Celebratio Mathematica — Choquet-Bruhat

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Y. Choquet-Bruhat, S. M. Paneitz, and I. E. Segal: “The Yang–Mills equations on the universal cosmos,” J. Funct. Anal. 53 : 2 (1983), pp. 112–150. MR 722506 Zbl 0535.58022 article

Global existence and regularity of solutions for the Yang–Mills equations on the universal cosmos \( \tilde{\mathbf{M}} \), which has the form \( R^1 \times S^3 \) for each of an 8-parameter continuum of factorizations of \( \tilde{\mathbf{M}} \) as \( \textit{time}\times\textit{space} \) are treated by general methods. The Cauchy problem in the temporal gauge is globally soluble in its abstract evolutionary form with arbitrary data for the \( \textit{field}\oplus\textit{potential} \) in \[ L_{2,r}(S^3)\oplus L_{2,r+1}(S^3) ,\] where \( r \) is an integer \( > \) 1 and \( L_{2,r} \) denotes the class of sections whose first \( r \) derivatives are square-integrable; if \( r= 1 \), the problem is soluble locally in time. When \( r \) is 3 or more the solution is identifiable with a classical one; if infinite, the solution is in \( C^{\infty}(\tilde{\mathbf{M}}) \). These results extend earlier work and approaches. Solutions of the equations on Minkowski space-time \( M \), extend canonically (modulo gauge transformations) to solutions on \( \tilde{\mathbf{M}} \) provided their Cauchy data are moderately smooth and small near spatial infinity. Precise asymptotic structures for solutions on \( \mathbf{M}_0 \), follow, and in turn imply various decay estimates. Thus the energy in regions uniformly bounded in direction away from the light cone is \( O(|x_0|^{-5}) \), where \( x_0 \) is the Minkowski time coordinate; analysis solely in \( \mathbf{M}_0 \), earlier yielded the estimate \( O(|x_o|^{-2}) \) applicable to the region within the light cone. Similarly it follows that the action integral for a solution of the Yang–Mills equuions in \( \mathbf{M}_0 \), is finite, in fact absolutely convergent.

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