Partial differential equations, Hamiltonian dynamical systems, and their applications

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The overall goals of this research proposal are to analyse the behavior and to address the singularities of solutions of a class of partial differential equations (PDEs) that govern a variety of physical phenomena involving wave propagation. Most of the principal equations which describe fundamental physical processes are differential equations; conservation laws in the form of evolution equations which describe the future state of a physical systems whose initial state is prescribed among a phase space of possible states. When a process involves wave propagation, these equations are generally PDEs, and those of physical or of geometric significance can often be formulated as Hamiltonian dynamical systems, whose phase space is naturally an infinite dimensional space of states. Examples include the nonlinear Schroedinger equation, many versions of nonlinear wave equations, the Euler equations for inviscid fluid dynamics, the problem of ocean waves, and the myriad of nonlinear dispersive equations such as the Korteweg deVries equation and the Boussinesq system that arise in fluid dynamics and plasmas. The first component of this research program focuses on extensions of analytic techniques of Hamiltonian mechanics to the setting of an infinite dimensional phase space, addressing problems such as KAM theory for PDEs, normal forms transformations, Nekhoroshev stability theorems, and possibly exploiting resonant situations to construct diffusing orbits. The second component is to study the equations for surface water waves, which are themselves Hamiltonian PDEs, which are in addition of serious physical and oceanographic significance. The third component of this proposal addresses several fundamental analytic questions about the Navier - Stokes equations, including to pursue the ramifications of our recent work on lower bounds on the microlocal singular set. The fourth component is to continue our extended collaborations with other physical scientists, which in the past have led to interesting and sometimes unexpected connections between fields.