{\mathbf {P}}}\) on \({\mathcal {N}}_0\) as in (1.24). From standard properties of \(L_0\) and (3.15), we can easily deduce the corresponding properties of L, namely the null space in (1.23), the spectral gap estimate in (1.25), and the existence of a unique inverse \(L^{-1}: {\mathcal {N}}^\perp \rightarrow {\mathcal {N}}^\perp \) in (3.17) which is defined via \(L^{-1}_0: {\mathcal {N}}_0^\perp \rightarrow {\mathcal {N}}_0^\perp \) with the identity
The inverse enjoys the following bound which turns out useful to prove Lemma 3.
Lemma 1
For \(0< \varrho < \frac{1}{4}\) and \(g \in {\mathcal {N}}_0^\perp \)
The proof is based on the well-known decomposition of \(L_0= \nu _0 -K_0\) and the compactness of \(K_0\): We first recall a standard decomposition
where \( \langle v\rangle \lesssim \nu _0 (v) \lesssim \langle v\rangle \). For (1.21) we have \(\nu (v)= \nu _0 (v-\varepsilon u)\) and \({\mathbf {k}}(v,v_*) = {\mathbf {k}}_0 (v-\varepsilon u, v_*-\varepsilon u)\). It is well-known (see (3.50) and (3.52) in [16]) that one can write \(K_0 g (v)= \int _{{\mathbb {R}}^3} {\mathbf {k}}_0 (v,v_*) g(v_*) \mathrm {d}v_*\) such that for some constants \(C_1, C_2>0\)
It is convenient to introduce a new notation, for \(\vartheta >0\),
Clearly \(|{\mathbf {k}}_0 (v,v_*)| \lesssim k_{\vartheta } (v,v_*)\) for \(0<\vartheta \le 1/8\).
Standard compactness estimates read as follows:
Lemma 2
For \(0<\varrho <2 \vartheta \) and \(C \in {\mathbb {R}}^3\), there exists \(C_{\varrho ,\vartheta }>0\) such that
Moreover
while the same bounds replacing |v| with \(|v_*|\) hold for integrations over v.
The proof of (3.22) relies on a fact that the exponent has a majorant \(- \vartheta {|v-v_*|^2} - \vartheta \frac{(|v|^2 - |v_*|^2)^2}{|v-v_*|^2}\le -2 \vartheta (|v|+|v_*|) ||v|- |v_*||\) which is a negative definite. Note that an exponent of \(\frac{e^{\varrho |v|^2 }}{e^{\varrho |v_*|^2 }}\) equals \({\varrho (|v|+ |v_*|) ||v| - |v_*|| } \) which can be absorbed as long as \(0< \varrho < 2 \vartheta \). This yields (3.22). We refer to a proof of Lemma 5 in [20] for details to show (3.23).
Proof of Lemma 1
We consider an operator \(g(v) \mapsto \nu _0^{-1} L_0 g (v):= \frac{1}{\nu _0(v)}L_0 g(v)\) on a restricted space of \(\{ g \in L^2({\mathbb {R}}^3): e^{\varrho |v|^2} g(v) \in L^2({\mathbb {R}}^3)\}\). First we claim that
From (3.19) we have \(\nu _0^{-1} L_0 g (v)= g(v) - \nu _0^{-1} e^{-\varrho |v|^2}\int _{{\mathbb {R}}^3} {\mathbf {k}}_0 (v,v_*) \frac{e^{\varrho |v|^2}}{e^{\varrho |v_*|^2}} e^{\varrho |v_*|^2}g(v_*) \mathrm {d}v_*\), and, using (3.23), for \(\varrho < 2 \vartheta \le 1/4\),
Therefore we prove (3.24) from
Now we view \(\{ g \in L^2({\mathbb {R}}^3): e^{\varrho |v|^2} g(v) \in L^2({\mathbb {R}}^3)\}\) as the Hilbert space with an inner product \(\langle e^{\varrho |v|^2} \cdot , e^{\varrho |v|^2} \cdot \rangle \). Then the compactness of \(\nu _0^{-1}K_0\) in this space is equivalent to the compactness of \(g \mapsto \int _{{\mathbb {R}}^3} {\mathbf {k}}_0 (v,v_*) \frac{e^{\varrho |v|^2}}{e^{\varrho |v_*|^2}} g(v_*) \mathrm {d}v_*\) in a usual \(L^2_v\). From Lemma 3.5.1 of [16], it suffices to prove that (i) \(\int _{{\mathbb {R}}^3} {\mathbf {k}}_0 (v,v_*) \frac{e^{\varrho |v|^2}}{e^{\varrho |v_*|^2}} \mathrm {d}v\) is bounded in \(v_*\), (ii) \({\mathbf {k}}_0 (v,v_*) \frac{e^{\varrho |v|^2}}{e^{\varrho |v_*|^2}} \in L^2 ( \{|v-v_*| \ge \frac{1}{n} \ \text {and}\ |v| \le n\})\) for all \(n\in {\mathbb {N}}\), and (iii) \(\sup _{v}\int _{{\mathbb {R}}^3} {\mathbf {k}}_0 (v,v_*) \frac{e^{\varrho |v|^2}}{e^{\varrho |v_*|^2}} \{ {\mathbf {1}}_{|v-v_*| \le \frac{1}{n}} + {\mathbf {1}}_{|v|\ge n} \}\mathrm {d}u\rightarrow 0\) as \(n\rightarrow \infty \). Both conditions (i) and (ii) come from the first bound of (3.23) directly. We prove (iii) from (3.22) and the first bound of (3.23). Now applying the Fredholm alternative to \(\nu _0^{-1} L_0=id -\nu _0^{-1}K_0 \) in the Hilbert space, we obtain an inverse map \((\nu _0^{-1} L_0)^{-1}\) which is a bounded operator of the Hilbert space. Note that \(L_0^{-1} (g)= (\nu _0^{-1} L_0)^{-1} (\nu _0^{-1} g)\). Hence we derive that
From the decomposition of \(L_0\), we have \(L^{-1}_0 g(v)= \nu _0(v)^{-1} g(v) + \nu _0(v)^{-1} KL^{-1}_0 g(v)\) for \(g \in {\mathcal {N}}_0^\perp \). Then we have
while \(\big |{\mathbf {k}}_0 (v,v_*) \frac{e^{\varrho |v|^2}}{e^{\varrho |v_*|^2}} \big |^2 \lesssim \frac{1}{|v-v_*|^2} e^{-2C_{\varrho } \frac{|v-v_*|^2}{2} } \in L^{\infty }_{v} L^{1}_{v_*}\) from (3.22). Hence we prove (3.18). \(\square \)
Equipped with Lemma 1 we provide bounds of \(A_{ij}\) in (1.34) and its derivatives:
Lemma 3
For \(0< \varrho < \frac{1}{4}\)
Proof
It is convenient to introduce a notation, with \(L_0\) in (3.14),
Then from (3.18) and (3.17) we can immediately prove the first bound in (3.27).
Recall the notations in (3.14) and (3.15). By taking a derivative to \(L_0\) (3.28), it follows that, from the decomposition of \(L_0A_{0,ij} (v) = \, \nu _0(v) A_{0,ij} (v)- \int _{{\mathbb {R}}^3} {\mathbf {k}}_0 (v, v-v_*) A_{0,ij} (v-v_*) \mathrm {d}v_* \) and (3.20),
From (3.20) and \(\nabla _v(|v|^2- |v-v_*|^2)^2= 4 v_{*} (|v| + |v-v_*|) (|v- |v-v_*|)\), it follows \(|\nabla _v [ {\mathbf {k}}_0 (v,v-v_*)]| \lesssim |v_*| \exp \{- \frac{ |v-v_*|^2+ |v*|^2}{8}\} + \frac{1}{|v_*|} \exp \{- \frac{|v_*|^2}{8} - \frac{1}{8} \frac{(|v |^2 - |v-v_*|^2)^2}{| v_*|^2}\}\). From the first bound of (3.23), it follows \(|\int _{{\mathbb {R}}^3} \partial _{v_k} [ {\mathbf {k}}_0 (v,v-v_*)] A_{0,ij} (v-v_*)\mathrm {d}v_*| \lesssim e^{-\varrho |v|^2}\) for any \(0< \varrho < 1/4\). Recall a projection \(\tilde{{\mathbf {P}}}\) on \({\mathcal {N}}_0\). Then \(|({\mathbf {I}}- \tilde{{\mathbf {P}}}) \text { r.h.s. of }(3.29)| \lesssim e^{-\varrho |v|^2}\). Now applying (3.18) to \( \partial _{v_k} A_{0,ij}= L_0^{-1} (({\mathbf {I}}- \tilde{{\mathbf {P}}}) \text { r.h.s. of }\)(3.29)) we derive
From (1.34) and (3.17), and the fact \({\tilde{\varphi }}_i=v_i\) for \(i=1,2,3\), and \({\tilde{\varphi }}_4=\frac{|v|^2}{3}\) (the notation \({\tilde{f}}\) is defined in (3.15)), we have
Therefore we prove the second and third bounds in (3.27) using the fact that \(\nabla _{x,t} A_{ij}(v)= -\varepsilon \nabla _{x,t} u \nabla _v A_{0,ij}(v-\varepsilon u)\).
Now we prove
By taking one more derivative to (3.29), we derive that
The terms in the first two lines in r.h.s are easily bounded above as \(e^{-\varrho |v|^2}\), recalling the fact \(|\nabla _v\nu _0(v)| + |\nabla ^2_v\nu _0(v)| \lesssim 1\). We only focus on the terms in the last line. From \(|\partial _{v_\ell } \nabla _v (|v|^2 - |v-v_*|^2)^2| \le \big |4 v_* \Big ( \frac{v_\ell }{|v|} + \frac{(v-v_*)_\ell }{|v-v_*|} \Big )(|v| - |v-v_*|) \big | + 4 \big | v_* (|v| + |v-v_*|)\Big ( \frac{v_\ell }{|v|} - \frac{(v-v_*)_\ell }{|v-v_*|} \Big )\big |\lesssim |v_*|^2 + |v_*||v|, \) we have \(| \partial _{v_\ell }\nabla _v [ {\mathbf {k}}_0 (v,v-v_*)]| \lesssim |v_*| \exp \{- \frac{ |v-v_*|^2+ |v*|^2}{8}\} + \frac{1+|v|}{|v_*|^2} \exp \{- \frac{|v_*|^2}{8} - \frac{1}{8} \frac{(|v |^2 - |v-v_*|^2)^2}{| v_*|^2}\}\). Using the second estimate of (3.23) with the first bound of (3.27), we have \(\big |\int _{{\mathbb {R}}^3} \partial _{v_k} \partial _{v_\ell }[ {\mathbf {k}}_0 (v,v-v_*)] A_{0,ij} (v-v_*) \mathrm {d}v_*\big |\lesssim e^{- \varrho |v |^2}\). From (3.30) and the first bound of (3.27), it follows that \(\big |\int _{{\mathbb {R}}^3} \partial _{v_k} [ {\mathbf {k}}_0 (v,v-v_*)] \partial _{v_\ell } [A_{0,ij} (v-v_*)] \mathrm {d}v_*\big | \lesssim e^{- \varrho |v |^2}\). Now we invert the operator \(L_0\) and use (3.18) to conclude (3.32).
Finally from \(\partial _t \nabla _x A_{ij} (v) = -\varepsilon \partial _{t} \nabla _x u \nabla _v A_{0,ij} (v-\varepsilon u) + \varepsilon ^2 \nabla _x u \partial _t u \nabla _v^2 A_{0,ij} (v-\varepsilon u)\), (3.30), and (3.32), we conclude the last estimate of (3.27). \(\square \)
For the estimates of \(\partial _t f_R\) we derive the commutator estimate of \(\partial _t L- L\partial _t \) and the corresponding one for \(\Gamma \) as follows.
