Nonlinear Hamiltonian PDE

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The proposed research is designed to advance the recent progress on nonlinearHamiltonian PDE towards three goals:1. Extend the local-in-time initial value methods to solve more complicated PDE.2. Adapt the initial value techniques to treat initial-boundary value problems. 3. Construct a global-in-time theory of nonlinear Hamiltonian PDE.The proposal identifies specific problems whose solutions contribute to the three goals for which there are methods of attack emerging from the last decades' spectacular progress.The studies for Goal 1 aim to extend the sharp 1-dimensional calculus techniques for proving multilinear estimates in Bourgain's Xs,b spaces by carrying out an incremental research plan, involvingsmall Xs,b denominators, spatial anisotropy and vanishing parameters.A technique for recasting initial-boundary value problems as initialvalue problems with boundary forcing has recently been developed, incollaboration with Kenig. The range of applicability of this method isthe main topic of the proposed investigations toward Goal 2. A reinterpretation of the L^2 conservation law for the KdV equation, obtained in collaboration with Keel, Staffilani, Takaoka and Tao, has led to a new method for showing global wellposedness by constructing almost conserved quantities using multilinear harmonic analysis and the local wellposedness machinery. The third thrust of the proposed research will exploit these quantities to understand the long-time behavior of nonlinear Hamiltonian PDE.No specific scientific or engineering application motivates the proposed studies; rather the intention 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.The proposed research is designed to advance the recent progress on nonlinearHamiltonian PDE towards three goals:1. Extend the local-in-time initial value methods to solve more complicated PDE.2. Adapt the initial value techniques to treat initial-boundary value problems. 3. Construct a global-in-time theory of nonlinear Hamiltonian PDE.The proposal identifies specific problems whose solutions contribute to the three goals for which there are methods of attack emerging from the last decades' spectacular progress.The studies for Goal 1 aim to extend the sharp 1-dimensional calculus techniques for proving multilinear estimates in Bourgain's Xs,b spaces by carrying out an incremental research plan, involvingsmall Xs,b denominators, spatial anisotropy and vanishing parameters.A technique for recasting initial-boundary value problems as initialvalue problems with boundary forcing has recently been developed, incollaboration with Kenig. The range of applicability of this method isthe main topic of the proposed investigations toward Goal 2. A reinterpretation of the L^2 conservation law for the KdV equation, obtained in collaboration with Keel, Staffilani, Takaoka and Tao, has led to a new method for showing global wellposedness by constructing almost conserved quantities using multilinear harmonic analysis and the local wellposedness machinery. The third thrust of the proposed research will exploit these quantities to understand the long-time behavior of nonlinear Hamiltonian PDE.No specific scientific or engineering application motivates the proposed studies; rather the intention 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.