Mark Keel, Tristan Roy, and I have just uploaded to the arXiv the paper “Global well-posedness for the Maxwell-Klein-Gordon equation below the energy norm“, submitted to Discrete and Continuous Dynamical Systems. This project started about eight years ago, and was in fact a partial result was essentially finished by 2002, but managed to get put on the backburner for a while due to many other priorities. Anyway, this paper applies the I-method to the Maxwell-Klein-Gordon system of equations in the Coulomb gauge (a simplified model for the hyperbolic Yang-Mills equations) in three spatial dimensions. Previously to this paper, it was known that the Cauchy problem was globally well-posed in the energy norm (which is essentially H^1({\Bbb R}^3)) and locally well-posed in H^s({\Bbb R}^3) for s > 1/2, with this condition being essentially best possible except for the endpoint. Here we manage to lower the regularity threshold for global wellposedness to s > \sqrt{3}/2 \approx 0.866.  (The partial result alluded to earlier was for s > 7/8 = 0.875; at one point we had announced an improvement to 5/6 \approx 0.833, but the argument turned out to be flawed.) This is part of what is now quite a large family of such “global well-posedness below the energy norm” results, but there are some notable technical features here which were not present in earlier works. Firstly, we can show that there is no smoothing effect in the nonlinearity, ruling out use of the Fourier truncation method. Secondly, due to our use of rescaling, supercritical quantities such as the L^2 norm are not under control, which necessitates some unusual treatment of the low frequency portions of the scalar and vector fields. Namely, they are estimated in L^p spaces rather than L^2 ones. This complicates a number of tasks, ranging from controlling the elliptic theory, to understanding the coercive nature of the Hamiltonian, to establishing the nonlinear commutator estimates underlying the almost conservation law.

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