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On the Existence and Regularity for Stationary Boltzmann Equation in a Small Domain

I-Kun Chen, Chun-Hsiung Hsia, Daisuke Kawagoe, Jhe-Kuan Su

Abstract

In this article, we study the stationary Boltzmann equation with the incoming boundary condition for the hard potential cases. Assuming the smallness of the domain and a suitable normal curvature condition on the boundary, we find a suitable solution space which is a proper subset of the $W^{1,p}$ space for $1 \leq p <3$.

On the Existence and Regularity for Stationary Boltzmann Equation in a Small Domain

Abstract

In this article, we study the stationary Boltzmann equation with the incoming boundary condition for the hard potential cases. Assuming the smallness of the domain and a suitable normal curvature condition on the boundary, we find a suitable solution space which is a proper subset of the space for .
Paper Structure (8 sections, 25 theorems, 166 equations, 4 figures)

This paper contains 8 sections, 25 theorems, 166 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary estimates
  3. Estimates for the linear integral kernel
  4. Estimates for the nonlinear cross section
  5. Some geometrical estimates on bounded convex domains
  6. Regularity for the linearized case
  7. Regularity for the nonlinear case
  8. The inclusion $\hat{L}^{\infty}_{\alpha} \subset W^{1, p}$

Key Result

Theorem 1.1

Suppose assumption_B1 holds. Given $0 \leq \alpha < (1-\rho)/2$, where $\rho$ is the constant in Property A, there exists a positive constant $\delta$ such that: For any domain $\Omega$ satisfying Assumption $\Omega$ with uniform circumscribed and interior radii $R$ and $r$ respectively, if then the equation Boltzmann equation with the incoming boundary condition boundary condition admits a solut

Figures (4)

  • Figure 2.1: A picture of the cross section of $\Omega$ and $B_R$ containing $O$, $x$ and $q^+(x,v)$ in Proposition \ref{['prop:geometric estimate C']}.
  • Figure 2.2: A picture of the cross section of $\Omega$ and $B_R$ containing $O$, $x$ and $q(x,v)$ in Proposition \ref{['prop:geometric estimate A']}.
  • Figure 2.3: A picture of $O$, $B$, $C$ and $q(x,v)$ in Proposition \ref{['prop:geometric estimate A']} (deleting the point $A$).
  • Figure 2.4: A picture of points $O$, $A$, $B$, $C$ and $D$ in Proposition \ref{['prop:geometric estimate B']} when the domain is a ball.

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Lemma 1.1
  • Lemma 1.2
  • Lemma 1.3
  • Lemma 2.1
  • Lemma 2.2
  • ...and 40 more