Analysis Seminar

In this talk, I will discuss two problems concerning random data Cauchy theory for nonlinear wave equations. The first, based on joint work with Luhrmann, focuses on nonlinear wave equations with defocusing energy-subcritical power-type nonlinearity on Euclidean space. Local well-posedness for these equations is by now well understood for initial data of subcritical or critical regularities, but techniques break down for initial data in the supercritical regime. Nonetheless, in recent years, probabilistic methods have been used to investigate the behavior of solutions in regimes where deterministic techniques fail. I will present an almost sure global existence result in the supercritical regime for these equations. Our proof involves introducing a randomization procedure for initial data in Sobolev spaces of low regularity, and energy estimates for a forced wave equation. The second problem focuses on energy critical nonlinear wave equations with null form nonlinearities in both the periodic and the Euclidean setting. In certain cases, there are counterexamples to the estimates in Sobolev spaces which yield a gap between the regularity of the initial data required in order to obtain local well-posedness, and the desired result in the critical energy space. We will show that for suitably chosen random initial data, by exploiting the null structure of these equations, we are able to prove estimates for solutions arising from initial data of supercritical regularity, in spite of the counterexamples. Consequently, we produce subsets of close to full measure in those spaces on which these equations are locally well-posed. This is part of a project in progress with Chanillo, Czubak, Nahmod and Staffilani.