Mathematical Sciences: Nonlinear Partial Differential Equations of Continuum and Fluid Mechanics

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PI: Sideris DMS-9500284 A class of isotropic, hyperelastic materials will be sought for which the corresponding equations of motion have smooth global solutions for small initial displacements. For planar, compressible, inviscid fluids, classes of initial data will be identified which delay the appearance of singularity formation. Smooth global solutions for the equations of motion for planar, inhomogeneous, incompressible, viscous fluids will be constructed. These problems have some features in common, and their analysis will require inter-related techniques. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations.