scholarly journals Generalized Gradient Flow Equation and Its Applications

2016 ◽  
Author(s):  
Kengo Kikuchi ◽  
Sinya Aoki ◽  
Tetsuya Onogi
Keyword(s):  
Gradient Flow ◽  
Flow Equation ◽  
Generalized Gradient

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

Global Solutions to the Gradient Flow Equation of a Nonconvex Functional

2006 ◽  
Vol 37 (5) ◽  
pp. 1657-1687 ◽  
Author(s):  
G. Bellettini ◽  
M. Novaga ◽  
E. Paolini
Keyword(s):  
Gradient Flow ◽  
Global Solutions ◽  
Flow Equation ◽  
Nonconvex Functional

scholarly journals A Generalization of Onsager’s Reciprocity Relations to Gradient Flows with Nonlinear Mobility

2016 ◽  
Vol 41 (2) ◽  
Author(s):  
Alexander Mielke ◽  
D. R. Michiel Renger ◽  
Mark A. Peletier
Keyword(s):  
Evolution Equations ◽  
Gradient Flow ◽  
Maximal Entropy ◽  
Gradient Flows ◽  
Generalized Gradient ◽  
Time Reversibility ◽  
Steepest Ascent ◽  
Reciprocity Relations ◽  
Generalized Symmetry ◽  
The Many

AbstractOnsager’s 1931 “reciprocity relations” result connects microscopic time reversibility with a symmetry property of corresponding macroscopic evolution equations. Among the many consequences is a variational characterization of the macroscopic evolution equation as a gradient-flow, steepest ascent, or maximal entropy production equation. Onsager’s original theorem is limited to close-to-equilibrium situations, with a Gaussian-invariant measure and a linear macroscopic evolution. In this paper, we generalize this result beyond these limitations and show how the microscopic time reversibility leads to natural generalized symmetry conditions, which take the form of generalized gradient flows.

scholarly journals Nonminimal gradient flows in QCD-like theories

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Marco Boers
Keyword(s):  
Degrees Of Freedom ◽  
Anomalous Dimension ◽  
Gradient Flow ◽  
Vacuum Expectation ◽  
Vacuum Expectation Value ◽  
Flow Equation ◽  
Gradient Flows ◽  
Leading Order ◽  
Flow Equations ◽  
Yang Mills

Abstract The Yang-Mills gradient flow for QCD-like theories is generalized by including a fermionic matter term in the gauge field flow equation. We combine this with two different flow equations for the fermionic degrees of freedom. The solutions for the different gradient flow setups are used in the perturbative computations of the vacuum expectation value of the Yang-Mills Lagrangian density and the field renormalization factor of the evolved fermions up to next-to-leading order in the coupling. We find a one-parameter family of flow systems for which there exists a renormalization scheme in which the evolved fermion anomalous dimension vanishes to all orders in perturbation theory. The fermion number dependence of different flows is studied and applications to lattice studies are anticipated.

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