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

本项目旨在研究某些双曲-抛物型 偏微分方程是由数学 物理学 要考虑的是：渐近行为的 粘性流向一般黎曼解，其中包括 压缩波和膨胀波;接触亚稳定性; 可压缩Navier-Stokes方程的不连续性， 粘性激波剖面的L_1非线性稳定性 粘性守恒定律系统;小的影响 基本波相互作用的耗散理论 非严格双曲系统的粘性激波（这是 由于出现了所谓的 过渡波，如超压和交叉冲击; 似乎这种波的可接受性应该包括 非线性稳定性分析）;平面非线性稳定性 多空间中守恒律方程组的粘性激波 尺寸;和粘性边界层的理论， 可压缩流体，特别是冲击波的相互作用 层和边界层，以及对 方程组弱熵解的初始数据 守恒定律 许多有趣的现象出现在物理学和其他 自然科学的各个分支可以用非线性偏微分方程来模拟。 双曲抛物型微分方程重要 例子包括欧拉和纳维尔-斯托克斯方程， 可压缩无粘和粘性流体，方程 电磁场动力学 可压缩粘性流体，弹性方程， 动力学理论中的非线性玻尔兹曼型方程，以及 燃烧、多相流和流体方程， 化学反应。

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.