Initial and Shock Layers for Boltzmann Equation

Abstract

We study an initial value problem of the Boltzmann equation with a Euler shock wave as initial data. Our analysis exhibits the presence of three singular layers, the initial layer, formation layer, and the shock layer. An approximate solution is constructed based on a solution of the Burgers-type equation for the formation layer time scale. The macroscopic conservation laws are preserved for the approximate solution. The Green’s function of the linearized Boltzmann equation around the approximate solution is constructed by a modification of the ${{\mathbb {T}} - {\mathbb {C}}}$ scheme introduced by Yu (Nonlinear wave propagation over a Boltzmann shock profile, 2013). With the Green’s function approach and a wave tracing method, one shows that the error of the approximate solution converges to zero with the convergent rate ${O(1) |\log{\varepsilon} | {\varepsilon} (1+t)^{-(1-\sigma_0)}}$ in pointwise norm ${\| \cdot \|_{L^\infty_{\xi,3}}}$ around the shock profile for ${\sigma_0 \in (0,\frac{1}{2})}$ and with the rate ${O(1) {\varepsilon }{{(1+t)}^{-1/2}}}$ outside the shock zone, where ${{\varepsilon}}$ is the strength of the weak hyperbolic shock wave.