Mathematical Sciences: Nonlinear Partial Differential Equations

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This research will examine several problems in nonlinear partial differential equations. Vortex methods for simulating time-dependent, three-dimensional, incompressible fluid flows will be extended. Model problems of nonlinear evolution, which are linearly ill-posed but still have predictable behavior, will be studied. The existence of exact solutions of the three- dimensional water wave equations representing obliquely interacting solitary waves or cnoidal water waves will be investigated. This research will extend theoretical understanding of partial differential equations and advance methodology for their numerical solution.