Mathematical Sciences: Nonlinear Evolutions and the Calculusof Variations

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93222617 Sternberg This award supports mathematical research on a range of problems involving nonlinear partial differential equations and the calculus of variations. The basis for much of the work is typically an energy arising from a model in continuum mechanics. Physical contexts include superconductivity and phase transitions. In all cases, the structure of minimizers of the energy is an issue of primary interest. Beyond this, work will be done on gradient flows associated with the energy. The flows are modeled by parabolic partial differential equations or systems whose solution tend to decrease the energy. A primary goal is to characterize the large-time behavior of these systems as they strive to achieve an equilibrium state. In all cases, certain geometric structures tend to characterize steady-state solutions such as minimal surfaces or vortices and one seeks to understand how such structures propagatein time during the late stages of the evolution. 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. Many equations arise by invoking physical principles, phrased in terms of minimizing certain quantities, such as energy. The resulting mathematical expression is a variational problems whose solution is expressed by a partial differential equation. ***