Harmonic Analysis and Hyperbolic Partial Differential Equations

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PI: Hart Smith, University of WashingtonDMS - 0140499Abstract:--------------------------------------------------------The investigator's research focuses on the behavior of solutions tohyperbolic equations in the setting of metrics of low regularity.The key tool is the construction, through wave packet techniques, ofapproximate solutions for linear wave equations with minimally regularmetrics, which are then used to establish Strichartz and related estimatesfor exact solutions. One application of this work is to well-posednessfor quasi-linear hyperbolic equations with initial data of low Sobolevregularity. In joint work the investigator has established a best possibleresult for general quasi-linear equations with quadratic growth in theinhomogeneity; proposed work investigates relaxing the regularity assumptionin special cases such as the Einstein vacuum equation, where a null conditionindicates that better results should hold. Wave packet techniques arealso being used to establish norm estimates for eigenfunctions onRiemannian manifolds with metrics of limited differentiability.The proposed research includes establishing best possible boundsfor compact manifolds with Lp pinched curvature. The investigatoris also adapting the above methods to establish norm estimateson solutions to mixed-type wave equations with Dirichlet conditions on aconvex obstacle. This is carried out by reflecting the metric across theboundary to obtain a Lipschitz metric on an open set. The geometry ofthe resulting geodesic flow suggests that wave packet techniques can beused to establish the same norm estimates on solutions as hold in thenon-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 waveschanges abruptly from point to point. By studying the properties of aspecial family of localized solitary waves, the investigator is able toanswer questions about the possible concentration of energy that can occurfor general waves traveling in such media. This work has importantapplications in the study of nonlinear wave equations; that is, situationswhere the wave can be considered to interact with itself. One such exampleis the gravitational field equation arising from Einstein's general theoryof relativity, where the geometry of space itself is the object of theequation. Rough solutions, and thus a rough media, necessarily arise whenconsidering what kind of singularities the theory can lead to. The researchalso has implications for investigating the fundamental vibrational modesin rough media. It is known that refraction in such media can lead to highconcentrations of energy that can be detected by examining these modes.Work is being done to relate the possible degree of concentration of energyto the roughness of the underlying media. The research finds applicationsas well in studying the reflection of waves off obstacles. The techniquesdeveloped to study rough media are being used to show that waves reflectingoff of convex obstacles must diffuse to the same degree as do wavestraveling without reflection.