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Finite energy solutions of the Yang-Mills equations in \(\mathbb{R}^{3+1}\). (English) Zbl 0827.53056
The authors consider Yang-Mills equations \[ D^\mu F_{\alpha \mu} = 0, \quad F_{\alpha \mu} = (d A + [A, A])_{\alpha \mu} \tag{1} \] for a \(\mathfrak g\)-valued 1-form \(A\) on the Minkowski space \(\mathbb{R}^{1,3}\) where \(\mathfrak g\) is a compact Lie algebra and \(D_\mu = \partial / \partial x^\mu + \text{ad }A_\alpha\). They prove that the initial value problem with finite energy has a unique global solution (in an appropriate class of solutions).
More precisely, an initial data set at \(t = x^0 = 0\) is defined as a couple \((\overline {A}, \overline {E})\) of \(\mathfrak g\)-valued 1-forms in \(\mathbb{R}^3\), verifying the constraint condition \[ \text{div } \overline {E} + [\overline {A}, \overline {E}] = 0. \] The energy of an initial data set is defined as \[ {\mathcal E} = \left( {1\over 2} \int_{\mathbb{R}^3} (|\overline {E} |^2 + |H (\overline {A})|^2) dx\right)^{{1\over 2}}, \] where \(H(A) = dA + [A, A]\). An initial data set \((\overline {A}, \overline {E})\) is called locally \(H^2\) if for any bounded domain \(D \subset \mathbb{R}^3\) the norm \[ |\overline {A} |_{H^2 (D)} + |\overline {E} |_{H^1 (D)} < \infty \] where \(H^s (D)\) is the standard \(W^{2,s} (D)\) Sobolev space.
A development of an initial data set \((\overline {A}, \overline {E})\) in the temporal gauge \((A_0 = 0)\) is a solution \(A = (0, {\mathbf A})\) of (1) such that at \(t = 0\) \[ {\mathbf A} = \overline {A}, \quad \partial_t {\mathbf A} = \overline {E}. \] The main global existence theorem for the smooth data in the temporal gauge states that any finite energy initial data set in \(\mathbb{R}^3\) which is locally \(H^2\) admits a unique global development \(A(t,x)\) in \([0, \infty) \times \mathbb{R}^3\) which satisfies some norm estimations.
A generalization of this result to the case of generalized solutions of Yang-Mills equation is also proved.

53Z05 Applications of differential geometry to physics
81T13 Yang-Mills and other gauge theories in quantum field theory
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