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

点击翻译按钮获取中文摘要

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.