Mathematical Sciences: Nonlinear Partial Differential Equations in Fluid Dynamics and Mechanics

本课题主要研究数学物理中的某些双曲-抛物型偏微分方程式。要考虑的问题有：粘性流动向一般Riemann解的渐近行为，其中包括压缩波和膨胀波；可压缩N-S方程和MHD中接触不连续性的亚稳定性；粘性守恒律系统粘性激波剖面的L_1非线性稳定性；小耗散对基波相互作用的影响；非严格双曲型系统的粘性激波理论(这是由于超压和交叉激波等所谓过渡波的出现而具有特别重要的意义；这种波的可容许性似乎应该涉及到非线性稳定性分析)；可压缩流体的粘性边界层理论，特别是激波层与边界层的相互作用，守恒律组的弱熵解对初值的连续依赖性。在物理学和自然科学的其他分支中出现的许多有趣的现象都可以用双曲-抛物型的非线性偏微分方程组来模拟。重要的例子包括可压缩无粘和粘性流体的Euler和Navier-Stokes方程，导电可压缩粘性流体的电磁场动力学方程，弹性方程，动力学理论中的非线性Boltzmann型方程，以及燃烧、多相流和具有化学反应的流体的方程。

This project aims study certain hyperbolic-parabolic type partial differential equations that arise form mathematical physics. To be considered are: the asymptotic behavior of viscous flows toward general Riemann solutions which consist of both compressive and expansive waves; metastability of contact discontinuities for compressible Navier-Stokes equations and in MHD; L_1-nonlinear stability of viscous shock profiles for systems of viscous conservation laws; the effects of small dissipation on the interactions of elementary waves; theory of viscous shock waves for nonstrictly hyperbolic systems (this is of particular significance due to the appearance of so-called transitional waves such as overcompressive and crossing shocks; it seems that the admissibility of such waves should involve nonlinear stability analysis); nonlinear stability of planar viscous shocks for systems of conservation law in several space dimensions; and the theory of viscous boundary layer for compressible fluids, in particular, the interaction of shock layer and boundary layers, and the continuous dependence on initial data of weak entropy solutions for systems of conservation laws. Many interesting phenomena arising in physics and other branches of natural sciences can be modeled by nonlinear partial differential equations of hyperbolic-parabolic type. Important examples include the Euler and Navier-Stokes equations for compressible inviscid and viscous fluids, the equations of electro-magneto field dynamics for electrically conducting compressible viscous fluids, the equations of elasticity, nonlinear Boltzmann type equations in kinetic theory, and equations for combustion, multiphase flows, and fluids with chemical reactions.