Lemma 4
Suppose \(\varepsilon |u|\lesssim 1\) in the definition of \(\mu \) in (1.16). For L and \(\Gamma \) in (1.20) and (1.21),
where
We have
Pointwise estimates are given as follows: for \(0< \varrho < 1/4\) and \(C \in {\mathbb {R}}^3\)
and
Proof
The decomposition (3.33) with (3.34) comes from a direct computation to (1.21) and \( \partial _t (Lf_R) = \partial _t (L( {\mathbf {I}}-{\mathbf {P}})f_R) =L ( {\mathbf {I}}-{\mathbf {P}})\partial _tf_R + L_t( {\mathbf {I}}-{\mathbf {P}})f_R + L (-{\mathbf {P}}_t f_R). \) On the other hand, from (3.20) it is easy to check that, for any \(0< \varrho < 1/4\),
These estimates above combining with (3.23) and \({\mathbf {P}}L=0\) yield the first two estimates of (3.35).
We derive \(\Gamma _t\) as in the third identity of (3.34) from a direct computation to (1.22) and \( \partial _t \sqrt{\mu (v_*)} = - \varepsilon \partial _t u \cdot \nabla _v \sqrt{\mu _0 (v_* - \varepsilon u )} = \frac{1}{2} \varepsilon \partial _t u \cdot (v_*-\varepsilon u ) \sqrt{\mu _0 (v_* - \varepsilon u )} . \) Then it is standard (see Lemma 2.13 in [13] for example) to have the last estimate in (3.35).
The first bound of (3.36) is a direct consequence of applying (3.22) and (3.23) to the first identity of (3.34). For the second bound of (3.36) we bound it as
For the last bound of (3.36) we recall a standard estimate (e.g. [16]) that
From the second equality of (3.15) we deduce the last bound of (3.36). A bound (3.37) is standard. \(\square \)
3.2 Proof of Proposition 6
We verify two statements of Section 1.1 Hilbert Expansion. Firstly, we will show that the solvability condition (1.33) implies the incompressible condition (1.14). From (1.16), (1.23), and direct computations we verify the first identity of (1.33). Then from the oddness of the integrand with respect to the variable \(\varphi _i\) we derive that \(\Big \langle \varphi _i \sqrt{\mu }, \frac{ \varepsilon ^{-1} (v- \varepsilon u )\cdot \nabla _x \mu }{\sqrt{\mu }} \Big \rangle =0\) for \(i=1,2,3\). For \(i=0,4\), we compute that
This shows that (1.33) implies (1.14).
Secondly, we will verify the following statement of Section 1.1: the leading order terms of the hydrodynamic part in (1.28) vanish by solving the Navier-Stokes equations (1.13)-(1.15). Consider (1.28). We set \({\mathbf {P}}f_2 =\{ {\tilde{\rho }} \varphi _0+ \sum _{\ell =1}^3 {\tilde{u}}_\ell \varphi _\ell +{\tilde{\theta }} \varphi _4 \}\sqrt{\mu }\) whose coefficients will be determined as in (3.1). Then the leading order term of (1.28)= \(\frac{1}{\delta }\) (1.35) can be decomposed as
while the lower order term consists of
First we focus on a leading order contribution of (3.38)\(_*\) in (3.38). A direct computation yields
Among many other choices we make a special choice \(({\tilde{\rho }}, {\tilde{u}}, {\tilde{\theta }})=(p,0,0)\) which is equivalent to (3.1). From (1.38), (3.41), and (3.1), it follows that for (u, p) solving (1.13)
which verifies the second statement of Section 1.1.
Now we turn to proving the estimates. While the leading order terms vanish in (1.28), the rest of terms of (1.28) are bounded as follows. Upon the choice of (3.1), the first term of (3.39) vanishes and the first line of (3.40) are bounded by
From (3.27) we deduce that the second term of (3.39) and the second line of (3.40) are bounded respectively by
In conclusion we end up with the following result: Assume (u, p) solves (1.13)-(1.15), and both (1.34) and (3.1) hold. Then
The term \(\partial _t (1.28)\) can be bounded similarly. The entire leading order term of \(\partial _t (1.28)\) can be decomposed as
Following the argument to get (1.38) and (3.41), we derive that
On the other hand,
Now the lower order term \(\partial _t (1.28)- (3.47)- (3.48)\) consists of
Since the lower order term of \(\partial _t (1.28)\) always contains \(|\partial _t (v-\varepsilon u)| \le \varepsilon |\partial _t u|\), they can be bounded by, from (3.27) and (3.1),
Now we consider (1.29). From (1.34), (3.1), and (3.27), we derive that
and, for \(0< \varrho < 1/4\),
Next we consider the last term in (1.29). From (1.16) and (1.14)
and hence we derive (3.12) and (3.13).
Applying (3.27) to (1.34), it follows that, for \(0< \varrho < \frac{1}{4}\),
From (3.1)
These estimates give (3.10) and (3.11).
The last term of (1.29) is bounded as
Lastly from (3.36), (3.1), and (1.34)
where we have used \(\Gamma ({\mathbf {P}} f_2, {\mathbf {P}}f_2 )=\Gamma (p \sqrt{\mu }, p \sqrt{\mu })=0\) to eliminate the contribution of \(p^2\) in (3.61).
Finally we wrap up the estimates of the source term of (3.2) to show (3.4) and (3.5). The term \(({\mathbf {I}} - {\mathbf {P}}) {\mathfrak {R}}_{1}\) consists of (3.39), which is bounded as (3.46) and hence we prove (3.4). The rest of terms form \({\mathfrak {R}}_{2}\), which can be proved to be bounded as (3.5), from (3.45), (3.53), (3.55), (3.59), and (3.61).
Now we consider the source term of (3.3). The term \(({\mathbf {I}} - {\mathbf {P}}) {\mathfrak {R}}_3\) consists of (3.48), which is bounded as (3.50). From (3.49), (3.52), (3.54)-(3.56), (3.12), (3.13), (3.60), (3.62), and (3.36), we prove (3.7).
4 A Priori Estimates for \(f_R\)
For each \(\varepsilon >0\) an existence of a unique solution in a time interval \([0,\infty )\) can be found in [13]. Thereby we only focus on a priori estimates of \(f_R\) in different spaces. For the sake of simplicity at times we will use simplified notations
Recall the boundary integral and the norms in (2.5). Also recall \({\mathfrak {w}}={\mathfrak {w}}_{\varrho , \ss }(x,v)\) in (2.3) and \({\mathfrak {w}}'={\mathfrak {w}}_{\varrho ', \ss }(x,v) \) for \(0<\varrho '<\varrho \).
4.1 \(L^2\)-Energy Estimate
Our starting point is a basic \(L^2\)-energy estimate for the Boltzmann remainder \(f_R\) and its temporal derivative \(\partial _t f_R\) in which the dissipation (1.31) plays an important role in the nonlinear estimate.
Proposition 7
Under the same assumptions in Proposition 6, we have
where
Here \(L^p_t=L^p_t([0,t])\) in particular. Note that a weighted \(L^\infty \)-bound of \(f_R\) is involved in this energy estimate, where the weight \({\mathfrak {w}}={\mathfrak {w}}_{\varrho , \ss }(x,v)\) is defined in (2.3).
We also have
where \(\partial _t f_R (0,x,v) := f_{R,t}(0,x,v)\) is defined in (2.6). Here
where \(0<\varrho ^\prime < \varrho \).
Remark 8
We utilize several different time-space norms to control the fluid source terms, which possess the initial-boundary and boundary layers as in Theorem 3.
The following trace theorem is useful to control the boundary terms.
Lemma 5
(Trace theorem)
where \(\gamma _+^N:= \{(x,v) \in \gamma _+: |n(x) \cdot v|> 1/N \ \text {and} \ 1/N<|v|<N \}\).
The proof is standard (for example see Lemma 3.2 in [13] or Lemma 7 in [6]).
Proof of Proposition 7
First we prove (4.2). An energy estimate to (3.2) and (3.8) reads as
Among others two terms (4.9) and (4.13) are most problematic.
We start with (4.7). From the spectral gap estimate in (1.25), we have
Now we consider (4.9), in which we need integrability gain of \({\mathbf {P}}f_R\) in \(L^6_x\) of the next sections. From decomposition \(f_R= {\mathbf {P}} f_R + ({\mathbf {I}} - {\mathbf {P}}) f_R\) and \(\Gamma =\Gamma _+- \Gamma _-\) in (1.22), we derive
From (3.4) and (3.5) we derive that
Next using (3.12) it follows that
Finally we control the boundary term (4.8) using a trace theorem (4.6). First we have, from (3.8),
where we have used the fact \(|P_{\gamma _+}f_R|_{L^2_{\gamma _+}}= |P_{\gamma _+}f_R|_{L^2_{\gamma _-}}\) from \(P_{\gamma _+}f_R(t,x,v)\) being a function of (t, x, |v|) due to \(u|_{\partial \Omega }=0\).
Now we estimate \(P_{\gamma _+}f_R\). Since \(P_{\gamma _+}\) in (3.8) is a projection of \(c_\mu \sqrt{\mu }\) on \(\gamma _+\), it follows \(\int _{\gamma _+} |P_{\gamma _+}f|^2 \le 2 \int _{\gamma _+^N} |P_{\gamma _+}f|^2\) for large enough \(N>0\), where \(\gamma _+^N:= \{(x,v) \in \gamma _+: |n(x) \cdot v|> 1/N \ \text {and} \ 1/N<|v|<N \}\). Setting \(h=|f|^2\) in (4.6) and using (3.2), (3.4), and (3.5) we derive
Furthermore from (3.8) and (4.20)
Finally we collect the terms as
We choose large N and then large C so that \(\frac{C_N}{4C}\ll \sigma _0\). Using Young’s inequality for products, and then moving contributions of \( \Vert \kappa ^{-\frac{1}{2}} \varepsilon ^{-1} \sqrt{\nu } ({\mathbf {I}} - {\mathbf {P}}) f_R \Vert _{L^2_{t,x,v}}^2\) to l.h.s., we derive (4.2).
Next we prove (4.4). An energy estimate to (3.3) and (3.9) lead to (4.4)
We consider the first term of (4.30). We decompose \(\partial _t f_R = {\mathbf {P}} \partial _t f_R + ({\mathbf {I}} - {\mathbf {P}}) \partial _t f_R\). The contribution of \({\mathbf {P}} \partial _t f_R\) can be bounded above as, from (3.12),
For the contribution of \(({\mathbf {I}} - {\mathbf {P}}) \partial _t f_R\) we utilize an extra decomposition \({\mathbf {1}}_{|v|\le \varepsilon ^{-1}} + {\mathbf {1}}_{|v|\ge \varepsilon ^{-1}}\) Then it is bounded as
For the second term of (4.30) using (3.13) we bound it by
Using (3.35) we bound (4.24) and (4.27) as
The rest of terms can be controlled similarly as in the proof of (4.2):
Lastly we estimate (4.23) and the first term of (4.30). As in (4.19) we derive that (4.23) is bounded from below by
where we have used \(|r_{\gamma _+} (g)|_{L^2(\gamma _-)}\lesssim \varepsilon \Vert \partial _t u \Vert _\infty | g|_{L^2(\gamma _-)}\) from (3.9). Now we bound \(P_{\gamma _+} \partial _t f_R\) using (4.6). Following the argument arriving at (4.20) and setting \(h=|\partial _t f|^2\) we derive
We conclude (4.4) by collecting the terms.\(\square \)
4.2 \(L^6_x\)-Integrability Gain for \({\mathbf {P}}f_R\)
Proposition 8
Under the same assumptions in Proposition 6, we have for all \(t \in [0,T]\)
where
Proof
For the sake of simplicity we use notations (4.1) throughout this subsection.
