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Here Fαβ = ∂aAβ − ∂bAb + [ Aα , Aβ ] represents the curvature of a connection 1form, or gauge field, A = Aαdx with values in the Lie algebra of a classical Lie group of matrices such as SU(N) or SO(N). The equations (1.1) are invariant, up to a conjugation, under the gauge transformations Aα −→ OAαO−1 − ∂αOO−1, with O elements of the corresponding group. They are obtained by considering the critical points corresponding to the Lagrangian 14 〈Fαβ , Fαβ〉 with 〈 , 〉 the positive definite Killing form of the Lie algebra. The equations have a finite number of conservation laws, among them the total energy,

On the optimal local regularity for the Yang-Mills equations in ℝ⁴⁺¹