Existence of Solutions to Hyperbolic Differential Equations in Mathematical Physics

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PI: Hans Lindblad, Univ. of Cal. San DiegoDMS-0200226Abstract:Lindblad's research concerns basic mathematical questions for systems of nonlinear hyperbolic equations in mathematical physics. These include several important equations in classical field theory and continuum mechanics as well as in classical physics, e.g. Einstein's equations in general relativity and the equation of fluids and elastic bodies. These basic questions are: (i) Do we have local existence and uniqueness of solutions in a certain class? (ii) Do we have blow-up of solutions? (e.g. black holes in general relativity) (iii) What is the long time behavior of solutions? More specifically, Lindblad is mainly working on two projects. One on proving the well-posedness for a class of problems that occur in fluid dynamics and general relativity, in particular proving the well-posedness for the free boundary problem of the motion of the surface of a fluid in vacuum. A long term goal is to study the long time behavior of astrophysical bodies such as gaseous stars as well as the surface of the ocean. Another project is to study global solutions of equations related to Einstein's equations. A long term goal is to simplify and generalize the global existence results for Einstein's equations. These two problems are related to the question of whether the fundamental equations in physics have global solutions. Solution to these questions could have important consequences. For instance, it is conceivable that one could use the knowledge obtained from the solutions of Einstein's equations to permit the use of gravitational waves to observe the Universe. Understanding the properties of and controlling the interface between two fluids could have industrial applications. To solve these problems Lindblad and his collaborators are developing new techniques that could be useful for studying many other problems as well. In particular, they are using geometric methods to study hyperbolic differential equations.