## Differential and Integral Equations

- Differential Integral Equations
- Volume 9, Number 5 (1996), 1119-1138.

### Uniqueness for the BGK-equation in $\mathbb{R}^N$ and rate of convergence for a semi-discrete scheme

#### Abstract

We prove estimates, in weighted $L^\infty$ spaces, for solutions of the BGK equation in the whole space and lower bound on the associated macroscopic density. $L^\infty$ bound on the macroscopic object $\rho, u$ and $T$ are deduced. Then we may show uniqueness of the solution of the BGK equation with $L^\infty$-bound assumption on the initial data, propagation of estimates on derivatives. As an application, with a BV-bound assumption on the initial data we get the convergence with rate $(\Delta t)^{1/2}$ of a time semi-discretized scheme to the solution.

#### Article information

**Source**

Differential Integral Equations, Volume 9, Number 5 (1996), 1119-1138.

**Dates**

First available in Project Euclid: 6 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367871533

**Mathematical Reviews number (MathSciNet)**

MR1392097

**Zentralblatt MATH identifier**

0852.76083

**Subjects**

Primary: 82C40: Kinetic theory of gases

Secondary: 65M06: Finite difference methods 76P05: Rarefied gas flows, Boltzmann equation [See also 82B40, 82C40, 82D05]

#### Citation

Mischler, Stéphane. Uniqueness for the BGK-equation in $\mathbb{R}^N$ and rate of convergence for a semi-discrete scheme. Differential Integral Equations 9 (1996), no. 5, 1119--1138. https://projecteuclid.org/euclid.die/1367871533