scholarly journals GRADIENT FLOWS OF HIGHER ORDER YANG–MILLS–HIGGS FUNCTIONALS

2021 ◽  
pp. 1-31
Author(s):  
PAN ZHANG
Keyword(s):  
Blow Up ◽  
Higgs Field ◽  
Gradient Flow ◽  
Higher Order ◽  
Energy Estimates ◽  
Four Dimensions ◽  
Yang Mills ◽  
Blow Up Analysis ◽  
Long Time Existence ◽  
Time Existence

Abstract In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the $L^2$ -bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgs k-functional with Higgs self-interaction, we show that, provided $\dim (M)<2(k+1)$ , for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local $L^2$ -derivative estimates, energy estimates and blow-up analysis.

The elastic flow of curves on the sphere

2018 ◽  
Vol 3 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Anna Dall’Acqua ◽  
Tim Laux ◽  
Lin ◽  
Paola Pozzi ◽  
Adrian Spener
Keyword(s):  
Critical Points ◽  
Elastic Energy ◽  
Gradient Flow ◽  
Closed Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Short Time ◽  
Time Existence

Abstract We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

scholarly journals The lifespan for 3-dimensional quasilinear wave equations in exterior domains

2014 ◽  
Vol 26 (6) ◽  
Author(s):  
John Helms ◽  
Jason Metcalfe
Keyword(s):  
Wave Equations ◽  
Energy Estimates ◽  
Exterior Domains ◽  
Small Initial Data ◽  
3 Dimensional ◽  
Long Time Existence ◽  
Homogeneous Wave ◽  
Long Time ◽  
Quasilinear Wave Equations ◽  
Time Existence

AbstractThis article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.

scholarly journals A speed preserving Hilbert gradient flow for generalized integral Menger curvature

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Jan Knappmann ◽  
Henrik Schumacher ◽  
Daniel Steenebrügge ◽  
Heiko von der Mosel
Keyword(s):  
Gradient Descent ◽  
Gradient Flow ◽  
Optimization Methods ◽  
The Self ◽  
Initial Curve ◽  
Long Time Existence ◽  
Sobolev Gradient ◽  
Long Time ◽  
Menger Curvature ◽  
Time Existence

Abstract We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Finite-action solutions of higher-order yang-mills-higgs theory in four dimensions

1986 ◽  
Vol 92 (1) ◽  
pp. 107-115 ◽  
Author(s):  
J. Burzlaff ◽  
D. H. Tchrakian
Keyword(s):  
Higher Order ◽  
Four Dimensions ◽  
Yang Mills ◽  
Finite Action

scholarly journals Long-time existence for Yang–Mills flow

2019 ◽  
Vol 217 (3) ◽  
pp. 1069-1147 ◽  
Author(s):  
Alex Waldron
Keyword(s):  
Yang Mills ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

Saint-Venant's principle in blow-up for higher-order quasilinear parabolic equations

2003 ◽  
Vol 133 (5) ◽  
pp. 1075-1119 ◽  
Author(s):  
V. A. Galaktionov ◽  
A. E. Shishkov
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Theory Of Elasticity ◽  
Higher Order ◽  
Energy Estimates ◽  
Quasilinear Parabolic Equations ◽  
Group Invariant ◽  
Blowing Up ◽  
Jacobi Equations ◽  
Quasilinear Parabolic

We prove localization estimates for general 2mth-order quasilinear parabolic equations with boundary data blowing up in finite time, as t → T−. The analysis is based on energy estimates obtained from a system of functional inequalities expressing a version of Saint-Venant's principle from the theory of elasticity. We consider a special class of parabolic operators including those having fixed orders of algebraic homogenuity p > 0. This class includes the second-order heat equation and linear 2mth-order parabolic equations (p = 1), as well as many other higher-order quasilinear ones with p ≠ 1. Such homogeneous equations can be invariant under a group of scaling transformations, but the corresponding least-localized regional blow-up regimes are not group invariant and exhibit typical exponential singularities ~ e(T−t)−γ → ∞ as t → T−, with the optimal constant γ = 1/[m(p + 1) − 1] > 0. For some particular equations, we study the asymptotic blow-up behaviour described by perturbed first-order Hamilton–Jacobi equations, which shows that general estimates of exponential type are sharp.

scholarly journals New bi-harmonic superspace formulation of 4D, $$ \mathcal{N} $$ = 4 SYM theory

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
I. L. Buchbinder ◽  
E. A. Ivanov ◽  
V. A. Ivanovskiy
Keyword(s):  
Effective Action ◽  
Harmonic Superspace ◽  
Four Dimensions ◽  
Yang Mills ◽  
Convenient Tool ◽  
Leading Term

Abstract We develop a novel bi-harmonic $$ \mathcal{N} $$ N = 4 superspace formulation of the $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills theory (SYM) in four dimensions. In this approach, the $$ \mathcal{N} $$ N = 4 SYM superfield constraints are solved in terms of on-shell $$ \mathcal{N} $$ N = 2 harmonic superfields. Such an approach provides a convenient tool of constructing the manifestly $$ \mathcal{N} $$ N = 4 supersymmetric invariants and further rewriting them in $$ \mathcal{N} $$ N = 2 harmonic superspace. In particular, we present $$ \mathcal{N} $$ N = 4 superfield form of the leading term in the $$ \mathcal{N} $$ N = 4 SYM effective action which was known previously in $$ \mathcal{N} $$ N = 2 superspace formulation.

BLOW-UP ANALYSIS FOR LIOUVILLE TYPE EQUATION IN SELF-DUAL GAUGE FIELD THEORIES

2005 ◽  
Vol 07 (02) ◽  
pp. 177-205 ◽  
Author(s):  
HIROSHI OHTSUKA ◽  
TAKASHI SUZUKI
Keyword(s):  
Gauge Field ◽  
Blow Up ◽  
Type Equation ◽  
Singular Part ◽  
Gauge Field Theories ◽  
Field Theories ◽  
Liouville Type ◽  
Blow Up Analysis ◽  
The Green Function ◽  
Dirac Measures

We study the asymptotic behavior of the solution sequence of Liouville type equations observed in various self-dual gauge field theories. First, we show that such a sequence converges to a measure with a singular part that consists of Dirac measures if it is not compact in W1,2. Then, under an additional condition, the singular limit is specified by the method of symmetrization of the Green function.

scholarly journals Long-Time Existence for Semi-linear Beam Equations on Irrational Tori

2021 ◽  
Author(s):  
Joackim Bernier ◽  
Roberto Feola ◽  
Benoît Grébert ◽  
Felice Iandoli
Keyword(s):  
Long Time Existence ◽  
Beam Equations ◽  
Long Time ◽  
Time Existence
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