We view (3.2) as a weak formulation for a test function \(\psi \)
The proof of the lemma is based on a recent test function method in the weak formulation ([12, 13]). We define
where \(a:= \langle f_R, \sqrt{\mu _0} \rangle , b:= \langle f_R,v \sqrt{\mu _0} \rangle \), and \(c:= \langle f_R, \frac{|v|^2-3}{\sqrt{6}}\sqrt{\mu _0} \rangle \). We choose a family of test functions as
where we choose \(\beta _a=10, \beta _b=1, \beta _c=5\) such that
Here,
A unique solvability to the above Poisson equations when \((a,b,c) \in L^6(\Omega )\) and an estimate
is a direct consequence of Lax-Milgram and suitable extension (extend \(a^5\) of (4.53) evenly in \(x_3 \in {\mathbb {R}}\), and \(b^5\) and \(c^5\) of (4.54) and (4.55) oddly in \(x_3 \in {\mathbb {R}}\), then solve the Poisson equation, and then restrict the whole space solutions to the half space \(x_3>0\)) and a standard elliptic estimate \((L^{\frac{6}{5}}(\Omega ) \rightarrow {\dot{W}}^{2,\frac{6}{5}} (\Omega ) \cap {\dot{W}}^{1,2} (\Omega ) \cap L^6(\Omega ))\).
From \(M_{1,\varepsilon u , 1}(v) = M_{1,0 , 1}(v) + O( \varepsilon ) |u| | v-\varepsilon u | M_{1,\varepsilon u , 1}(v) \) we can easily check that
Therefore we have
Note that \( \Vert ( {\mathbf {I}}-{\mathbf {P}}) f_R (t) \Vert _{L^6_{x,v}} \le \Vert ( {\mathbf {I}}-{\mathbf {P}}) f_R (t) \Vert _{L^\infty _{x,v}} ^{2/3} \Vert ( {\mathbf {I}}-{\mathbf {P}}) f_R (t) \Vert _{L^2_{x,v}} ^{1/3} \lesssim o(1) (\kappa \varepsilon )^{1/2} \Vert {\mathfrak {w}} f_R (t) \Vert _{L^\infty _{x,v}} + (\kappa \varepsilon )^{-1} \Vert ( {\mathbf {I}}-{\mathbf {P}}) f_R (t) \Vert _{L^2_{x,v}}\). Hence to prove the lemma and (4.44) it suffices to prove the same bound for \(\Vert {\tilde{P}}f_R \Vert _{L^6_{x,v}}:=\Vert ( a,b,c) \Vert _{L^6_{x }} \).
Following the direct computations in the proof of Lemma 2.12 in [13] we derive that
For \(\Vert b_i\Vert _{6}^6\), using the second and third estimate of (4.59) we deduce that
Now we consider the boundary term \((4.46)_2\). From (4.48)-(4.51) and (4.52)
Here we have used the Neumann boundary condition of (4.53) for \(\psi _a\), and the last identity in (4.52) for \(\psi _c\). For \(\psi _{b,1}^{i,j}\) or \(\psi _{b,2}^{i,j}\) we used the fact that the integrands are odd in v. From (3.8), we decompose \(f|_{\gamma }= P_{\gamma _+} f + {\mathbf {1}}_{\gamma _+} (1- P_{\gamma _+})f - {\mathbf {1}}_{\gamma _-}\frac{\varepsilon }{\delta } (1- P_{\gamma _+}) f_2\). From (4.61) together with (3.57) and (3.58) we have
where we have used \(|\int _{\gamma _+} \psi (1-P_{\gamma _+}) f|\lesssim |\nabla _x \varphi |_{L^{ {4} / {3}}(\partial \Omega ) } |(1-P_{\gamma _+}) f|_{4, \gamma _+}\) at the last line. Here \(\varphi \in \{ \varphi _a, \varphi _b, \varphi _c\}\). For the first term of (4.62) we interpolate
For the second term of (4.62), we use (4.56) and a trace theorem \(({\dot{W}}^{1,\frac{6}{5}} ({\mathbb {T}}^2\times {\mathbb {R}}_+) \cap L^2({\mathbb {T}}^2\times {\mathbb {R}}_+) \rightarrow {W}^{1- \frac{1}{6/5},\frac{6}{5}} ({\mathbb {T}}^2 )) \), and the Sobolev embedding \((W^{\frac{1}{6},\frac{6}{5}}( {\mathbb {T}}^2) \rightarrow L^{4/3} (\mathbb {T}^2))\) to conclude that
Next we consider \((4.46)_3\). For \(\psi \) of (4.48)-(4.51) and \(\varphi \) of (4.53)-(4.55), using (4.56), it follows that
Lastly we consider the right hand side of (4.46). From (1.21), (3.23), (3.37), and (4.56), it follows
Note that, from (3.37), \(|\Gamma (\frac{\varepsilon }{\kappa } {f_2}, f_R) |\lesssim \frac{\varepsilon }{\kappa } \Vert {\mathfrak {w}}_{\varrho , \ss } f_2 \Vert _{\infty } {\mathfrak {w}}_{\varrho , \ss }(v)^{-1} \big [\nu (v) f_R(v)+ \int _{{\mathbb {R}}^3} k_{\vartheta } (v,v_*) f_R(v_*) \mathrm {d}v_*\big ].\) Then from (3.57) and (3.58)
For the contribution of \(\Gamma (f_R,f_R)\) we decompose \(f_R= {\mathbf {P}} f_R + ({\mathbf {I}} - {\mathbf {P}}) f_R\). From (3.37) (or (3.36))
Then from (4.48)-(4.51), (3.23), and the Hölder’s inequality (\(1=1/2+ 1/3+ 1/6\))
where we have used an interpolation \(\Vert Pf_R\Vert _{L^3}\le \Vert Pf_R\Vert _{L^6}^{1/2} \Vert Pf_R \Vert _{L^2}^{1/2}\) and (4.56) at the last step. A contribution of the rest of terms in the r.h.s of (4.46) can be easily bounded as, from (3.4) and (3.5),
In conclusion, collecting the terms from (4.59) with (4.60), (4.62) with (4.63) and (4.64), (4.65), (4.66), (4.67), (4.69), (4.70), and utilizing (4.58), and two facts from (A.1):
we prove (4.44).\(\square \)
4.3 Average in Velocity
We prove a version of velocity lemma when a suitable bound for source terms is only known in a finite time interval. In this section we often specify domains in which an \(L^p\)-norm is taken while the simplified notation (4.1) will be used only when the domain is \([0,T] \times \Omega \times {\mathbb {R}}^3\).
Proposition 9
Assume the same assumptions in Proposition 6. Then we have, for \(2<p<3\),
with
and for \(\varrho '<\varrho \)
with
where both bounds are uniform-in-p for \(2<p<3\).
We prove the proposition by several steps.
Step 1: Extension. We define a subset
We regard \({\tilde{\Omega }}\) as an open subset but not a periodic domain as \(\Omega \). Without loss of generality we may assume that \(f_R(0,x,v)\) is defined in \({\mathbb {R}}^3 \times {\mathbb {R}}^3\) and \(\Vert f_R(0) \Vert _{L^p({\mathbb {R}}^3 \times {\mathbb {R}}^3)} \lesssim \Vert f_R(0) \Vert _{L^p({\tilde{\Omega }} \times {\mathbb {R}}^3)}\) for all \(1 \le p \le \infty \). Then we extend a solution for whole time \(t \in {\mathbb {R}}\) as
where a smooth non-negative function \(\chi _1\) satisfies \(\chi _1 (t)\equiv 1\) for \(t \in [-1,0]\), \(\chi _1 (t) \equiv 0\) for \(t<-2\), and \(0 \le \frac{d}{dt}\chi _1 \le 4\).
A closure of \({\tilde{\Omega }}\) is given as \(cl({\tilde{\Omega }})= [0,2\pi ]\times [0,2\pi ] \times [0,\infty )\). Let us define \({\tilde{t}}_B(x,v) \in {\mathbb {R}}\) for \((x,v) \in ({\mathbb {R}}^3 \backslash {\tilde{\Omega }}) \times {\mathbb {R}}^3\). We consider \({\tilde{B}}(x,v):=\{s \in {\mathbb {R}}:x+ s v \in {\mathbb {R}}^3 \backslash cl({\tilde{\Omega }}) \}\). Clearly if \({\tilde{B}}(x,v) \ne \emptyset \) then \( \{s>0\}\subset {\tilde{B}} (x,v)\) or \( \{s<0\}\subset {\tilde{B}} (x,v)\) exclusively. If \( \{s>0\} \subset {\tilde{B}} (x,v)\), let \( I_+\) be the largest interval such that \( \{s > 0 \} \subset I_+ \subset {{\tilde{B}}}(x,v) \). And if \( \{s < 0\} \subset {\tilde{B}} (x,v)\), let \( I_-\) be the largest interval such that \( \{s > 0 \} \subset I_- \subset {{\tilde{B}}}(x,v)\).
We define
Using (4.78) we define
It is easy to see that \(\varepsilon \partial _t f_E + v\cdot \nabla _x f_E=0\) in the sense of distributions.
Next we define two cutoff functions. For any \(N>0\) we define smooth non-negative functions as
We denote
We define an extension of cut-offed solutions
We note that in the sense of distributions \({\bar{f}}_R\) solves
Here we have used the fact that \({\bar{f}}_R\) in (4.84) is continuous along the characteristics across \(\partial {\tilde{\Omega }}\) and \(\{t=0\}\). We derive that, using (4.84),
Recall \({\tilde{\varphi }}_i \in \{{\tilde{\varphi }}_0, \cdots {\tilde{\varphi }}_4\}\) in (4.47). From (4.83) we note that
From (1.24), we decompose
We consider the right hand side of above terms. From (4.57), \(\int {\tilde{\varphi }}_i{\tilde{\varphi }}_j \mu _0 = \delta _{ij}\), and (4.81), the first term can be bounded below by \(\big (1-O( \varepsilon ) \Vert u\Vert _\infty - O(\frac{1}{N})\big ) \big \Vert \chi _2 {P} f_R \big \Vert _{L^2_t ((0,T); L^p_x ({\tilde{\Omega }}))}\). For the second term we use (4.57), \(L^2_t (0,T) \subset L^p_t( 0,T )\), and \(L^1(\{|v|\le N\}) \subset L^p(\{|v|\le N\})\) to bound it above by \(C_{T,N} \Vert ({\mathbf {I}} - {\mathbf {P}}) f_R \Vert _{L^p((0, T) \times {\tilde{\Omega }} \times {\mathbb {R}}^3)}+ \big ( O( \varepsilon ) \Vert u\Vert _\infty + O(\frac{1}{N})\big ) \big \Vert {P} f_R \big \Vert _{L^2_t( (0,T) ;L^p_x({\tilde{\Omega }}))}\). Hence we derive
Step 2: Average lemma. Recall \({\tilde{\varphi }}_i \in \{{\tilde{\varphi }}_0, \cdots {\tilde{\varphi }}_4\}\) in (4.47). We choose \({\tilde{\varphi }}(v)\) such that
Lemma 6
We define
Then, for \(p<3\) and \(1\ll N\),
where the bound (4.90) only depends on N but can be independent on \(p<3\).
