Special Session 108: 

Global Well-Posedness and Higher Sobolev Norm Bounds for Non-Focusing Schrodinger Equations on Mixed Domains

Nathan D Totz
University of Miami
USA
Co-Author(s):    
Abstract:
We consider the long time well-posedness of the Cauchy problem with large Sobolev data for a class of nonlinear Schr\odinger equations (NLS) on mixed flat/periodic domains of spatial dimension at least 3, and with power nonlinearities of arbitrary odd degree. Specifically, the method applies to those NLS equations having either elliptic signature with a defocusing nonlinearity, or else having an indefinite signature. We argue by contradiction that, if any scaling-subcritical Sobolev norm of a solution increases faster than a certain threshold of exponential growth, we can directly construct a perturbation of the solution that grows slower than this exponential growth rate, violating classical stability results. This establishes unconditional global well-posedness with exponential bounds on all subcritical homogeneous Sobolev norms. The perturbed NLS solution at the core of the argument is constructed as a modulational limit of a specific artificial evolution equation.