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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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