Existence and Blow-Up of Solutions to Systems of Nonlinear Wave Equations

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Lindblad's research concerns basic mathematical questions for systems of nonlinear wave equations which include several important equations in physics, for instance, in classical field theory and continuum mechanics as well as in classical physics, e.g. Einstein's equations, the Yang-Mills equation and the equation of nonlinear elasticity. 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 working on two projects. One with Demetri Christodoulou on proving the well-posedness for a class of problems that occurr in fluid dynamics and general relativity, in particular proving the well-posedness for the free boundary problem of the surface of a fluid in vacuum. Another project is to study how regular initial data needed to ensure the existence of a local solution to nonlinear wave equations, in particular constructing counterexamples to local existence. 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. To solve these problems Linblad and his collaborators are developing completely new techniques that could be useful for studying many other problems as well, e.g. the properties of the interface between two fluids. Such a result would have industrial applications.