Shock Waves in Macroscopic and Microscopic Models

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0104019LiuIt is proposed to study the shock wave theory for conservation laws and the Boltzmann equation in the kinetic theory. Shock waves occurs in many natural phenomena, such as supersonic flight, sonar wind, earthquakes and hurricanes. There is now a satisfactory mathematical theory for the inviscid plane motion. It is planned to study the multi-dimensional gas motion as well as the dissipation effects. The Boltzmann equation models the gas motion on the microscopic level. This is necessary for important physical phenomena such as the thermal effects of the boundary on the gas motion. The nonlinear waves and the boundary effects for the Boltzmann equation will be studied.The Euler equations and Navier-Stokes equations for the gas dynamics are considered. The project will include the study of the nonlinear stability of shock waves for these and more general systems of hyperbolic-parabolic conservation laws. The approach combines the pointwise estimates and the energy estimates and is based on the new understanding of the structure of the Green's function for the equations linearized about the shock. For the Boltzmann equation, the positivity of Boltzmann shocks has been shown and the boundary effects such as the thermal creep are being studied. These are based on the new macro-micro decomposition of the Boltzmann equation and time-asymptotic analysis.