We remark that from (4.85) and (4.89) \( \int _{{\mathbb {R}}^3}{\bar{f}}_R(t,x,v) {\tilde{\varphi }}_i (v) \mathrm {d}v \le S({\bar{g}})(t,x)\).
Proof of Lemma 6
We prove (4.90) by a \(TT^*\)(\(SS^*\) for our case) method. First we derive a dual of S in the following equalities:
where we have defined
Here, in the second equality of (4.91) we have used the Fubini theorem for changing order of s and t integrations, and then used a change of variables \(x \mapsto x- \frac{t-s}{\varepsilon } v\). In the third equality of (4.91) we have used a change of variable \((t,s) \mapsto (s,t)\) and the fact \(\text {Supp} (g) \subset (-\infty ,T] \times U \times V\).
On the other hand, for \(1/p+1/q=1\) , following the argument of (4.91) with \(h(t,x) = {\mathbf {1}}_{x \in {\tilde{\Omega }}} h(t,x)\) we derive that
It is important to check the integral region in space of the last term of (4.93). From (4.92), we note that if \(x + \frac{\tau -t}{\varepsilon } v \notin cl( {\tilde{\Omega }})\) for all \(\tau \in [t,T]\) then the last term would vanish since \(\text {supp} (h) \subset (-\infty ,T] \times {\tilde{\Omega }}\). Therefore we can exclude (t, x, v) from the last integration in (4.93) if \(L(t,x,v) \cap {\tilde{\Omega }} = \emptyset \) for \(L(t,x,v):=\{x + \frac{\tau -t}{\varepsilon } v : \tau \in [t,T] \}\). Now we define
Then we can write
Therefore to prove (4.90) it suffices to show that
Note that since \(\text {supp}(h) \subset (-1,T] \times U\) and \(\text {supp} ({\tilde{\varphi }}) = V\) for \((x,v) \in U\times V\), we have, with \(x= (x_1,x_2,x_3), v=(v_1,v_2,v_3)\)
Hence we can rewrite (4.92) as
On the other hand, from (4.91), we have for \(\text {supp}(h) \in (-1,T] \times {\tilde{\Omega }}\),
Therefore to show (4.96) (which will imply (4.90)) we only need to prove that, for \(\text {supp}(h) \subset (-1,T] \times {\tilde{\Omega }}\),
Now we prove (4.98). From (4.89) and (4.97), we read
Now for the same reason to restrict \(\tau \)-integration in (4.97) we rewrite the above expression as
We consider a map with the change of variables
Now we apply (4.100) to (4.99) and derive that
First using the Minkowski’s inequality and the Young’s inequality to a convolution in y with \(1+ 1/p=1/q + 1/(p/2)\) we have
From the properties of \({\tilde{\varphi }} \in C_c^\infty \), it follows that
where \({\tilde{y}}= \frac{\varepsilon }{|\tau -t|}(y-x)\) with \(\mathrm {d}{\tilde{y}}= \frac{\varepsilon ^3 }{|\tau -t|^3}\mathrm {d}y\). Therefore we derive that
Using the Minkowski’s inequality and the Young’s inequality, finally we prove (4.98) as
\(\square \)
Step 3: Applying Lemma 6. Now we apply Lemma 6 to (4.85) and derive that
where we have used (4.83), (4.77), (4.79), and the fact that \(|v\cdot \nabla _x \chi _2 (x)| \lesssim _N 1\) on \(v \in V\).
First we consider (4.104). We split the cases of (4.104) according to (4.78). For \(x \in \partial {\tilde{\Omega }}\), which has a zero measure in \(L^2 ((-1,T]\times (U \backslash {\tilde{\Omega }})\times V)\), we have \({\tilde{t}}_B(x,v)=0\) from the first line of (4.78). If \({\tilde{B}}(x,v) = \emptyset \) and \(x \notin \partial {\tilde{\Omega }}\) then \({\tilde{t}}_B(x,v)=-\infty \) from the last line of (4.78) and hence \({\bar{f}}_R(-\infty )=0\) since \(\chi _1(-\infty )=0\) in (4.77). Therefore we derive that
We need a special attention to (4.106). Since \((t,x,v) \in {\mathfrak {D}}_T\) we know that \(\inf \{\tau \ge t: x+ \frac{\tau - t}{\varepsilon } v \in cl({\tilde{\Omega }})\} \le T.\) If \(\{s<0\} \subset {\tilde{B}} (x,v)\) then, from the third line of (4.78), \({\tilde{t}}_B(x,v) = \sup {\tilde{B}} (x,v)= \sup \{s \in {\mathbb {R}}: x+ sv \in {\mathbb {R}}^3 \backslash cl ({\tilde{\Omega }})\} \le (T-t)/\varepsilon \). Therefore the argument of \(f_I\) in (4.106) is confined as
For (4.107), from the second line of (4.78), \({\tilde{t}}_B (x,v)= \inf {\tilde{B}}(x,v) = \inf \{s \in {\mathbb {R}}: x+ sv \in {\mathbb {R}}^3 \backslash cl ({\tilde{\Omega }})\} \le 0\). Therefore \(t+ \varepsilon {\tilde{t}}_B (x,v) \le t \le T\) and hence the argument of \(f_I\) in (4.107) is confined as in (4.108). Now we apply the Minkowski’s inequality in time, change of variables \(t+ \varepsilon {\tilde{t}}_B (x,v) \mapsto t\), and use (4.108) to derive that
Let us define an outward normal \({\tilde{n}}(x)\) on \(\partial {\tilde{\Omega }}\). More precisely
From (4.82) we have therefore \((x ,v) \in (U \backslash {\tilde{\Omega }}) \times V\) then \(|{\tilde{n}}( {\tilde{x}}_B (x,v) ) \cdot v|\ge 1/N\). We consider maps
Note that if \(v \in V\) of (4.82) then \(|v_i| \ge 1/N\) for all \(i=1,2,3.\) We define
We apply the change of variables (4.111) to (4.109):
We recall the trace theorem:
We apply (5.33) with \(h=f^2\) and derive an estimate
Finally we conclude a bound of (4.104) as below via (4.106), (4.107), (4.109), (4.113), and (4.115)
Next we estimate (4.105) (and \((4.116)_*\)). Using (4.84) and (3.2) we conclude that
Following the arguments of (4.15)-(4.18), and (3.4), (3.5), we derive that
where we further bound
Step 4. Proof of (4.72). First we use (4.87) and then (4.104) and (4.105). We bound (4.104) via (4.109) and (4.113), which are bounded by (4.115) and (4.117) respectively. These conclude that, for \(p<3\),
Then we move a contribution of \(\Vert P f_R \Vert _{L^2_t ((0,T); L_x^{p}(\Omega ))}\) to the l.h.s and use (4.118). This concludes (4.72).
Step 5: Sketch of proof for (4.74). We follow the same argument for (4.72). Thereby we only pin point the difference of the proof of (4.74). Recall \(\partial _t f_R (0,x,v) = f_{R,t}(0,x,v)\) from (2.6). We regard \({\tilde{\Omega }}\) as an open subset but not a periodic domain as \(\Omega \). Without loss of generality we may assume that \(f_{R,t}(0,x,v)\) is defined in \({\mathbb {R}}^3 \times {\mathbb {R}}^3\) and \(\Vert f_{R,t}(0) \Vert _{L^p({\mathbb {R}}^3) \times {\mathbb {R}}^3} \lesssim \Vert f_{R,t}(0) \Vert _{L^p({\tilde{\Omega }}) \times {\mathbb {R}}^3}\) for all \(1 \le p \le \infty \). Then we extend a solution for whole time \(t \in {\mathbb {R}}\) as
Using \({\tilde{t}}_B(x,v) \) in (4.78) we define
We define an extension of cut-offed solutions
We note that in the sense of distributions \({\bar{f}}_{R,t}\) solves
Here we have used the fact that \({\bar{f}}_{R,t}\) in (4.123) is continuous along the characteristics across \(\partial {\tilde{\Omega }}\) and \(\{t=0\}\). We derive that, using (4.123),
Now we apply Lemma 6 to (4.124) and derive that, for \(p<3\),
Following the same argument of (4.116)-(4.117) we deduce that
From (4.31)-(4.33), the last term of (4.126) is bounded above by
Here the most singular term comes from \(\frac{1}{\varepsilon ^2 \kappa } L({\mathbf {P}}_t f_R)\) in the r.h.s. of (3.3) .
On the other hand from (4.122) and the argument of (4.86) we derive
Here we have used
Combining (4.128), (4.125), (4.126), and (4.127) and choosing \(N\gg 1\) we conclude (4.74).
4.4 \(L^\infty \)-Estimate
In this section we develop a unified \(L^\infty \)-estimate in the local Maxwellian setting. We devise the weight functions to control an extra growth in |v| comes from \(\frac{(\partial _t + \varepsilon ^{-1} v\cdot \nabla _x) \sqrt{\mu }}{\sqrt{\mu }}\) and its temporal derivative:
where \({\mathfrak {z}}_{\ss }: {\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+\) is defined as, for \(\ss >0\)
We often abuse the notation of \({\mathfrak {w}}_{\varrho , \ss }\) and \({\mathfrak {w}}\). We compute to have
where we have used \(\partial _{x_3} {\mathfrak {z}}_{\ss } (x_3)= {\mathbf {1}}_{[ \ss ^{-1}-1,\infty )} (x_3) \frac{-1}{(1+ x_3)^2}\). The last term, the sole term without a sign, can be bounded as
Therefore we conclude that
We consider
An equation for h can be written from (3.2) and (3.8) as
For (4.131), we have \(r=- \frac{\varepsilon }{\delta } {\mathfrak {w}} (1- P_{\gamma _+}) f_2\) and \({\mathcal {S}}_h: = \frac{\delta }{\kappa \varepsilon } \Gamma _{ {{\mathfrak {w}} }}( {h} , {h} ) +\frac{2}{\kappa } \Gamma _{{\mathfrak {w}}} ( {\mathfrak {w}} f_2,h) + {\mathfrak {w}} ({\mathbf {I}} - {\mathbf {P}}) {\mathfrak {R}}_1 +{\mathfrak {w}} {\mathbf {R}}_2, \) and
where we denote \(\Gamma _{{\mathfrak {w}}}(\cdot , \cdot )(v) := {\mathfrak {w}} (v)\Gamma (\frac{\cdot }{{\mathfrak {w}}}, \frac{\cdot }{{\mathfrak {w}}})(v)\) and \(K_{{\mathfrak {w}}}(\cdot ):= {\mathfrak {w}} K( \frac{\cdot }{{\mathfrak {w}}})\).
If we have
then for sufficiently small \(\varepsilon , \kappa >0\), from (4.130),
From (1.20), (1.22), and (2.3)
From (3.20) clearly we have
As in (3.23) we derive
Proposition 10
Recall \({\mathfrak {w}}_{\varrho ,\ss }\) in (2.3). Assume the same assumptions in Proposition 6. In addition we assume (4.135), and the conditions of \(\varrho \) and \(\ss \) in (2.3). Then
where
Proposition 11
Assume the same assumptions of Proposition 10. We denote
Let \(p<3\). Then
with
In the proof of propositions, for simplicity, we often use \(\Vert \ \cdot \ \Vert _\infty \) for \(\Vert \ \cdot \ \Vert _{L^\infty _{t,x,v}}\), \(\Vert \ \cdot \ \Vert _{L^\infty _{x,v}}\) or \(\Vert \ \cdot \ \Vert _{L^\infty _{x}}\) if there would be no confusion.
