报告地点：行健楼学术活动室526
邀请人：吴奕飞教授
摘要：We establish the acoustic limit from the Boltzmann equation with a general cutoff collision kernel in the half space, subject to the incoming boundary condition with vertical evaporation-condensation velocity $\xi$, in the framework of renormalized solutions. The boundary conditions to the acoustic system are derived from analyzing the kinetic and fluid boundary layers equations, which depend on the comparison between $\xi$ and the sound speed $c$. More specifically, three characteristic speeds $\xi= 0, c, -c$ separate the real line into four intervals $(-\infty, -c)$, $(-c,0)$, $(0,c)$ and $(c, \infty)$, totally seven cases.
When the boundary velocities are characteristic, i.e. $\xi=0, c, -c$, both the Knudsen and the viscous (also called Prandtl) layers are needed, while for $\xi \neq 0, c, -c$, only the Knudsen layer is needed. The main novelty lies in proving that the derived boundary energies are either conserved or dissipated, thereby ensuring the well-posedness of the acoustic system. This is achieved by quantitatively analyzing the coefficients in the boundary conditions which come from the Knudsen boundary layer equations. Another contribution is the application of the duality arguments to the rigorous justification of weak convergence. We carefully designed the test functions and established a connection between the dual Knudsen layer equation and the dual of the derived boundary conditions for the acoustic system. This is a joint work with K. Aoki, F. Golse and Yulong Wu.
