Convergence Rate to Stationary Solutions for Boltzmann Equation with External Force*

Abstract

For the Boltzmann equation with an external force in the form of the gradientof a potential function in space variable, the stability of its stationary solutions as localMaxwellians was studied by S. Ukai et al. (2005) through the energy method. Based onthis stability analysis and some techniques on analyzing the convergence rates to station-ary solutions for the compressible Navier-Stokes equations, in this paper, we study theconvergence rate to the above stationary solutions for the Boltzmann equation which is afundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components andthe dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained byconstructing some energy functionals.

Keywords

Convergence rate / Boltzmann equation with external force / Energy functionals / Stationary solutions / 76P05 / 35B35 / 35F20

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