Proof of Proposition 10
We define backward exit time and position as
Since the characteristics for (4.132) are given by \((x- \frac{t-s}{\varepsilon }v, v)\), we have, for \(0 \le t-s < t_{{\mathbf {b}}}(x,v)\),
Here \(e^{-\int ^t_s \frac{\nu _{\ss } }{\varepsilon ^2 \kappa } }=e^{-\int ^t_s \frac{\nu _{\ss } (\tau , x- \frac{t-\tau }{\varepsilon }v, v )}{\varepsilon ^2 \kappa } \mathrm {d}\tau }\). We regard \((x_1- \frac{t-s}{\varepsilon } v_1, x_2- \frac{t-s}{\varepsilon } v_2) \in {\mathbb {R}}^2\) belongs to \({\mathbb {T}}^2\) without redefining them in \([- \pi , \pi ]^2\).
Now we represent h using (4.146) and (4.133) as
Since the integrand of (4.148) reads on the boundary, using the boundary condition (4.133) and (4.146) again, we represent it as
where \(r= - \frac{\varepsilon }{\delta } {\mathfrak {w}} (1- P_{\gamma _+}) f_2\) and \(e^{- \int ^{t-t_{{\mathbf {b}}}(x,v)}_0 \frac{\nu _{\ss } }{\varepsilon ^2 \kappa } }:= e^{-\int ^{t-t_{{\mathbf {b}}}(x,v)}_0 \frac{1}{\varepsilon ^2 \kappa } {\nu _{\ss } (\tau , x- \frac{ t_{{\mathbf {b}}}(x,v) }{\varepsilon }v - \frac{t-t_{{\mathbf {b}}}(x,v) -s }{\varepsilon } {\mathfrak {v}} , {\mathfrak {v}} )}\mathrm {d}\tau }\).
Note that, from (3.4), (3.5), (3.57), (3.58), and (4.137),
We derive a preliminary estimate as
We note that \(|h(s,x- \frac{t-s}{\varepsilon }, v_*)|\) has the same upper bound. Then we bound (4.152) by a summation of (4.151) and
and importantly
We consider (4.155). We decompose the integration of \(\tau \in [0,s] = [0, s- o(1)\varepsilon ^2 \kappa ] \cup [s- o(1)\varepsilon ^2 \kappa , s]\). The contribution of \(\int ^s_{s-o(1) \varepsilon ^2 \kappa } \cdots \mathrm {d}\tau \) is bounded as
For the rest of term we decompose \({\mathbf {k}}_{{\mathfrak {w}} }(v_*,v_{**})= {\mathbf {k}}_{{\mathfrak {w}},N}(v_*,v_{**}) + \{ {\mathbf {k}}_{{\mathfrak {w}} }(v_*,v_{**})- {\mathbf {k}}_{{\mathfrak {w}},N}(v_*,v_{**})\}\) where \({\mathbf {k}}_{{\mathfrak {w}},N}(v_*,v_{**}):={\mathbf {k}}_{{\mathfrak {w}}}(v_*,v_{**})\) \( \times {\mathbf {1}}_{ \frac{1}{N}<|v_*- v_{**}|< N \ \& \ |v_*|< N }. \) From (4.139), \(\int _{{\mathbb {R}}^3} {\mathbf {k}}_{{\mathfrak {w}}}(v_*,v_{**}) {\mathbf {1}}_{|v_*|\ge N} \mathrm {d}v_{**} \lesssim 1/N\). Also from the fact \({\mathbf {k}}_{{\mathfrak {w}}}(v_*,v_{**}) \le \frac{e^{-C |v_*- v_{**}|^2}}{|v_*- v_{**}|} \in L^1(\{ v_*- v_{**} \in {\mathbb {R}}^3 \})\), \(\sup _{v_*}\int _{{\mathbb {R}}^3} {\mathbf {k}}_{{\mathfrak {w}}}(v_*,v_{**}) \{{\mathbf {1}}_{\frac{1}{N}\ge |v_*- v_{**}| } + {\mathbf {1}}_{ |v_*- v_{**}|\ge N } \}\mathrm {d}v_{**} \downarrow 0\) as \(N \rightarrow \infty \). Hence for \(N\gg 1\)
where we have used the fact \(\sup _{x}{\mathbf {k}}_{{\mathfrak {w}} } (v_*, v_{**}) {\mathfrak {w}}_{\varrho , \ss }(v_{**}) \le C_N<\infty \) when \(\frac{1}{N}<|v_*- v_{**}|< N\) and \(|v_*|< N\) (then \(|v_{**}|< 2N\)).
Now we decompose \(f_R={\mathbf {P}}f_R+ ({\mathbf {I}} -{\mathbf {P}})f_R\). We first take integrations (4.157) over \(v_{*}\) and \(v_{**}\) and use Holder’s inequality with \(p=6, p=2\) in \(1/p+ 1/p^\prime =1\) for \({\mathbf {P}}f_R , ({\mathbf {I}} -{\mathbf {P}})f_R\) respectively to derive
Now we consider a map
We note that this mapping is not one-to-one and the image can cover \(\Omega \) at most N times. Therefore we have
Therefore we conclude that
where we have used (A.1) the Sobolev embedding in 1D at the last line.
Now we consider (4.153) and (4.154). We decompose \(s \in [0, t-t_{{\mathbf {b}}}] = [0, t-t_{{\mathbf {b}}}- o(1) \varepsilon ^2 \kappa ] \cup [t-t_{{\mathbf {b}}}- o(1) \varepsilon ^2 \kappa , t-t_{{\mathbf {b}}}]\). The contribution of \(\int ^{t-t_{{\mathbf {b}}}}_{t-t_{{\mathbf {b}}}- o(1) \varepsilon ^2 \kappa } \cdots \) is bounded as
For \(s \in [0, t-t_{{\mathbf {b}}}- o(1) \varepsilon ^2 \kappa ]\) we consider a map as (4.159)
Following the argument to have (4.158) we bound
In conclusion, we bound |h(t, x, v)| by (4.151), (4.160), (4.161), (4.163) and conclude (4.140) by choosing small enough o(1) in (4.160) and (4.161). \(\square \)
Proof of Proposition 11
Since many parts of the proof are overlapped with the proof of Proposition 10 we only pin point the differences. An equation for \({\mathfrak {w}}^\prime \partial _t f_R\) takes the similar form of (4.132) and (4.133). We can read (3.3) for
as (4.132) and (4.133) replacing
where \(r _{\gamma _+}(g) \) has been defined in (3.9).
We have the same equality of (4.147), (4.148) with (4.149) for h of (4.164) but replacing \(S_h\) and r of (4.165). Note that \(\frac{{\mathfrak {w}}^\prime (x,v)}{{\mathfrak {w}} (x,v)}\lesssim e^{- (\varrho - \varrho ^\prime )|v|^2}\) and hence \(\Big |\partial _t \Big ( \frac{( \partial _t + \varepsilon ^{-1} v\cdot \nabla _x) \sqrt{\mu }}{\sqrt{\mu }} \Big ) \frac{{\mathfrak {w}}^\prime }{{\mathfrak {w}} }\Big | \lesssim (3.13) \) from (3.13). From (3.36), (3.6), (3.7), (3.57), (3.12), (3.13), (3.58), and (4.137), we bound terms of (4.165)
Then as in (4.151)-(4.155) we derive a preliminary estimate as
As (4.154) and (4.155), we bound (4.169) by a summation of (4.168) and
Then we follow the argument of (4.158)-(4.160) to derive that, for \(p<3\),
Now we use the Young’s inequality for temporal convolution twice to derive that, for \(p<3\),
As in (4.163), for (4.172) we use (4.162) to derive that, for \(p<3\),
where we have used the Young’s inequality for temporal convolution.
In conclusion, we bound \(\Vert h\Vert _{L^2_t L^\infty _{x,v}}\) by \(\Vert (4.168)\Vert _{L^2_t L^\infty _{x,v}},\) (4.176), (4.173), (4.177) and conclude (4.143) by choosing small enough o(1) in (4.173). \(\square \)
4.5 Proof of Theorem 2
An existence of a unique global solution F for each \(\varepsilon >0\) can be found in [12,13,14,15]. Thereby we only focus on the (a priori) estimates (2.13).
Step 1. Define \(T_*>0\) as
where \(d_2, d_{2,t}, d_6, d_3, d_{3,t}, d_\infty , d_{\infty , t}\) are defined in (4.3), (4.5), (4.45), (4.73), (4.75), (4.141) and (4.144).
From (2.10) and (4.178) we read all the estimates of Proposition 7, Proposition 8, Proposition 10, Proposition 9, and Proposition 11 in terms of \({\mathcal {E}} (t)\) and \({\mathcal {D}} (t)\) as follows.
From (4.140), (4.178), and (2.10)
Now applying (4.179) to (4.44) we derive that
From (4.179), (4.180), and (4.178) and (2.10) we conclude that
From (4.72), (4.181), (4.178) and (2.10)
Using (4.181) and (2.9), from (4.74) and (4.143), we deduce that, for \(p<3\) and \(\varrho ^\prime <\varrho \),
Step 2. Using the estimates of the previous step we will close the estimate ultimately in the basic energy estimates (4.2) and (4.4) via the Gronwall’s inequality. We note that from (2.9) the multipliers of \(\int ^t_0 \Vert Pf_R(s) \Vert _{L^2_x}^2 \mathrm {d}s\) in (4.2) and \(\int ^t_0 \Vert P \partial _t f_R(s) \Vert _{L^2_x}^2 \mathrm {d}s\) in (4.4) are bounded above by
where we have used (2.11).
We can check that the multiplier of \( \Vert \varepsilon ^{-1}\kappa ^{-1/2} \sqrt{\nu } ({\mathbf {I}} -{\mathbf {P}}) f_R \Vert _{L^2_{t,x,v}}^2\) in (4.4) is bounded as, from (2.10) and (4.181),
Applying (4.181), (4.182), (4.183), (4.185) to (4.2)+o(1)(4.4), using the above bound and (2.11), and collecting the terms, we derive that
Under the assumption of
we derive that, for some constants \({\mathfrak {C}}_1>0\) and \({\mathfrak {C}}_2>0\),
Note that among others the last condition condition is the strongest in (4.187), which can be read as, from \(\delta =\sqrt{\varepsilon }\) of (2.11),
Applying the Gronwall’s inequality to (4.188) (we may redefine \({\mathcal {E}}(t)\) as \(\sup _{0 \le s \le t}{\mathcal {E}}(s)\) if necessary), we derive that
Applying this estimate to the last term of (4.188) and using the fact \({\mathfrak {P}}^\prime < {\mathfrak {P}}\) we derive that, after redefining \({\mathfrak {C}}_1\) if necessary,
under the assumptions of (2.10), (4.178), and (4.187).
Step 3. Now we find out the ranges of \(\delta , \kappa , \varepsilon \) satisfying the assumptions of (4.178) and (4.187). From (2.12) and (4.190), if we choose \(\delta \) as
then we can achieve (4.189) and hence all conditions of (4.187). Clearly (2.11) and (2.12) ensure (4.191).
Now from (4.190) and (2.11) we derive (2.13), which implies
These imply \(\min \{d_2, d_{2,t}, d_6, d_3, d_{3,t}, d_\infty , d_{\infty , t}\} \ge \frac{1}{4}\) and \(\frac{\delta \varepsilon ^{1/2}}{\kappa } \sqrt{{\mathcal {D}}(t)}\ll 1\) from (4.3), (4.5), (4.45), (4.73), (4.75), (4.141) and (4.144).
Then by the standard continuation argument we can verify all assumptions (4.178) up to \(t\le T\) and \(T=T_*\). The estimate (2.13) follows easily.
5 Navier-Stokes Approximations of the Euler Equations
In this section we prove Theorem 3. The proof of the theorem relies on the integral representation of the solution to the Navier-Stokes equations using the Green’s function for the Stokes problem in the same spirit of [47].
