On the Global Regularity Properties of Nonlinear Wave Equations

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9706754 Klainerman One of the central issues in the theory of nonlinear Partial Differential Equations of evolution is that of regularity or break-down of solutions. The best known problems of this type, in fluid mechanics, concern the regularity of the three dimensional Euler and Navier-Stokes equations. Another well known problem is that of finding a correct concept of global generalized solutions, including shock discontinuities, to compressible Euler equations. Similar questions arise also in General Relativity. The problem of regularity has been formulated, by Penrose, under the code name of "Cosmic Censorship". It states, roughly, that the stable solutions of the Einstein Field Equations should exclude "naked singularities", i.e. singularities not hidden by black holes. Most of these problems are today out of reach. My objective is to help develop tools for attacking these fundamental problems by concentrating my attention on the simplified situation of the Yang-Mills and Wave Maps field equations. As is known, the Yang-Mills equations are at the heart of our most successful particle physics theory, the so called standard model. The Wave Maps equations appear naturally when we look for special solutions of the Einstein equations. The question of break-down of solutions to the Euler and Navier-Stokes equations governing the behavior of fluids and gases and of the Einstein field equations of General Relativity is of fundamental significance. In the case of the former, this problem is intimately tied to the highly unstable behavior of fluids and has bearing on the practical problems of taming turbulence and predicting the weather. In the case of General Relativity, the problem has deep significance to our understanding of the Universe, since black holes and singularities are the most important, and least understood, predictions of General Relativity. While from a physical point of view these theories seem very different, mathematically they bear a lot of similarities. Due to the non-linear character of these problems, they have led, throughout this century, to the development of new qualitative techniques in which mathematicians have played a leading role. Some of these techniques are now routinely used in practice. The final goals remain distant, and the investigator would consider this proposal a ringing success if it can help make a dint on these difficult problems.