Mathematical Sciences: Systems of Nonlinear Hyperbolic Partial Differential Equations

所提出的研究包括研究二维和三维空间中控制塑性流动的方程组的不稳定性和病态性，以及研究一维空间中模拟欠压激波演化的非严格双曲系统。解决技术将从动力系统理论和分岔理论中汲取，并计划用数值方法解决一些问题。许多重要的物理现象都是用方程系统来模拟的，其中耗散效应如粘度可以安全地忽略。为了研究塑料流动和某些类型激波的动力学问题，申请者将使用一整套分析和数值技术来研究这种“双曲”系统。

The proposed research involves the study of instability and illposedness for systems of equations that govern plastic flow in two and three space dimensions and the study of nonstrictly hyperbolic systems in one space dimension that model the evolution of undercompressive shock waves. Solution techniques will be drawn from the theory of dynamical systems and bifurcation theory, and there are plans to attack some of the problems numerically. Many important physical phenomena are modelled by systems of equations in which dissapative effects such as viscosity can be safely neglected. The proposer will study such "hyperbolic" systems using a panoply of analytical and numerical techniques in order to investigate problems in the flows of plastics and in the dynamics of certain types of shock waves.