Nonlinear hamiltonian partial differential equations

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An explosion of insight into basic nonlinear Hamiltonian partial differential equations (PDE) occurred in the last fifteen years through a systematic application of multilinear harmonic analysis and other tools. A robust local-in-time theory for initial value problems with rough initial data is now in place. This research proposal is designed to advance the recent progress towards a maximal-in-time theory for nonlinear Hamiltonian PDE under optimal regularity properties. In particular, the proposed research contributes along three lines: 1. Clarify optimal regularity for local-in-time well-posedness. 2. Establish global-in-time well-posedness and long time behavior at low regularity. 3. Describe finite time blowup behavior at low regularity. These lines of investigation addresses mathematically common features of physical evolution equations in plasma physics, optics, fluid dynamics, quantum mechanics, etc., with engineering applications in semiconductors, telecommunications, aerospace, and beyond. No specific scientific or engineering application motivates the proposed studies rather the aim is to contribute toward a general rigorous theory of nonlinear phenomena including turbulence, singularity formation, scattering and recurrence. The widespread applicability of Hamiltonian PDE, across diverse fields of current scientific and technological significance, demonstrates the central prominence of the proposed research to our science and engineering infrastructure.