## 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.

### MSC:

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

### Keywords:

Yang-Mills equations; initial value problem; finite energy
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