EVENT DETAILS AND ABSTRACT

Title: Nonlinear Boundary Layers of the Boltzmann Equation
Speaker: Professor Tong Yang
Speaker Info: City University of Hong Kong
Brief Description:
Special Note:
Abstract:

This is a joint work with Seiji Ukai and Shih-Hsien Yu. We study the half-plane problem of the nonlinear Boltzmann equation, assigning the Dirichlet data for outgoing particles at the boundary and a Maxwellian as the far field. We will show that the solvability condition of the problem changes with the Mach number ${\mathscr M}^\infty$ of the far field Maxwellian. If ${\mathscr M}^\infty<-1$, there exists a unique smooth solution connecting the Dirichlet data and the far field Maxwellian for any Dirichlet data sufficiently close to the far field Maxwellian. Otherwise, such a solution exists only for the Dirichlet data satisfying certain admissible conditions. The set of admissible Dirichlet data forms a smooth manifold of co-dimension 1 for the case $-1<{\mathscr M}^\infty<0$, 4 for $0<{\mathscr M}^\infty<1$ and 5 for ${\mathscr M}^\infty>1$, respectively. We also show that the same is true for the linearized problem at the far field Maxwellian, and the manifold is, then, a hyperplane. The proof is essentially based on the macro-micro or hydrodynamic-kinetic decomposition of solutions combined with an artificial damping term and a spatially exponential decay weight.