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 incoming 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 finite dimensional operator based on macroscropic projection. Then we demonstrate that the solvability of the problem varies with the Mach number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> </mml:math> . If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> <mml:mo>&lt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , a unique solution exists connecting the Dirichlet data and the far field Maxwellian when the boundary data is sufficiently close to the Maxwellian. If <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">M</mml:mi> </mml:mrow> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , such a solution exists only if the inflow 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> <mml:mi>p</mml:mi> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>≫</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> ) and spatially exponential decay weight is chosen to carry out the energy estimates. The result extends the previous work (Ukai et al 2003 Commun. Math. Phys. 236 373–93) to the relativistic problem.