9703703 Hoff This is a proposal to continue the investigation of various questions relating to the existence, stability, regularity, large-time behavior, and numerical approximation of solutions of systems of certain partial differential equations arising in various areas of continuum mechanics. Discontinuous solutions are of particular interest, and will play a unifying role in the project. A fairly well-developed theory has been attained for flows in the whole space which are not subject to external forces and which are small. These results will be extended to regions with boundaries and to flows with large initial data and with forcing terms. A continuous-dependence theory will be developed, sufficient to provide a framework in which numerical procedures for approximating these solutions can be studied. The entire analysis will be extended to systems of differential equations arising in related physical problems, such as magnetohydrodynamics and viscoelasticity. Finally, recent work concerning the pointwise behavior of Navier-Stokes diffusion waves will be continued: the derivation and pointwise analysis of diffusion waves will be carried out for related systems of physical interest, and a stability analysis of planar viscous shock waves with respect to multidimensional perturbations will be given. The proposer will study various mathematical questions concerning important models of compressible fluids and materials. These models arise in a broad range of applications, including supersonic flight, dynamic meteorology, semiconductor theory, and the design and use of viscoelastic materials. While the main goal in constructing these models is to achieve a predictive capability, they are far too complicated to be "solved" in any explicit sense. On the other hand, adequate approximate solutions can frequently be generated by computer methods. The intelligent design of such methods depends crucially, however, on a rigorous understanding of why solutions do e xist, in what sense, and in what ways they are sensitive to noise in the data. The primary goal of this project is therefore to provide such a rigorous mathematical analysis for these models; a secondary objective is to apply these mathematical insights to the intelligent design and analysis of algorithms for generating approximate solutions.
