Shock Wave Theory

9803323刘泰平我们建议研究一般双曲型守恒律组的两个基本问题：第一个是适定性问题。这是和童阳一起做的。为此，我们已经为两个守恒律构造了一个非线性泛函，并正在为一般系统构造一个非线性泛函。第二个问题是研究多维气体流动。Lien和作者成功地证明了经过锥体的三维自相似气体流动是非线性稳定的。我们正在推广将局部自相似流叠加到其他问题的新方法。我们期望这种方法能为多维气体流动的非线性稳定性和不稳定性提供新的定性信息。我们的第二个项目是研究粘性守恒定律。与Zeng一起，我们正在研究双曲-抛物型方程组，如可压缩的Navier-Stokes方程的非线性波的稳定性。正在考虑燃烧和MHD中的其他问题，以及激波计算的数值分析。尤其令人感兴趣的是耗散参数对非线性波的结构和稳定性的作用。作者介绍的逐点方法对于这些目的是有效的。作者计划研究气体动力学和力学的无粘性和粘性解的基本问题。其中一个基本问题是关于气体动力学的欧拉方程的有效性。这些方程是由欧拉在十八世纪推导出来的。直到最近，作者和他的合作者才能证明这些方程在数学上是有效的，因为数据中的小误差会导致解中的小误差。这在实际应用中具有重要意义，因为无论是在实验数据还是在数值计算中都存在小的误差。作者感兴趣的其他问题包括耗散参数的敏感作用。在燃烧、磁流体力学、非线性弹性和其他连续介质物理中，耗散机制，如粘度、导热系数和物种扩散，是重要的。非线性波的几何性质和稳定性可能敏感地依赖于这些参数。作者最近介绍了一种研究波的非线性相互作用的有效方法--逐点方法。耗散参数的作用正在用物理学中的基本守恒定律来研究。

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.