The Yang-Mills equations on the universal cosmos

Abstract

Global existence and regularity of solutions for the Yang-Mills equations on the universal cosmos M̃, which has the form R 1 × S 3 for each of an 8-parameter continuum of factorizations of M̃ as time × 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 field ⊕ potential in L 2, r ( S 3) ⊕ 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 ∞( M̃). These results extend earlier work and approaches [1–5]. Solutions of the equations on Minkowski space-time M 0 extend canonically (modulo gauge transformations) to solutions on M̃ provided their Cauchy data are moderately smooth and small near spatial infinity. Precise asymptotic structures for solutions on 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 M 0 [8,9] earlier yielded the estimate O(¦x 0¦ −2) applicable to the region within the light cone. Similarly it follows that the action integral for a solution of the Yang-Mills equations in M 0 is finite, in fact absolutely convergent.