Mathematical Sciences: Systems of Nonlinear Hyperbolic Partial Differential Equations

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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.