Shock Waves in Macroscopic and Microscopic Models

提出了研究激波守恒律理论和动力学中的Boltzmann方程。冲击波发生在许多自然现象中，如超音速飞行、声纳风、地震和飓风。对于无粘平面运动，现在已有了令人满意的数学理论。计划研究多维气体运动以及耗散效应。玻尔兹曼方程在微观水平上模拟气体运动。这对于重要的物理现象是必要的，例如边界对气体运动的热效应。我们将研究Boltzmann方程的非线性波和边界效应，并考虑气体动力学的Euler方程和Navier-Stokes方程。该项目将包括研究这些和更一般的双曲-抛物守恒律系统的冲击波的非线性稳定性。该方法结合了逐点估计和能量估计，并基于对激波线性化方程的绿色函数结构的新认识。对于玻尔兹曼方程，玻尔兹曼激波的正性已被证明，边界效应如热蠕变正在研究中。这些都是基于新的宏观-微观分解的玻尔兹曼方程和时间渐近分析。

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.