Decay and Well-Posedness of Solutions to Hyperbolic Partial Differential Equations

The PI’s research is in the field of dispersive partial differential equations, in particular in questions regarding decay and well-posedness for wave equations on curved backgrounds. The PI has been working on understanding decay of solutions to the linear wave equation on Schwarzschild and Kerr spacetimes, which occur naturally in mathematical physics. Even the linear problem is quite difficult, since the complicated background geometry affects the dispersion properties in nontrivial ways. In compact regions one must deal with high frequency wave packets that linger along trapped geodesics for a long time, while at infinity the non-Euclidean character of the metric affects the pointwise rates of decay. The very delicate (and unstable) nature of the trapped set in particular requires tools coming from harmonic analysis, differential equations, and differential geometry. The PI proposes to use the robust ways of measuring decay (e.g. local energy estimates and Strichartz estimates) developed for linear waves to tackle nonlinear problems.