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The Non-cutoff Boltzmann Equation in Bounded Domains

Dingqun Deng

Abstract

The initial-boundary value problem for the inhomogeneous non-cutoff Boltzmann equation is a challenging open problem. In this paper, we study the stability and long-time dynamics of the Boltzmann equation near a global Maxwellian without angular cutoff assumption in a general $C^3$ bounded domain $Ω$ (including convex and non-convex cases) with physical boundary conditions: inflow boundary and Maxwell-reflection boundary with accommodation coefficient $\al\in(0,1)$. We obtain the global-in-time existence, which has an exponential decay rate towards the global Maxwellian for both hard and soft potentials. The crucial methods are the forward-backward extension of the boundary problem to the whole space by Vlasov-type equations, a level-function trace lemma, an improved velocity averaging lemma with less regularity but without cutoff in velocity, and an extra damping provided by the advection operator, followed by the De Giorgi iteration and the $L^2$--$L^\infty$ energy method.

The Non-cutoff Boltzmann Equation in Bounded Domains

Abstract

The initial-boundary value problem for the inhomogeneous non-cutoff Boltzmann equation is a challenging open problem. In this paper, we study the stability and long-time dynamics of the Boltzmann equation near a global Maxwellian without angular cutoff assumption in a general bounded domain (including convex and non-convex cases) with physical boundary conditions: inflow boundary and Maxwell-reflection boundary with accommodation coefficient . We obtain the global-in-time existence, which has an exponential decay rate towards the global Maxwellian for both hard and soft potentials. The crucial methods are the forward-backward extension of the boundary problem to the whole space by Vlasov-type equations, a level-function trace lemma, an improved velocity averaging lemma with less regularity but without cutoff in velocity, and an extra damping provided by the advection operator, followed by the De Giorgi iteration and the -- energy method.
Paper Structure (93 sections, 52 theorems, 982 equations, 1 figure)

This paper contains 93 sections, 52 theorems, 982 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Model and bounded domain
  3. Reformulations and boundary condition
  4. Notations, weight function and function spaces
  5. Miscellany
  6. Weight function
  7. $L^p$ spaces
  8. Bessel potential
  9. Besov space
  10. Lorentz space
  11. Kernel of $L$
  12. Dissipation norm
  13. Inflow and outflow regions
  14. Main result: inflow boundary
  15. Main result: Maxwell boundary
  16. ...and 78 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^3_x$ be a bounded domain satisfying Omega, Bnboundary and naxi1. Let $-\frac{3}{2}<\gamma\le 2$, and $s\in(0,1)$. Fix any $l\ge \gamma+10$. Let $l_0=l_0(l,s)$ be a large constant and fix any $\tilde{C}>0$. Then there exists a generic constant $c_0=c_0(\gamma,s)>0$ and su then there exist a global-in-time solution $f(t)$$(t\ge 0)$ to the Boltzmann equation B1 with inflo

Figures (1)

  • Figure 1: Inflow and outflow regions

Theorems & Definitions (104)

  • Theorem 1.1: Stability of Boltzmann equation with inflow boundary condition
  • Theorem 1.2: Stability of Boltzmann equation with Maxwell boundary condition
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • ...and 94 more