scholarly journals A Cahn-Hilliard-Biot system and its generalized gradient flow structure

2021 ◽  
pp. 107799
Author(s):  
Erlend Storvik ◽  
Jakub Wiktor Both ◽  
Jan Martin Nordbotten ◽  
Florin Adrian Radu
Keyword(s):  
Flow Structure ◽  
Gradient Flow ◽  
Generalized Gradient ◽  
Biot System

scholarly journals Convergence of a fully discrete and energy-dissipating finite-volume scheme for aggregation-diffusion equations

2020 ◽  
Vol 30 (13) ◽  
pp. 2487-2522
Author(s):  
Rafael Bailo ◽  
José A. Carrillo ