Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions

Feehan, Paul M. N.; Maridakis, Manousos

doi:10.48550/arxiv.1510.03815

Cited by 7 publications

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“…In [379], Wilkin has extended the results of Daskalopoulos [98] and Råde [293] for Yang-Mills gradient flow to the case of Yang-Mills-Higgs gradient flow on a Hermitian vector bundle over a closed Riemann surface; those results are extended to the case of a principal G-bundle over a closed Riemann surface by Biswas and Wilkin in [40]. The Yang-Mills-Higgs gradient flow is a type of coupled Yang-Mills gradient flow -see Feehan and Maridakis [138] and references cited therein for further details for further details and applications of the Lojasiewicz-Simon gradient inequality.…”

Section: Fluid Dynamicsmentioning

confidence: 86%

“…Just as in Case 1, Proposition 23.4 guarantees that the map, A ∞ + X ∋ A → M 1 (A) ∈ L (H A , H ), is continuous. Proposition 23.14 ensures that the gradient map, M : [138] to prove a refinement of Theorem 23.1 that holds for all d ≥ 2 and p ∈ [2, ∞) obeying p ≥ d/3, with A and A ∞ of class W 2,q and q > d/3. We digress in order to identify a dual space arising in the proof of Theorem 23.17 below.…”

Section: Proof By the Expression (238) For M (A) And Writingmentioning

confidence: 99%

“…Seiberg-Witten gradient flow. The Seiberg-Witten gradient flow is the gradient flow for the Seiberg-Witten action functional, and thus a type of coupled Yang-Mills gradient flowsee Feehan and Maridakis [138] and Hong and Schabrun [185] and references cited therein for further details and applications of the Lojasiewicz-Simon gradient inequality. 7.9.…”

Section: Fluid Dynamicsmentioning

confidence: 99%

“…In particular, it does not follow from standard references as in the proof of Theorem 23.1. However, we can apply our [138,Theorem 10] to give, for constants ε 0 = ε 0 (A 1 , A ∞ , g, p, q) ∈ (0, 1] and…”

Section: Given a Bounded Linear Operatormentioning

confidence: 99%

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

Feehan

2014

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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 ...

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Section: Fluid Dynamicsmentioning

confidence: 86%

Section: Proof By the Expression (238) For M (A) And Writingmentioning

confidence: 99%

Section: Fluid Dynamicsmentioning

confidence: 99%

Section: Given a Bounded Linear Operatormentioning

confidence: 99%

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

Feehan

2014

Preprint

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“…Remark 2.13 (Further applications). The proof of Proposition 2.12 could be adapted to extend [13,Lemma 41.1] and [18,Theorem A.1] from the case of elliptic partial differential operators with C ∞ coefficients to those with suitable Sobolev coefficients.…”

Section: By Definition Of Analyticity Of the Composition Fmentioning

confidence: 99%