Shock Wave Theory

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DMS-9803323 Tai-Ping Liu We propose to study two fundamental questions for general systems of hyperbolic conservation laws: The first is the well-posedness problem. This is done with Tong Yang. For this, we already constructed a nonlinear functional for the two conservation laws and are in the process of doing so for the general systems. The second problem is to study multi-dimensional gas flows. Lien and the author have succeeded in showing that three-dimensional self-similar gas flow past a cone is nonlinearly stable. We are generalizing the new approach of superpositioning local self-similar flows to other problems. We expect this approach to yield new qualitative information on nonlinear stability and instability of multi-dimensional gas flows. Our second project is to study the viscous conservation laws. With Zeng, we are in the process of studying the stability of nonlinear waves for hyperbolic-parabolic systems such as the compressible Navier-Stokes equations. Other problems in combustion and MHD, and numerical analysis of shock calculations are being considered. Of particular interest is the role of dissipative parameters on the structure and stability of nonlinear waves. The pointwise approach the author introduced is effective for these purposes. The author plans to study basic questions concerning the invisicd and viscous solutions for gas dynamics and mechanics. One of the fundamental questions concerns the validity of the Euler equations for the gas dynamics. The equations were derived by Euler in the eighteenth century. Only until recently the author and his collaborator are able to show that the equations are mathematically valid in that small errors in the data yield small errors in the solutions. This is of obvious importance in the applications because there is always small error either in the experimental data or numerical computations. Other problems of interest to the author include the sensitive role of di ssipation parameters. In combustion, magneto-hydrodynamics, nonlinear elasticity and other continuum physics, dissipative mechanisms, such as viscosity, heat conductivity, and species diffusions are important. The geometric properties and stability of the nonlinear waves may depend sensitively on these parameters. The author has recently introduced a pointwise approach, which is effective in studying the nonlinear interactions of waves. The role of dissipation parameters are being studied using the basic conservation laws in physics.