E-grāmata: Yang-Mills Heat Equation with Finite Action in Three Dimensions

"The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R3 and over a bounded open convex set in R3. The initial data is taken to lie in the Sobolev space of order one half, which is the critical Sobolev index for this equation over a three dimensional manifold. The existence is proven by solving first an augmented, strictly parabolic equation and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. The gauge functions needed to carry out this procedure lie in the critical gauge group of Sobolev regularity three halves, which is a complete topological group in a natural metric but is not a Hilbert Lie group. The nature of this group must be understood in order to carry out the reconstruction procedure. Solutions to the Yang-Mills heat equation are shown to be strong solutions modulo these gauge functions. Energy inequalities and Neumann domination inequalities are used to establish needed initial behavior properties of solutions to the augmented equation"--

Gross proves the existence and uniqueness of solutions to the Yang-Mills heat equation over R3 and over a bounded open convex set in R3. He finds his initial data in the Sobolev space of order one half, which, he says, is the critical Sobolev index for this equation over a three-dimensional manifold. He proves the existence by solving first an augmented, strictly parabolic equation, and then gauge transforming the solution to a solution of the Yang-Mills heat equation itself. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)