Nonlinear Evolution Equations

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The proposed research will focus on mathematical models involving systems of nonlinear partial differential equations that arise in fluid mechanics and in the study of wave propagation. The emphasis of the work will be on the qualitative properties of solutions to the initial value problem associated with any such system. Unique continuation properties of nonlinear dispersive models will be investigated by exploiting the relationship between the dispersive relation and the structure of the nonlinearity. A specific problem will be to establish the uniqueness of a solution, as well as its possible reconstruction from some incomplete information given at two different times. Other problems concerning the time evolution of several forms of the vorticity will be also considered. As a consequence of the classical principle of determinism, the evolutionary laws of a physical system together with an initial configuration should delineate the state of the system at all future times. To rephrase this in mathematical terms, the laws of physics are encoded in a system of partial differential equations whose solutions should exist and depend uniquely and continuously on the initial configuration. Hence, the development of a mathematical framework in which the existence, uniqueness, and stability of the solution to a given system can be verified and quantified is a problem of fundamental importance. Mathematical analysis offers a powerful tool for investigating the properties of solutions to these models well beyond the point of purely phenomenological assumptions. Much of the research in this project is devoted to such basic questions for models describing the dynamics of water waves and incompressible fluids.