Mathematical Problems in Compressible Fluid Flow

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9986658HoffThis is a proposal to study various questions relating to the existence, stability, regularity, and large-time behavior of solutions of certain partial differential equations arising in the theory of viscous, compressible fluid flow. Previous work on the existence of discontinuous solutions of the Navier-Stokes equations in three space dimensions will be extended to finite regions, requiring new insight into the production of compressible vorticity at boundaries. A rigorous analysis of the propagation of singularities and the regularity of discontinuity surfaces in solutions of these equations will be given. A continuous-dependence theory will be developed, sufficient to provide a framework in which numerical procedures for approximating solutions can be studied. The dimension of the attractor for the Navier-Stokes equations in one space dimension will be estimated in terms of the size of the applied force and scale-invariants of the system. Finally, previous work on the multidimensional stability of large-amplitude viscous shocks for scalar conservation laws will be extended to the more applicable case of systems.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 exist, 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.