By Spencer J. Bloch, R. Keith Dennis, Eric M. Friedlander, Micahel Stein (ed.)

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Progress in Nonlinear Differential Equations and Their Applications, Vol. 29 © 1997 Birkhauser Verlag Basel/Switzerland On the Regularity of Classical Field Theories in Minkowski Space-Time R3+ 1 S. KLAINERMAN Department of Mathematics, Princeton University Princeton, N J 08544 ABSTRACT. One of the central issues in the theory of nonlinear partial differential equations is that of regularity or break-down of solutions to the important physical examples. This is intimately tied to the basic mathematical question of what we actually mean by solutions and, from a physical point of view, to the understanding of the very limits of validity of the corresponding physical theories.

We wish to study Mre near this trivial solution. ; - q( 1/J), Dw1/J) which vanishes precisely on the solutions to the Seiberg-Witten equations. Its linearization L is a Fredholm map given by the matrix which we view as having domain ker d* EB r(V+). ' in the second coordinate is Clifford multiplication). After splitting off ker L from the domain and coker L from the range, L becomes an isomorphism. Let P1 be the projection of n~ x r(V-) orthogonal to coker L, and let Po be the complementary projection onto coker L.

The problem of global regularity for the Yang-Mills equation in Minkowski space-time, for sufficiently regular initial data, was solved in a beautiful paper by Eardley-Moncrief [E-M]. The proof required an insightful observation concerning the structure of the nonlinear terms of the Yang-Mills equations expressed in the Cronstrom gauge. It also uses in an essential way the explicit form of the fundamental solution of the wave equation in the fiat Minkowski space-time. Finally, in view of the same simple minded scaling analysis we can easily check that, relative to the total ADM mass 12 , the Einstein field equations are supercritical 13 .