[Paper Review] Stationary solutions to the boundary value problem for relativistic BGK model in a slab

Byung-Hoon Hwang, Seok-Bae Yun|arXiv (Cornell University)|Jan 25, 2018

Gas Dynamics and Kinetic Theory29 references3 citations

TL;DR

This paper establishes the existence and uniqueness of stationary solutions to the relativistic BGK model of Marle type in a one-dimensional slab with fixed inflow boundary conditions. Using a fixed-point argument in a carefully constructed function space, the authors prove that for sufficiently small collision frequency $ w $, a unique mild solution exists under integrability and positivity conditions on the boundary data.

ABSTRACT

In this paper, we are concerned with the boundary value problem in a slab for the stationary relativistic BGK model of Marle type, which is a relaxation model of the relativistic Boltzmann equation. In the case of fixed inflow boundary conditions, we establish the existence of unique stationary solutions.

Motivation & Objective

• To establish the existence of stationary solutions for the relativistic BGK model of Marle type in a bounded spatial domain.
• To address the challenge of the nonlinear, nonlocal structure of the relativistic Maxwellian in the collision term.
• To prove existence and uniqueness of mild solutions under fixed inflow boundary conditions in a slab geometry.
• To analyze the dependence of the solution on the collision frequency $ w $, showing existence for small $ w $.
• To ensure physical consistency by preserving conservation laws and the H-theorem through the BGK relaxation model.

Proposed method

• Formulates the stationary relativistic BGK equation as a kinetic equation with a relaxation operator toward a local relativistic Maxwellian.
• Defines a mild solution via integral formulation involving exponential decay factors and boundary data $ f_L, f_R $.
• Introduces a function space $ \Omega $ with constraints on particle density, velocity, and temperature to ensure physical consistency.
• Applies a fixed-point argument using the Banach contraction principle in $ L^1 $-based norms to prove existence and uniqueness.
• Establishes Lipschitz continuity of the Maxwellian $ J_f $ with respect to macroscopic fields $ n, u, \beta $, enabling contraction estimates.
• Uses estimates on the $ L^1 $-norm of differences in $ f $ to control the nonlinear dependence of $ J_f $, leading to a contraction mapping.

Experimental results

Research questions

• RQ1 Does a unique stationary solution exist for the relativistic BGK model in a slab with fixed inflow boundary conditions?
• RQ2 How does the solution depend on the collision frequency $ w $, and for which values of $ w $ does a solution exist?
• RQ3 Can the nonlinear coupling between the distribution function and the local Maxwellian be handled via a fixed-point argument in $ L^1 $?
• RQ4 What conditions on the boundary data $ f_L, f_R $ ensure the existence of a physically meaningful solution?
• RQ5 Is the solution stable under small perturbations in the distribution function, as implied by the contraction mapping argument?

Key findings

• For sufficiently small collision frequency $ w < \varepsilon $, a unique mild solution $ f \in L^1([0,1] \times \mathbb{R}^3) $ exists.
• The solution satisfies the constraints $ \int f \frac{1}{q_0^2} dq \geq a_l $, $ \int f dq \leq a_u $, and $ \frac{K_1}{K_2}(\beta) \leq \sqrt{\lambda} $, ensuring physical consistency.
• The existence relies on the positivity of $ a_l = \int e^{-w/|q_1|} f_{LR} \frac{1}{q_0^2} dq > 0 $, which prevents division by zero in the definition of $ \lambda $.
• The contraction mapping argument yields a contraction constant $ \alpha < 1 $ for small $ w $, with the bound depending on $ a_l, a_u $, and model parameters.
• The solution is shown to be stable in the $ L^1_q $-norm, with the difference between two solutions bounded by a constant times $ \|f - g\|_{L^1_q} $.
• The proof establishes that the nonlinear operator mapping $ f \mapsto J_f $ is Lipschitz continuous in $ L^1 $, enabling the contraction argument.