Mathematical Sciences: Nonlinear Partial Differential Equations

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Primary emphasis of this mathematical work will be directed at three problem areas: nonlinear transport equations, regularity theory in the calculus of variations, and geometric properties for motion hypersurfaces by mean curvature. All, in one way or another, are concerned with nonlinear problems in analysis. Nonlinear transport is modeled by relatively simple partial differential equations (e.g. scalar conservation laws or the porous medium equation). The goal here is to use the newer concept of viscosity solution and expand its role from spaces metrized by maximum norms to the Lebesgue space of integrable functions, in an effort to understand solutions. The key is the introduction of weak solutions of differential operators satisfying an accretive condition, with the expectation that an interpretation of non-smooth solutions will result. A long-term goal of this work is that of analyzing the hydrodynamical limit of stochastic interacting particle systems. A second line of research concerns the question of partial regularity for vector-valued problems in the calculus of variations - in particular, for problems of nonlinear elasticity. The basic issue here is that of performing variations within the constraint which requires that the determinant of the differential be positive. Work will proceed first with more tractable questions involving minimizers of convex integrands for mappings between manifolds. Considerable activity is presently under way investigating geometric problems of motion by mean curvature. The associated partial differential equations, which give the motion of level sets (of a manifold) moving normally at a velocity proportional to the mean curvature, may have solutions with singularities. The first concern of this work will be to show that the level sets are smooth except possibly for negligible singular sets.