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Existence of solutions to Dirichlet boundary value problems of the stationary relativistic Boltzmann equation

Yi Wang, Li Li, Zaihong Jiang

Abstract

In this paper, we study the Dirichlet boundary value problem of steady-state relativistic Boltzmann equation in half-line with hard potential model, given the data for the outgoing particles at the boundary and a relativistic global Maxwellian with nonzero macroscopic velocities at the far field. We first explicitly address the sound speed for the relativistic Maxwellian in the far field, according to the eigenvalues of an operator based on macro-micro decomposition. Then we demonstrate that the solvability of the problem varies with the Mach number $\mathcal{M}^\infty$. If $\mathcal{M}^\infty<-1$, a unique solution exists connecting the Dirichlet data and the far field Maxwellian when the boundary data is sufficiently close to the Maxwellian. If $\mathcal{M}^\infty>-1$, such a solution exists only if the outgoing boundary data is small and satisfies certain solvability conditions. The proof is based on the macro-micro decomposition of solutions combined with an artificial damping term. A singular in velocity (at $p_1=0$ and $|p|\gg 1$) and spatially exponential decay weight is chosen to carry out the energy estimates. The result extends the previous work [Ukai, Yang, Yu, Comm. Math. Phys. 236, 373-393, 2003] to the relativistic problem.

Existence of solutions to Dirichlet boundary value problems of the stationary relativistic Boltzmann equation

Abstract

In this paper, we study the Dirichlet boundary value problem of steady-state relativistic Boltzmann equation in half-line with hard potential model, given the data for the outgoing particles at the boundary and a relativistic global Maxwellian with nonzero macroscopic velocities at the far field. We first explicitly address the sound speed for the relativistic Maxwellian in the far field, according to the eigenvalues of an operator based on macro-micro decomposition. Then we demonstrate that the solvability of the problem varies with the Mach number . If , a unique solution exists connecting the Dirichlet data and the far field Maxwellian when the boundary data is sufficiently close to the Maxwellian. If , such a solution exists only if the outgoing boundary data is small and satisfies certain solvability conditions. The proof is based on the macro-micro decomposition of solutions combined with an artificial damping term. A singular in velocity (at and ) and spatially exponential decay weight is chosen to carry out the energy estimates. The result extends the previous work [Ukai, Yang, Yu, Comm. Math. Phys. 236, 373-393, 2003] to the relativistic problem.

Paper Structure

This paper contains 14 sections, 32 theorems, 273 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Some notations and properties
  3. Notations in the relativistic system
  4. Properties of $L$ and $\Gamma$
  5. Properties of $P_0\hat{p}_1P_0$
  6. Existence for linear problem with damping
  7. Weighted estimates for the linearized problem with damping
  8. Existence of the nonlinear problem with damping
  9. Existence of the problem without damping
  10. Classification of $P_0^+ \hat{p}_1 \mathbb{I}^\gamma(a_0) = 0$
  11. Classification of $P_0^+ \hat{p}_1 \mathbb{J}^\gamma(a_0) = 0$
  12. Appendix
  13. Lorentz transformation
  14. Moments of the relativistic Maxwellian

Key Result

Theorem 1

Suppose that the scattering kernel $\sigma$ satisfies eq:sigma, eq:ab and eq:gamma, $\eta$ defined in eq:eta belongs to $[\frac{2}{3}, 1]$, $\beta > 2$ and $\mathcal{M}^\infty\ne 0,\pm1$. Then there exist positive numbers $\epsilon_0$, $\tau$, $c_0$ and a $C^1$ map such that the following holds. (i) For any $F_0$ satisfying and problem eq:BVP has a unique solution $F$ in the class (ii) The set

Theorems & Definitions (60)

  • Remark 1.1
  • Theorem 1
  • Remark 1.2
  • Proposition 1
  • proof
  • Lemma 1: Glassey:Strauss
  • Lemma 2: Glassey:Strauss
  • Proposition 2
  • proof
  • Proposition 3
  • ...and 50 more