This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This project conducts research in the area of dispersive partial differential equations. Its intent is twofold. On the one hand, it is concerned with specific questions about the behavior of solutions to certain dispersive equations. On the other, it is focused on developing new theoretical tools that naturally address a wide range of problems in the area Hamiltonian partial differential equations. These two objectives feed into each other by way of multilinear harmonic analysis techniques. The main problems in the area are associated with solutions that tend to behave differently depending on whether their regularity is subcritical, critical, or supercritical with respect to a symmetry of the equation. In particular, the hope is that the project will supplement and broaden our understanding about the properties of mass critical, mass subcritical, and energy supercritical equations. This project looks closely at the qualitative behavior of such solutions. This behavior is based on properties that include local/global existence of solutions, uniqueness, regularity, smoothing effect, finite-in-time blow up, and asymptotic behavior of solutions.
Dispersive equations model certain wave phenomena that occur in nature. Their solutions tend to be waves that spread out spatially on unbounded domains. They have received a great deal of attention from mathematicians, in particular because of their applications to nonlinear optics, water wave theory, and plasma physics. One famous example from this class is the nonlinear Schrodinger equation. Building on the previous advances, the goal of the project is to extend the known results to cases where the analysis is harder, the terrain is unknown, and existing theoretical results lag behind conjecture for explaining the properties that we expect and have observed numerically. The intent is to provide analytical testing grounds for these physical observables.