5.1 Elliptic Estimates and Nonlinear Estimates
In this section, we prove the estimates of the solutions of incompressible Navier-Stokes equations in large Reynolds numbers with the no slip boundary condition satisfying (1.13)-(1.15) based on recent Green’s function approach using the vorticity formulation of (2.16)-(2.18) applied to the inviscid limit problem [38, 44, 47, 54]. An advantage of working with analytic function spaces is the Cauchy estimates useful for recovery of the loss of derivatives. We recall the spaces, norms, and terminology we have defined in Section 2.
Lemma 7
([47, 54], Embeddings and Cauchy estimates) The following holds
-
(1)
\({\mathfrak {B}}^{\lambda , \kappa t} \subset {\mathfrak {L}}^{1,\lambda }\) and \({\mathfrak {B}}^{\lambda , \kappa } \subset {\mathfrak {L}}^{1,\lambda }\).
-
(2)
\(\Vert g_1 g_2\Vert _{*,\lambda }\lesssim \Vert g_1\Vert _{\infty ,\lambda } \Vert g_2\Vert _{*,\lambda }\).
-
(3)
\( \sum _{|\beta |=1} \Vert D^\beta g\Vert _{*,\lambda } \lesssim \frac{\Vert g\Vert _{*,{\tilde{\lambda }}}}{{{\tilde{\lambda }}} -\lambda }\), for any \(0< \lambda <{{\tilde{\lambda }}}\).
For (2) and (3), \(\Vert \cdot \Vert _{*,\lambda }\) can be either \(\Vert \cdot \Vert _{\infty ,\lambda ,\kappa }\) or \( \Vert \cdot \Vert _{\infty ,\lambda ,\kappa t}\) or \(\Vert \cdot \Vert _{\infty , \lambda , 0}\) or \( \Vert \cdot \Vert _{1,\lambda }\).
Lemma 8
([47, 54], Elliptic estimates) Let \(\phi \) be the solution of \(-\Delta \phi = \omega \) with the zero Dirichlet boundary condition, and let \(u=\nabla \times \phi \). Then
Proof
Here we only sketch the proofs. For full justification we refer to Proposition 2.3 in [47] for 2D and Section 4 of [54] for 3D and the proofs therein. From \((|\xi |^2 - \partial _z^2 )\phi _\xi = \omega _\xi \) and \(\phi _\xi (0)=0\) we write
The first two estimates of (5.1) can be easily derived from this explicit form. For the third estimate of (5.1), we write \(u_1 = \partial _2 (-\Delta )^{-1}\omega _3 - \partial _3 (-\Delta )^{-1} \omega _2\) and \(\partial _3u_1 =\partial _3 \partial _2 (-\Delta )^{-1}\omega _3 -\partial _3 \partial _3 (-\Delta )^{-1} \omega _2\). Then the third estimate of (5.1) follows from the identity
Next we prove the last estimate. Note that
From (5.2) we read that for \(i=1,2\)
From the identity and estimate above we conclude the last bound of (5.1). \(\square \)
As a consequence of Lemma 8, we have the following nonlinear estimates.
Lemma 9
([47, 54]) Let u and \({{\tilde{u}}}\) be the velocity field associated with \(\omega = \nabla _x \times u\) and \({{\tilde{\omega }}}= \nabla _x \times {{\tilde{u}}}\) respectively. Then
Moreover
where \(\Vert \cdot \Vert _{*, \lambda }\) can be either \(\Vert \cdot \Vert _{\infty , \lambda , \kappa }\) or \(\Vert \cdot \Vert _{\infty , \lambda , \kappa t}\).
Furthermore
where \(\Vert (1+ |\nabla _h|^k) g \Vert _* = \sum _{\ell =0}^k \Vert \nabla _h^\ell g \Vert _*\).
Proof
Again we refer to Proposition 2.3 in [47] for 2D and Section 4 of [54] for the full justification. The bounds (5.3) and (5.4) directly follow from Lemma 8. The proof of the first estimate of (5.5) is an outcome of applying (5.1) to an easy bound
For the second estimate of (5.5) it suffices to prove the bound for \(\omega _h \cdot \nabla _h {{\tilde{u}}}_3\). From \(|\zeta (z)(1+ \phi _\kappa (z) )|\lesssim 1\) or \(|\zeta (z)(1+ \phi _\kappa (z) + \phi _{\kappa t} (z))|\lesssim 1\),
Then we use the last bound of (5.1) to finish the proof. \(\square \)
We finally record the crucial estimate of nonlinear forcing terms \(N=-u\cdot \nabla \omega + \omega \cdot \nabla u\), as an outcome of Lemma 9, that will be also crucially used to control \(B= [\partial _{x_3} (-\Delta )^{-1} (-u \cdot \nabla \omega + \omega \cdot \nabla u ) ] \, |_{x_3=0}\) in the vorticity formulation (2.16) and (2.18).
Lemma 10
([47, 54], Nonlinear estimate) Let \(\lambda \in (0,\lambda _0-\gamma s)\) be given. We have the following:
For \([[ \ \cdot \ ]]_{*,\lambda }\) to be either \([[ \ \cdot \ ]]_{\infty , \lambda , \kappa }\) or \([[ \ \cdot \ ]]_{\infty , \lambda , \kappa t}\),
The proof relies on Lemma 9. We refer to Lemma 4.2 and Lemma 4.5 in [54] for the detailed proof.
5.2 Green’s Function and Integral Representation for the Vorticity Formulation
By taking the Fourier transform of (2.16)-(2.18) in \(x_h \in {\mathbb {T}}^2\), we obtain
with \(\omega _\xi |_{t=0} = {\omega _0}_\xi \) for \(\xi \in {{\mathbb {Z}}^2}\). Here
and
Here \((-\Delta _\xi )^{-1}\) denotes the inverse of \(-\Delta _\xi \) with the zero Dirichlet boundary condition at \(x_3=0\).
We give the integral representation and present key estimates on Green’s function for the Stokes problem. As shown in [47, 54], letting \(G_{\xi }(t,x_3,y)\) be the Green’s function for (5.10)-(5.11), the solution can be represented by the integral formula via Duhamel’s principle:
where
with \(G_{\xi h}\) of (5.19) and \(G_{\xi 3}\) of (5.22): for \(G_{\xi *}\) can be either \(G_{\xi h}\) or \(G_{\xi 3}\)
The following estimates and properties for \(G_{\xi }\) will be useful to show the propagation of analytic norms of \(\omega \), \(\partial _t\omega \) and \(\partial _t^2\omega \).
Lemma 11
-
(1)
(Bounds on \(G_{\xi h}\)) The Green’s function \(G_{\xi h}\) for the Stokes problem (5.16) and (5.17) is given by
$$\begin{aligned} G_{\xi h} = {\tilde{H}}_\xi + R_\xi , \end{aligned}$$(5.19)where \({\tilde{H}}_\xi \) is the one dimensional Heat kernel in the half-space with the homogeneous Neumann boundary condition which takes the form of
$$\begin{aligned} {\tilde{H}}_\xi (t,x_3,y)= & {} H_\xi (t,x_3-y) + H_\xi (t,x_3+y) \nonumber \\= & {} \frac{1}{\sqrt{\kappa \eta _0 t}} \bigg ( e^{- \frac{|x_3-y|^2}{4 \kappa \eta _0 t}} + e^{- \frac{|x_3+y|^2}{4 \kappa \eta _0 t}} \bigg ) e^{- \kappa \eta _0 |\xi |^2 t}, \end{aligned}$$(5.20)and the residual kernel \(R_\xi \) due to the boundary condition satisfies
$$\begin{aligned} |\partial _{x_3}^k R_\xi (t,x_3, y)| \lesssim b^{k+1} e^{- \theta _0 b (x_3 + y)} + \frac{1}{(\kappa \eta _0 t)^{(k+1)/2}} e^{- \theta _0 \frac{|x_3+y|^2}{\kappa \eta _0 t}} e^{- \frac{\kappa \eta _0 |\xi |^2 t}{8}} ,\nonumber \\ \end{aligned}$$(5.21)with \(b= |\xi | + \frac{1}{\sqrt{\kappa \eta _0}}\) and \(R_\xi (t,x_3,y) = R_\xi (t, x_3 + y)\).
-
(2)
(Formula of \(G_{\xi 3}\)) The Green’s function \(G_{\xi 3}\) for the Stokes problem (5.16) and (5.18) is given by one dimensional Heat kernel in the half-space with the homogeneous Dirichlet boundary condition as
$$\begin{aligned} G_{\xi 3} (t,x_3,y)= & {} H_\xi (t,x_3-y) -H_\xi (t,x_3+y)\nonumber \\= & {} \frac{1}{\sqrt{\kappa \eta _0 t}} \bigg ( e^{- \frac{|x_3-y|^2}{4 \kappa \eta _0 t}} - e^{- \frac{|x_3+y|^2}{4 \kappa \eta _0 t}} \bigg ) e^{- \kappa \eta _0 |\xi |^2 t}. \end{aligned}$$(5.22) -
(3)
(Complex extension) The Green’s function \(G_{\xi }\) has a natural extension to the complex domain \({\mathcal {H}}_\lambda \) for small \(\lambda >0\) with similar bounds in terms of \(\text {Re}\, y\) and \(\text {Re}\, z\) (cf. (3.16) in [47]). The solution \(\omega _\xi \) to (5.10)-(5.11) in \({\mathcal {H}}_\lambda \) has a similar representation: for any \(z\in {\mathcal {H}}_\lambda \), let \(\sigma \) be the positive constant so that \(z\in \partial {\mathcal {H}}_\lambda \), then \(\omega _\xi \) satisfies
$$\begin{aligned} \begin{aligned} \omega _\xi (t,z) =&{\int _{\partial {\mathcal {H}}_\lambda } G_{\xi } (t,z, y) \omega _{0\xi } (y) \mathrm {d}y} + {\int ^t_0\int _{\partial {\mathcal {H}}_\lambda } G_{\xi } (t-s, z, y) N_\xi (s,y ) \mathrm {d}y \mathrm {d}s}\\&- {\int ^t_0 G_{\xi } (t-s, z, 0) (B_\xi (s), 0) \mathrm {d}s} . \end{aligned} \end{aligned}$$
The proof of Lemma 11 can be found in Proposition 3.3 and Section 3.3 of [47]. The next lemma concerns the convolution estimates.
