Global existence and convergence of solutions to gradient systems and applications to Yang-Mills gradient flow

Abstract

Preface xv Acknowledgments xix Chapter 1. Introduction 7.1. Anti-self-dual curvature flow over four-dimensional manifolds 7.2. Chern-Simons gradient flow over three-dimensional manifolds 7.3. Donaldson heat flow 7.4. Fluid dynamics 7.5. Knot energy gradient flows 7.6. Lagrangian mean curvature flow 7.7. Mean curvature flow 7.8. Seiberg-Witten gradient flow 7.9. Yang-Mills gradient flow over cylindrical-end manifolds 7.10. Yang-Mills-Higgs gradient flow over Kähler surfaces 7.11. Additional gradient flows in mathematical physics and applied mathematics: Cahn-Hilliard, Ginzburg-Landau, Kirchoff-Carrier and related energy functionals Chapter 3. Preliminaries 8. Classical and weak solutions to the Yang-Mills gradient flow equation 9. On the heat equation method 10. Classification of principal G-bundles, the Chern-Weil formula, and absolute minima of the Yang-Mills energy functional 11. Critical points of the Yang-Mills energy functional and asymptotic limits of Yang-Mills gradient flow Chapter 4. Linear and nonlinear evolutionary equations in Banach spaces 12. Linear evolutionary equations in Banach spaces 12.1. Analytic semigroups and sectorial operators 12.2. Fractional powers and interpolation spaces 12.3. Solution concepts and the variation of constants formula 12.4. Solutions via analytic semigroup theory 12.5. Weak solutions to evolutionary linear equations in Hilbert spaces 13. Local existence for a nonlinear evolution equation in a Banach space 13.1. Local existence and uniqueness results for mild solutions 13.2. Strong solutions 13.3. Maximally defined solutions 13.4. Continuous dependence of solutions 13.5. A standard result for long time existence 13.6. Regularity in space and time 13.7. Existence and uniqueness of mild solutions to a nonlinear evolution equation in a Banach space with initial data of minimal regularity I 13.8. Existence and uniqueness of mild solutions to a nonlinear evolution equation in a Banach space with initial data of minimal regularity II Chapter 5. A priori estimates, existence, uniqueness, and regularity for elliptic and parabolic partial differential systems on manifolds 14. Elliptic partial differential systems and analytic semigroups on L p , C 0 , and L 1 Banach spaces 14.1. Sobolev embedding and multiplication theorems for real derivative exponents 14.2. L p theory for scalar elliptic partial differential and pseudo-differential operators on R d and applications to elliptic systems x CONTENTS 18. Critical-exponent parabolic Sobolev spaces and linear parabolic operators on sections of vector bundles over compact manifolds 19. Local well-posedness for the Yang-Mills heat equation with initial data of minimal regularity 19.1. L 2 (0, T ; L 2 (X)) estimates for quadratic and cubic terms arising in the Yang-Mills heat equation 19.2. Local existence of solutions to the Yang-Mills heat equation over a closed manifold with dimension less than or equal to four and small initial data in H 1 19.3. Local existence of solutions to the Yang-Mills heat equation over a closed manifold with ...

A Conformally Invariant Gap Theorem in Yang–Mills Theory

“…By employing Lojasiewicz-Simon arguments (cf. [26] Proposition 7.2, [36], [14] Theorem 7, all based on the classic [32] Theorem 2) one can improve this C ∞ Uhlenbeck subsequential convergence to convergence of the entire flow line.…”

A state space for 3D Euclidean Yang-Mills theories

“…Remark 3.22. By Proposition 3.19, we can use (ZDDS) to smooth out rough initial data (even more, see [24,Section 19.7] for a local existence result for L 3 initial data). Moreover, this smoothing procedure is gauge invariant, in the sense that if B is a solution to (ZDDS) on some interval [0, T ), then for all 0 < s < t < T , we have that [B(t)] = Φ t−s ([B(s)]) (by Lemmas 3.21 and 3.10).…”

The Yang-Mills heat flow with random distributional initial data

“…Since these results concern smooth initial data, they are not new. For instance, the local existence and regularity results stated below can be obtained by combining the various general results of [18,Sections 17.4,17.5,and 20.1]. However, we will still prove these results, both because we were not able to find any short, direct proofs in the literature, and because several parts of the arguments will be needed later on.…”

“…Indeed, one of the main methods for showing local existence of (YM) for various types of initial data is to first show it for (ZDDS), and then use a well-known procedure to obtain solutions to (YM) out of solutions to (ZDDS) (see, e.g., [9,Section 1.3]). By now, local existence of solutions to (ZDDS) has been established for various classes of initial data -again, see the survey [18], as well as [9,19]. However, as far as we can tell, there are no results for distributional initial data.…”