论文标题
Boltzmann方程的不可压缩的Euler极限，具有分析数据的扩散边界条件
Incompressible Euler limit from Boltzmann equation with Diffuse Boundary Condition for Analytic data
论文作者
论文摘要
从玻尔兹曼方程中具有无渗透边界条件的不可压缩欧拉方程的严格推导，具有弥漫性反射边界条件是一个挑战性的开放问题。当流体的初始数据在3D半空间中的真实分析空间中得到充分准备时，我们就会以肯定的方式解决这个开放的问题。作为此进步的关键，我们捕获了$$ \ textit {viscosity}的navier-stokes方程 \ sim \ frac {\ textIt {knudsen number}} {\ textit {mach number}} $$满足不滑动边界条件，作为$ \ textit {intermediary} $近似Euler方程的近近似，通过与纽希尔伯特（New Hilbert-type）相互扩展的效率扩展，以差异为条件。为了证明近似值，我们建立了一种新颖的定量$ l^p $ - $ l^\ infty $估计，对这种粘性近似的本地麦克斯韦人的玻璃体扰动，以及换向因子的估计以及各个空间中水力动力学部分的可积分性增益；我们还使用最近的绿色功能方法在较高的初始和边界层权重的帮助下建立了更高规律性的Navier-Stokes方程的直接估计。不可压缩的Euler限制是我们框架的副产品。
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 $$\textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}}$$ satisfying the no-slip boundary condition, as an $\textit{intermediary}$ approximation of the Euler equations through a new Hilbert-type expansion of 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.