Lemma 12
Let \(T>0\) be given. Recall the norms defined in Section 2. For any \(0\le s < t\le T\) and \(k\ge 0\), there exists a constant \(C_T>0\) so that the following estimates hold: for \(G_{\xi *}\) can be either \(G_{\xi h}\) or \(G_{\xi 3}\)
-
(1)
(\({\mathcal {L}}^1_\lambda \) estimates)
$$\begin{aligned} \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j \int _0^\infty G_{\xi *}(t, z, y ) g_\xi (y) \mathrm {d}y \right\| _{{\mathcal {L}}^1_\lambda } \le C_T \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j g_\xi \right\| _{{\mathcal {L}}^1_\lambda }, \end{aligned}$$(5.23)$$\begin{aligned} \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j \int _0^\infty G_{\xi *}(t-s,z, y ) g_\xi (y) \mathrm {d}y \right\| _{{\mathcal {L}}^1_\lambda } \le C_T \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j g_\xi \right\| _{{\mathcal {L}}^1_\lambda }. \end{aligned}$$(5.24) -
(2)
(\({\mathcal {L}}^\infty _{\lambda ,\kappa t}\) estimates)
$$\begin{aligned}&\sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j \int _0^\infty G_{\xi *} (t, z, y ) g_\xi (y) \mathrm {d}y \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa t}} \nonumber \\&\quad \le C_T \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j g_\xi \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa }}, \end{aligned}$$(5.25)$$\begin{aligned}&\sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j \int _0^\infty G_{\xi *}(t-s, z, y ) g_\xi (y) \mathrm {d}y \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa t}} \nonumber \\&\quad \le C_T \sum _{j=0}^k \sqrt{\frac{t}{s}} \left\| (\zeta (z)\partial _{z})^j g_\xi \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa s}}. \end{aligned}$$(5.26) -
(3)
(\({\mathcal {L}}^\infty _{\lambda ,\kappa }\) estimates) For either \(\kappa =0\) or \(\kappa >0\)
$$\begin{aligned} \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j \int _0^\infty G_{\xi *}(t,z, y ) g_\xi (y) \mathrm {d}y \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa }} \le C_T \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j g_\xi \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa }}, \end{aligned}$$(5.27)$$\begin{aligned} \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j \int _0^\infty G_{\xi *}(t-s, z, y ) g_\xi (y) \mathrm {d}y \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa }} \le C_T \sum _{j=0}^k \left\| (\zeta (z)\partial _{z})^j g_\xi \right\| _{{\mathcal {L}}^\infty _{\lambda ,\kappa }}. \end{aligned}$$(5.28)
Proof
We only give a proof for \(G_{\xi h}\) since \(G_{\xi 3}\) can be handled easier than the other. The proof of (1) and (2) can be found in Propositions 3.7 and 3.8 of [47]. Here we present the detail for (3), the second inequality. We consider real values \(y,z\in {\mathbb {R}}_+\) only as the complex extension follows similarly (cf. (3) in Lemma 11). Note that in view of (5.19), (5.20) and (5.21), it suffices to show
where \(R(t,z,y)= b e^{- b (y+z)}\) and \(H(t,z,y) = \frac{1}{\sqrt{\kappa t}} e^{-\frac{|y-z|^2}{M\kappa t}}\) for some \(M>0\). We start with (5.29). Let \(k=0\) first. First note that
since \(\phi _\kappa \) is a decreasing function. The last integral is uniformly finite for all \(|\xi |\) and \(\kappa \). Hence,
For \({{\bar{\alpha }}}>0\), if \(\kappa < \frac{1}{4 \eta _0 {{\bar{\alpha }}}^2}\) then
where the last bound follows from the fact that
For \(k\ge 1\), since \(|\zeta (z)\partial _z R |\lesssim be^{-\frac{b z}{2}}\), the derivative estimates follow analogously. Therefore, (5.29) holds true.
We move onto (5.30). Let \(k=0\) first. Note that, for \(0 \le s < t \le T\) and \(\kappa \lesssim 1\)
and thus
For the last integral, we divide the integral into two: \(\int _0^\infty = \int _0^{\frac{z}{2}} + \int _{\frac{z}{2}}^\infty \). For the latter, since \(\phi _\kappa \) is decreasing and the kernel is in \(L^1_y\), we deduce
For \( \int _0^{\frac{z}{2}} \mathrm {d}y\), note \(|y-z|\ge \frac{z}{2}\) and \(1+\phi _\kappa (y) \le 1+\phi _\kappa (0)\) for \(y\in (0,\frac{z}{2})\). Hence
Then
A similar argument as in (5.31) shows that \(\frac{1+\phi _\kappa (0)}{1+\phi _\kappa (z)}e^{-\frac{|z|^2}{16M\kappa (t-s)}}\) is uniformly finite in \(\kappa \). This shows (5.30) for \(k=0\).
For the derivative estimate, by splitting the integral into two parts and using \(\partial _z H (t,z,y)= - \partial _y H(t,z,y)\), we rewrite
For the first integral, since \( |y-z|\ge \frac{z}{2}\) for \(y\in (0,\frac{z}{2})\),
Hence, by the same argument as in \(k=0\) leads to the desired bound. For the second term,
which leads to the desired bound. For the last integral, note that \(\zeta (z) \le 2 \zeta (y)\) for \(y\ge \frac{z}{2}\). Therefore the corresponding integral can be treated in the same way as in \(k=0\) with \(g_\xi (y)\) replaced by \(\zeta (y)\partial _yg_\xi (y)\). This shows (5.30) for \(k=1\). Other \(k\ge 2\) can be estimated analogously. \(\square \)
The next result concerns the estimates for the trace kernel.
Lemma 13
Let \(a_\xi (s)= [\partial _{z}(- \Delta _\xi )^{-1} g_\xi ] \, |_{z=0}\). Then for any \(0\le s<t\le T\) and \(k\ge 0\), we have the following
Proof
Note that from (5.19), (5.20) and (5.21), the conormal derivatives \((\zeta (z)\partial _z)^j\) of \(G_{\xi h}(t-s,z,0)\) enjoy the same bounds as \(G_{\xi h}(t-s,z,0)\): for some small constant \(c_0\),
Therefore, it suffices to show the bounds for \(k=0\). We first recall the representation formula for \(a_\xi \) (cf. (4.29) of [38] or (4.2) of [54]):
from which we have \(\Vert a_\xi \Vert _{{\mathcal {L}}^\infty _\lambda } \lesssim \left\| g_\xi \right\| _{{\mathcal {L}}^1_\lambda }\). Since the above upper bound of \(G(t-s,z,0)\) is integrable in z, (5.33) follows. To show (5.34), we compute \(\Vert G_{\xi h}(t-s,z, 0 ) \Vert _{{\mathcal {L}}^\infty _{\lambda ,\kappa }}\):
It is a routine to check that both supremum norms are uniformly bounded in \(\kappa \) and \(|\xi |\). Therefore (5.34) is obtained. \(\square \)
5.3 Proof of Theorem 3
Our goal is to show that \(\omega (t)\) indeed belongs to \(C^1([0,T];{\mathfrak {B}}^{\lambda , \kappa })\) without the initial layer under the compatibility condition (2.34), and that \(\partial _t^2\omega \) in \({\mathfrak {B}}^{\lambda , \kappa t}\) with the initial layer. The existence of \(\omega (t)\) in \(C^1([0,T];{\mathfrak {B}}^{\lambda , \kappa t})\) under the assumption of Theorem 3 can be proved by following the argument of [47] and [54]. For the 2D case, Theorem 1.1 of [47] indeed ensures the existence of \(\omega (t)\) in \(C^1([0,T];{\mathfrak {B}}^{\lambda , \kappa t})\) under the assumption of Theorem 3. Such a result follows from Lemma 7, Lemma 11, Lemma 12, Lemma 8, Lemma 9. A 3D result can be obtained analogously. Hence, it suffices to show the propagation of the analytic norms in (2.35).
Step 1: Propagation of analytic norms for \(\omega \). It is convenient to define
The estimation of \(\omega \) follows from the nonlinear iteration using the representation formula (5.14).
The estimates for the \(L^1\)-based norm \({\left| \left| \left| \omega (t) \right| \right| \right| }_1\) are already available in Section 5 of [54] (for 2D see Section 4.1 of [47]): From (5.23), (5.24) and (5.33), we have that for \(k=0,1,2\)
For \(k=1\), after summing up over \(\xi \in {\mathbb {Z}}^2\), we deduce that
Using (5.6), (5.7), and the definition of \({\left| \left| \left| \ \cdot \ \right| \right| \right| }_s\) in (5.36) we derive that
The second order derivatives can be treated similarly except for the contributions of N for which we apply the analyticity recovery estimate using (3) of Lemma 7 while other terms are estimated in the same way. More precisely, we have
while we choose \({{\tilde{\lambda }}}=\frac{\lambda +\lambda _0-\gamma _0 s}{2}\) in particular. We note that still \({\tilde{\lambda }}< \lambda _0 - \gamma _0 s\) if \(\lambda < \lambda _0 -\gamma _0 s\) and hence from (5.6) and (5.7)
Therefore we derive that for \(t< \frac{\lambda _0}{2\gamma _0}\) and \(\lambda < \lambda _0 - \gamma _0 t\)
Therefore, we conclude that, from (5.37) with (5.38), and (5.40)
The propagation of the boundary layer norm \({\left| \left| \left| \omega (t) \right| \right| \right| }_{\infty ,\kappa }\) can be shown analogously using \({\mathcal {L}}^\infty _{\lambda , \kappa }\) estimates of Lemma 12 and Lemma 13: For \(k=0,1,2\) and \(\kappa >0\) for \(i=1,2\) and \(\kappa =0\) for \(i=3\) we have
Let \(k=1\). After summing up over \(\xi \in {\mathbb {Z}}\) and \(i=1,2\) (with \(\kappa >0\)) and \(i=3\) (with \(\kappa =0\)) , we deduce that
Using the definition of \({\left| \left| \left| \ \cdot \ \right| \right| \right| }_s\) in (5.36), and applying Lemma 10 with (5.6), (5.8), and (5.9), we derive
Therefore we derive that
Now we control the second order derivatives similarly except for the N. As in (5.39) we use the analyticity recovery estimate using Lemma 7
while again we choose \({{\tilde{\lambda }}}=\frac{\lambda +\lambda _0-\gamma _0 s}{2}\) in particular. We note that still \({\tilde{\lambda }}< \lambda _0 - \gamma _0 s\) if \(\lambda < \lambda _0 -\gamma _0 s\) and hence from (5.8) and (5.9)
Therefore we derive that for \(t< \frac{\lambda _0}{2\gamma _0}\) and \(\lambda < \lambda _0 - \gamma _0 t\)
Therefore we conclude that, from (5.42) and (5.44),
In conclusion, from (5.41), (5.45), and by a standard continuity argument we obtain for sufficiently large \(\gamma _0\)
Step 2: Propagation of analytic norms for \(\partial _t\omega \). The continuity of \(\omega (t)\) in t follows from the mild solution form (5.14) of \(\omega _\xi (t)\). We claim that \(\omega (t) \in C^1([0,T]; {\mathfrak {B}}^{\lambda ,\kappa })\) and moreover \({\left| \left| \left| \partial _t\omega (t) \right| \right| \right| }_{t} \) is bounded. To this end, we first derive the mild form of \(\partial _t\omega _\xi \) from (5.14):
where we recall \(\partial _t \omega _0\) in (2.32). To justify this formula, we first recall (5.16)-(5.18). We start with the horizontal part of the formula (5.47) for \(\partial _t\omega _{\xi ,h}\). From Lemma 11, \(G_{\xi h} (t,x_3,y) = H_\xi (t, x_3-y) +H_\xi (t,x_3+y)+ R_\xi (t,x_3+y)\). Then by using the fact that \(H^\prime _\xi (t, \cdot )\) is an odd function, we see \( \partial _{x_3} G_{\xi h} (t,x_3,y)|_{x_3=0} = R^\prime _\xi (t, y)\). Now we read (5.17) as
where we have used that \(H_\xi (t, \cdot )\) is an even function for the second relation. On the other hand, since we also have \( \partial _{y_3} G_{\xi h} (t,x_3,y)|_{y=0} = R^\prime _\xi (t, x_3)\), we deduce that
It is straightforward to see \(\Delta _\xi G_{\xi h} = \partial _{x_3}^2G_{\xi h} - |\xi |^2G_{\xi h} = \partial _y^2 G_{\xi h} - |\xi |^2 G_{\xi h}.\)
We now take \(\partial _t\) of (5.14):
and
Therefore we obtain
Next we show that the first line in the right-hand side is 0. From (5.49)
and hence the first line of (5.50) reads
which is zero due to the first compatibility condition of (2.34). Recalling \(\partial _t \omega _0\) in (2.32), the formula (5.47) for \(\partial _t\omega _{\xi ,h}\) has been established. We may follow the same procedure to verify the vertical part of the formula (5.47) for \(\partial _t\omega _{\xi ,3}\) by noting that the second compatibility condition of (2.34) removes the term \(- \kappa \eta _0 \partial _y G_{\xi 3} (t,x_3,0)\omega _{0\xi ,3}(0)\) which would create the initial layer otherwise because \( \partial _y G_{\xi 3} (t,x_3,0)\) does not vanish.
