Incompressible Euler Limit from Boltzmann Equation with Diffuse Boundary Condition for Analytic Data

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

$$\begin{aligned} \textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}} \end{aligned}$$

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.

Appendix A. Sobolev Embedding in 1D

Often we have used a standard 1D embedding: For $T>0$,

$$\begin{aligned} |g(t)|^2 \lesssim _T \int _0^T |g(s)|^2 \mathrm {d}s + \int _0^T |g^\prime (s)|^2 \mathrm {d}s \ \ \text {for } t \in [0,T]. \end{aligned}$$

(A.1)

A proof is based on an equality:

$$\begin{aligned} |g(t)|^2= \frac{1}{T/2 }\int ^{t+T /2}_t \Big ( g(s) - \int ^s_t g^\prime (\tau ) \mathrm {d}\tau \Big )^2 \mathrm {d}s . \end{aligned}$$

For $0< t \le T/2$,

$$\begin{aligned} \begin{aligned} |g(t)|^2&\le \frac{1}{T/2}\int ^{t+T/2}_t \Big ( 2 |g(s)|^2 + 2\Big | \int ^s_t g^\prime (\tau ) \mathrm {d}\tau \Big |^2 \Big ) \mathrm {d}s\\&\le \frac{1}{T/2}\int ^{t+T/2}_t \Big ( 2 |g(s)|^2 + 2 |s-t| \int ^s_t |g^\prime (\tau )|^2 \mathrm {d}\tau \Big ) \mathrm {d}s\\&\le \frac{2}{T/2}\int ^{t+T/2}_t |g(s)|^2 \mathrm {d}s + \frac{2}{T/2} \int ^{t+ T/2}_t |s-t| \int ^s_t |g^\prime (\tau )|^2 \mathrm {d}\tau \mathrm {d}s\\&\le \frac{2}{T/2}\int ^{t+T/2}_t |g(s)|^2 \mathrm {d}s+ T\int ^{t+T/2}_t |g^\prime (s)|^2 \mathrm {d}s\\&\lesssim _T \int _0^T |g(s)|^2 \mathrm {d}s + \int _0^T |g^\prime (s)|^2 \mathrm {d}s. \end{aligned} \end{aligned}$$

For $T/2<t \le T$, using

$$\begin{aligned} |g(t)|^2= \frac{1}{T/2 }\int _{t-T /2}^t \Big ( g(s) - \int ^s_t g^\prime (\tau ) \mathrm {d}\tau \Big )^2 \mathrm {d}s , \end{aligned}$$

we derive that

$$\begin{aligned} \begin{aligned} |g(t)|^2&\le \frac{1}{T/2}\int _{t-T/2}^t \Big ( 2 |g(s)|^2 + 2\Big | \int ^s_t g^\prime (\tau ) \mathrm {d}\tau \Big |^2 \Big ) \mathrm {d}s\\&\le \frac{2}{T/2}\int _{t-T/2}^t |g(s)|^2 \mathrm {d}s+ T\int _{t-T/2}^t |g^\prime (s)|^2 \mathrm {d}s\\&\lesssim _T \int _0^T |g(s)|^2 \mathrm {d}s + \int _0^T |g^\prime (s)|^2 \mathrm {d}s. \end{aligned} \end{aligned}$$