Critical Dispersive Partial Differential Equations

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The main objective of this project is to improve the understanding of dispersive partial differential equations. Dispersive partial differential equations include the wave, Schrodinger, and Korteweg de-Vries equation. These equations are ubiquitous in physics, modeling phenomena ranging from the behavior of subatomic particles to interstellar gravity waves. Of particular interest are questions of long time behavior of solutions. In other words, given certain initial data, does a solution to the equation exist? If a solution does exist, does it exist for all time? What is the behavior of the solution as time approaches either infinity or the maximum time for which the solution exists? Can we catalogue the various long time behaviors and obtain a complete description of the possible phenomena? The project provides research training opportunities for graduate students. In this project, the Principal Investigator and his collaborators study the long time behavior of dispersive partial differential equations with initial data in a critical norm. Many diverse dispersive partial differential equations have a scaling symmetry, and a solution to the equation gives an entire family of solutions. Often, the scaling symmetry completely describes the local behavior of the equation completely: the equation is well-posed for initial data in the critical space, but it is ill-posed for data in a less regular (subcritical) space. We wish to understand the long time behavior for such equations at the critical regularity, where it is known that local well-posedness occurs. Additionally, in many such equations, the set where ill-posedness occurs in a subcritical space is often a set of measure zero. Thus, we hope to describe the nature of the set of initial data for which ill-posedness occurs. The specific problems that are addressed in this project are the Schrodinger maps problem, the focusing, mass-critical nonlinear Schrodinger equation, the energy subcritical nonlinear wave and Schrodinger equations, and the one dimensional cubic nonlinear Schrodinger equation. For the defocusing, energy subcritical problems, we expect scattering to occur for initial data in the critical Sobolev space. Solitons are known to occur for the Schrodinger map problem and the mass-critical nonlinear Schrodinger equation. In both cases, scattering is known to occur for initial data below the soliton (for Schrodinger maps this is only in the equivariant case). We wish to understand the solution for initial data slightly above the soliton for these problems. Finally, for the one dimensional nonlinear Schrodinger equation, the aim is to understand the long time behavior for initial data that does not decay at infinity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.