We may now repeat Step 1 for \(\partial _t\omega \) using the representation formula (5.47). The estimates are obtained in the same fashion. For the nonlinear terms, since \(\partial _t N = - u \cdot \nabla \partial _t \omega - \partial _t u\cdot \nabla \omega + \omega \cdot \nabla \partial _t u+\partial _t \omega \cdot \nabla u \), the structure of \(\partial _t N\) with respect to \(\partial _t\omega \) is consistent with the one of N with respect to \(\omega \) and we can use the bilinear estimates (5.3) and (5.4). In summary, one can derive that for \(t < \frac{\lambda _0}{2 \gamma _0}\)
which lead to the desired bounds for \(\partial _t\omega (t)\) by choosing sufficiently large \(\gamma _0\).
Step 3: Propagation of analytic norms for \(\partial _t^2\omega _\xi \). As a consequence of Step 2, \(\partial _t\omega _\xi (t,x_3)\) solves the following system
with \(\partial _t\omega _\xi |_{t=0} = \partial _t \omega _{0 \xi }\) for \(\xi \in {\mathbb {Z}}^2\) where \(\partial _t \omega _0\) is defined in (2.32). Then as done in Step 2, by using the properties of \(G_\xi \) and integration by parts and by the last compatibility condition of (2.34), we can derive the representation formula for \(\partial _t^2\omega \):
where we recall \(\partial _t^2\omega _0\) in (2.32). As we do not require higher order compatibility condition for the horizontal vorticity, a new term representing the initial-boundary layer emerges. We first examine \(G_\xi (t,z,0)\). Recall (5.35).
Similar to Lemma 13, we have for \(C_0<\infty \)
Hence an \({L}^1\)-based analytic norm is easily obtained as
Now we move to the \(L^\infty \)-based analytic norm bound. We compute \(\Vert G_\xi (t,z, 0 ) \Vert _{{\mathcal {L}}^\infty _{\lambda ,\kappa t}}\):
It is a routine to check that both supremum norms are uniformly bounded in \(\kappa \) and \(|\xi |\). Hence (5.58) shows that the kernel \(G_\xi (t,z,0)\) is well-behaved in \({\mathcal {L}}^1_\lambda \) and the initial-boundary layer analytic space \({\mathcal {L}}^\infty _{\lambda ,\kappa t}\). Then we run the same argument as in Step 2 but with \({\mathcal {L}}^\infty _{\lambda ,\kappa t}\) in place of \({\mathcal {L}}^\infty _{\lambda ,\kappa }\). Thanks to (5.58), the estimates of the first term in (5.57) are bounded by the initial norm (2.33):
Other three terms are estimated in the same way as in [47] or [54] and we arrive that
Finally combining (5.59) and (5.60) and then choosing sufficiently large \(\gamma _0\) we derive a desired estimate for \({\left| \left| \left| \partial _t^2 \omega (t) \right| \right| \right| }_t\) for \(t \in (0, \frac{\lambda _0}{2\gamma _0})\).
Altogether from (5.46), (5.52), (5.53), (5.59), and (5.60), we finish the proof of the estimate (2.35).
Step 4: Estimate (1), vorticity estimates. Both (2.36) and (2.37) are direct consequences of (2.35). To show (2.38), we first note that the boundedness of \(\omega (t)\) norms implies \(|\partial _{x_3}\omega _\xi (t,x_3)|\lesssim e^{-{\bar{\alpha }} x_3}e^{-\lambda |\xi |}\) for all \(|\xi |\) and \(x_3 \ge 1\) (away from the boundary). When \(x_3\le 1\), we draw on the equation (5.10) to rewrite \( \partial _{x_3}^2\omega _{\xi ,h} = \frac{1}{\kappa \eta _0}\{ \partial _t \omega _{\xi ,h} + \kappa \eta _0 |\xi |^2 \omega _{\xi ,h} - N_{\xi ,h}\}\) and the boundary condition (5.11):
We now appeal to \(|B_\xi (t)| \le \Vert N_\xi (t)\Vert _{{\mathcal {L}}^1_\lambda }\) and \(\sum _{0\le \ell \le 1} ( {\left| \left| \left| \partial _t^\ell \omega (t) \right| \right| \right| }_{\infty ,\kappa }+ {\left| \left| \left| \partial _t^\ell \omega (t) \right| \right| \right| }_{1}) <\infty \) to obtain that for all \(x_3\in {\mathbb {R}}_+\)
which proves (2.38) for \(\omega _h\) and \(\ell =0\). The remaining case can be estimated similarly. Near O(1) boundary, from (5.54) and (5.55), we derive
Together with \(\sum _{0\le \ell \le 1} {\left| \left| \left| \partial _t^\ell \omega (t) \right| \right| \right| }_{\infty ,\kappa }+ \sum _{0\le \ell \le 2} {\left| \left| \left| \partial _t^\ell \omega (t) \right| \right| \right| }_{1}<\infty \) we deduce (2.38) for \(\omega _h\) and \(\ell =1\). For \(\omega _3\) we use \(\nabla \cdot \omega =0\) to write \(\partial _3 \omega _3 = - \partial _1\omega _1- \partial _2\omega _2\). Now (2.38) for \(\omega _3\) follows from (2.36).
Step 5: Estimate (2), velocity estimates, except (2.42). From (5.2)
For \(|\beta |= |\beta _h| + \beta _3=1\) we bound (5.64) by \(e^{-\lambda |\xi |} \Vert \partial _t^\ell \omega (t) \Vert _{1,\lambda }\). Then from (2.35) we conclude (2.39).
For \(|\beta |\ge 2\) and \(\beta _3 \le 1\), we bound (5.64) by
where we have used \(|\xi ||y-z| + \frac{{\bar{\alpha }}}{2} \text {Re}\, y\ge \min (1, \frac{{\bar{\alpha }}}{2} ) x_3\) for \(|\xi |\ge 1\) and (2.35).
For \(\beta _3=2,3\) we use \(\partial _z^2 \partial _t^\ell \phi _\xi = |\xi |^2 \partial _t^\ell \phi _\xi + \partial _t^\ell \omega _\xi \). Then following the same argument of (5.65), we derive
Finally from (5.65) and (5.66) we conclude (2.40) and (2.41).
Step 6: Estimate (3), pressure estimates and (2.42). We next turn to the pressure. Taking the divergence to (1.13) and using (1.14), we deduce
We obtain the boundary condition of p by reading the third component of (1.13), and then using (1.14) and (1.15),
where \(\omega _1= \partial _2 u_3 - \partial _3 u_2\) and \(\omega _2 = - \partial _1 u_3 + \partial _3 u_1\).
In the Fourier side we read the problem as
A representation of \(p_\xi (t,x_3)\) is given by
which is valid for all \(\xi \ne 0\). When \(\xi =0\), by integrating (5.69) and by using the boundary conditions \(\partial _3 p_0(t,0)=0\), \(u(t,x_h,0)=0\) and the divergence free condition \(\nabla \cdot u=0\), we first obtain
where we have used the integration by parts and \(\nabla \cdot u=0\) at the last step.
Observe that \(\partial _3p_0\) decays exponentially in \(x_3\), and in particular \(\int _0^\infty | \partial _3 p_0(t,x_3) | \mathrm {d}x_3 <\infty \). The integration yields
Since \(p_0(t,0)\) is a free constant in \(x_3\), we fix \(p_0(t,x_3)\) by choosing
such that
The pressure p is then recovered by
where \( p_0 (t,x_3) \) and \(p_\xi (t,x_3)\) are given in (5.72) and (5.70).
Now the pressure estimate follows readily from the velocity and vorticity estimates. To show (2.43), we first note from (2.39) and (2.40) \(|p_0(t,x_3)|\lesssim |u_3(t,x)| \int _\Omega |\partial _3 u_3 (t,x)| \mathrm {d}x \lesssim 1\) and from Lemma 8
from which we deduce \(|p(t,x_h,x_3)|\lesssim 1\). The estimation of \(\partial _t p \) and \(\partial _t^2p\) follows analogously.
For the decay estimates (2.44), we start with \(\ell =0\) and \(\beta =0\). Due to our choice of \(p_0(t,x_3)\) in (5.72), using (2.39) and (2.40), we have the spatial decay for \(p_0(t,x_3)\):
For \(\xi \ne 0\), we use another estimate for \(|g_\xi |\) and Lemma 8
from which we deduce that \(| p_\xi (t,x_3) |\lesssim \kappa ^{-\frac{1}{2}} e^{-\min (1,\frac{{{\bar{\alpha }}}}{2}) x_3} \). Hence (2.44) holds for \(\ell =0\) and \(\beta =0\). For the pressure gradient estimate when \(|\beta |=1\), from (5.71) and (2.40) we first note
For \(\xi \ne 0\), by (5.74) it is easy to see that \(|\xi p_\xi (t,x_3) |\lesssim \kappa ^{-\frac{1}{2}} e^{-\min (1,\frac{{{\bar{\alpha }}}}{2}) x_3} \). Note that \(\partial _3 p_\xi (t,x_3)\) has a similar integral form as \(|\xi | p_\xi (t,x_3)\) and the estimate follows in the same way, which results in \( |\partial _3 p_\xi (t,x_3) |\lesssim \kappa ^{-\frac{1}{2}} e^{-\min (1,\frac{{{\bar{\alpha }}}}{2}) x_3}\). This finishes (2.44) for \(\ell =0\) and \(|\beta |=1\). The remaining cases for \(\ell =1\) and \(|\beta |=0,1\) can be treated in the same way.
For the decay estimate of \(\partial _t^2p\), we take into account the initial layer which occurs at \(\partial _t^2 \omega \) and \(\nabla \partial _t^2 u\). First using (2.39), (2.40) and (2.41) we have
while for \(|\xi |\ne 0\) we have
from which we deduce (2.45).
The last estimate for \(\partial _t^\ell u\) for \(\ell =1,2\) follows from the equation: \(\partial _t u = \kappa \eta _0 \Delta u - u \cdot \nabla u - \nabla p \) and \(\partial _t^2 u = \kappa \eta _0 \Delta \partial _t u - u \cdot \nabla \partial _t u - \partial _t u \cdot \nabla u - \nabla \partial _t p \).
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Acknowledgements
Part of this work was conducted while the authors were participating in the INdAM worksop “Recent advances in kinetic equations and applications” organized by Francesco Salvarani in Rome. We thank the institute and the organizer for its generous hospitality and support. JJ was supported in part by the NSF Grants DMS-1608494, DMS-2009458, and by the Simons Fellowship (Grant # 616364). CK was supported in part by National Science Foundation under Grant Nos. 1501031, 1900923, 2047681, and the Wisconsin Alumni Research Foundation.
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Appendix A. Sobolev Embedding in 1D
Appendix A. Sobolev Embedding in 1D
Often we have used a standard 1D embedding: For \(T>0\),
A proof is based on an equality:
For \(0< t \le T/2\),
For \( T/2<t \le T\), using
we derive that
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Jang, J., Kim, C. Incompressible Euler Limit from Boltzmann Equation with Diffuse Boundary Condition for Analytic Data. Ann. PDE 7, 22 (2021). https://doi.org/10.1007/s40818-021-00108-z
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DOI: https://doi.org/10.1007/s40818-021-00108-z