Harmonic Analysis and Hyperbolic Partial Differential Equations

University Of Washington, Seattle WA

Investigators

Abstract

PI: Hart Smith, University of Washington DMS - 0140499 Abstract: -------------------------------------------------------- The investigator's research focuses on the behavior of solutions to hyperbolic equations in the setting of metrics of low regularity. The key tool is the construction, through wave packet techniques, of approximate solutions for linear wave equations with minimally regular metrics, which are then used to establish Strichartz and related estimates for exact solutions. One application of this work is to well-posedness for quasi-linear hyperbolic equations with initial data of low Sobolev regularity. In joint work the investigator has established a best possible result for general quasi-linear equations with quadratic growth in the inhomogeneity; proposed work investigates relaxing the regularity assumption in special cases such as the Einstein vacuum equation, where a null condition indicates that better results should hold. Wave packet techniques are also being used to establish norm estimates for eigenfunctions on Riemannian manifolds with metrics of limited differentiability. The proposed research includes establishing best possible bounds for compact manifolds with Lp pinched curvature. The investigator is also adapting the above methods to establish norm estimates on solutions to mixed-type wave equations with Dirichlet conditions on a convex obstacle. This is carried out by reflecting the metric across the boundary to obtain a Lipschitz metric on an open set. The geometry of the resulting geodesic flow suggests that wave packet techniques can be used to establish the same norm estimates on solutions as hold in the non-obstacle case. The proposed research involves the study of waves traveling in rough media; a rough medium being one where the physics which governs the speed of waves changes abruptly from point to point. By studying the properties of a special family of localized solitary waves, the investigator is able to answer questions about the possible concentration of energy that can occur for general waves traveling in such media. This work has important applications in the study of nonlinear wave equations; that is, situations where the wave can be considered to interact with itself. One such example is the gravitational field equation arising from Einstein's general theory of relativity, where the geometry of space itself is the object of the equation. Rough solutions, and thus a rough media, necessarily arise when considering what kind of singularities the theory can lead to. The research also has implications for investigating the fundamental vibrational modes in rough media. It is known that refraction in such media can lead to high concentrations of energy that can be detected by examining these modes. Work is being done to relate the possible degree of concentration of energy to the roughness of the underlying media. The research finds applications as well in studying the reflection of waves off obstacles. The techniques developed to study rough media are being used to show that waves reflecting off of convex obstacles must diffuse to the same degree as do waves traveling without reflection.