The Yang-Mills Heat Equation with Finite Action in Three Dimensions

Description

The existence and uniqueness of solutions to the Yang-Mills heat equation is proven over R 3 and over a bounded open convex set in R 3 . 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. Cover Title page Chapter 1. Introduction Chapter 2. Statement of results 2.1. Strong solutions 2.2. Gauge groups 2.3. Main theorem 2.4. The ZDS procedure and the augmented equation Chapter 3. Solutions for the augmented Yang-Mills heat equation 3.1. The integral equation and path space 3.2. Free propagation lies in the path space \P_{𝑇}^{𝑎} 3.3. Contraction estimates 3.4. Proof of existence of mild solutions 3.5. 𝐶(⋅) has finite action 3.6. Mild solutions are strong solutions Chapter 4. Initial behavior of solutions to the augmented equation 4.1. Identities 4.2. Differential inequalities and initial behavior 4.3. Initial behavior, order 0 4.4. Initial behavior, order 1 4.5. Initial behavior, order 2 4.6. The case of infinite action 4.7. High 𝐿^{𝑝} bounds via Neumann domination Chapter 5. Gauge groups 5.1. Notation and statements 5.2. Multiplier bounds for 𝐴𝑑𝑔 5.3. \G_{1,𝑝} and \G₁₊ₐ are groups 5.4. \G_{1,𝑝} is a topological group 5.5. \G₁₊ₐ is a topological group if 𝑎≥1/2 5.6. Completeness Chapter 6. The conversion group 6.1. The ZDS procedure 6.2. 𝑔 estimates 6.3. The vertical projection 6.4. Integral representation of 𝑔_{𝜀}⁻¹𝑑𝑔_{𝜀} 6.5. Estimates for 𝑔_{𝜀}⁻¹𝑑𝑔_{𝜀} 6.6. Convergence of 𝑔_{𝜀}⁻¹𝑑𝑔_{𝜀} 6.7. Smooth ratios 6.8. Proof of Theorem 6.2 Chapter 7. Recovery of 𝐴 from 𝐶 7.1. Construction of 𝐴 7.2. Initial behavior of 𝐴 7.3. Uniqueness of 𝐴 Bibliography Back Cover