Global solutions of semilinear and quasilinear dispersive equations

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The PI will study questions related to global existence and long-term dynamics of smooth solutions of certain evolution equations. More precisely, the PI will consider several quasilinear models, such as the Euler-Maxwell system, the Euler-Poisson system, and the irrotational water wave problem in dimensions 2 and 3. The main problems to be considered have to do with the global stability of certain equilibrium solutions of these equations. The PI will also continue his work on the global existence of Schrodinger maps and other spin field systems, in the case of large data.The equations considered in the project describe physical phenomena, such as plasma evolutions, ferromagnetic models, and fluid dynamics, and their relevance is often verified numerically. The PI proposes to study the solutions of these equations rigorously, and recover quantitative and qualitative information about their behavior as mathematical theorems. An important aspect of this proposal is to support the training of graduate students and to foster collaborations with researchers in related fields, such as Physics and Engineering.