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On the optimal local regularity for the Yang-Mills equations in \(\mathbb{R}^{4+1}\). (English) Zbl 0924.58010
The Yang-Mills equations are considered in the form: \[ \phi_{tt}^{I} - \triangle \phi^{I} = N^{I} (\phi, \phi). \tag{1} \] Each \(N^{I}\) is a linear combination with constant real coefficients of terms of the type \(| D_{x} |^{-1} Q_{ij} (\phi^{I}\), \(\phi^{J})\) and \(Q_{ij} ( | D_{x} |^{-1} \phi^{I},\phi^{J},)\) where the “null quadratic” forms \(Q_{ij}\) are defined by: \[ Q_{ij} (u,v) = \partial_{i} u \partial_{j} v - \partial_{j} u \partial_{i} v . \tag{2} \] The authors concentrate their attention on the model problem in \(\mathbb{R}^{n+1}, n \geq 4 \). The aim of the paper is to develop the main Fourier analysis techniques which are needed in the study of optimal well-posedness and global regularity properties of the Yang-Mills equations in Minkowski spacetime \(\mathbb{R}^{n+1}\), for the case of the critical dimension \(n=4\). The main result is as follows:
Theorem. Suppose that \(s \geq s_{c} + 4 \delta \). Then the semilinear equation (1) is locally well posed for small initial data in \( H^{s} \times H^{s-1} \).
Reviewer: L.G.Vulkov (Russe)

MSC:
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
35B65 Smoothness and regularity of solutions to PDEs
35Q40 PDEs in connection with quantum mechanics
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