Nonlinear wave propagations over a Boltzmann shock profile

Abstract

In this paper we study the wave propagation over a Boltzmann shock profile and obtain pointwise time-asymptotic stability of Boltzmann shocks. We design a T {\mathbb T} - C {\mathbb C} scheme to study the coupling of the transverse and compression waves. The pointwise information of the Green’s functions of the Boltzmann equation linearized around the end Maxwellian states of the shock wave provides the basic estimates for the transient waves. The compression of the Boltzmann shock profile together with a low order damping allows for an accurate energy estimate by a localized scalar equation. These two methods are combined to construct an exponentially sharp pointwise linear wave propagation structure around a Boltzmann shock profile. The pointwise estimates thus obtained are strong enough to study the pointwise nonlinear wave coupling and to conclude the convergence with an optimal convergent rate O ( 1 ) [ ( 1 + t ) ( 1 + ε t ) ] − 1 / 2 O(1)[(1+t)(1+\varepsilon t)]^{-1/2} around the Boltzmann shock front, where ε \varepsilon is the strength of a shock wave.