On the optimal local regularity for gauge field theories

Abstract

In [5], [6] we have proved that the Maxwell-Klein-Gordon equations, respectively Yang-Mills equations, in R 3+1 are well posed in the energy norm.The result relied on the precise structure of the equations relative to the Coulomb gauge, and on the space-time estimates for null forms proved earlier in [4].However, the results proved in [5], [6] are not sharp; based on the scaling properties of the equation we expect that the Maxwell-Klein-Gordon and Yang-Mills equations in R n+1 are well posed in the Sobolev spaces H s (R n ) for all s > n 2 2 , or even s n 2 2 in a suitable sense, at least when n 4. In particular, for the critical case n = 4 the optimal exponent s = 1 corresponds to the energy norm.In this case local well-posedness for finite energy solutions would also imply global well-posedness.In this paper we present the main estimates which allow one to prove local well posedness in H s for all s > n 2 2 in the case of the M.K.G. equations in 4 + 1 dimensions.For simplicity we will demonstrate the relevance of our estimates for a model problem which presents the main di culties manifest in the Maxwell-Klein-Gordon expressed relative to the Coulomb gauge.We consider systems of wave equations in R n+1 with quadratic nonlinearities in = ( I ) I=1,...,M , = ( I ) I=1,...,N written symbolically in the form,subject to the initial conditions, at t = 0, = f 0 , @ t = f 1 and = g 0 , @ t = g 1 .(1.1c)

Locations

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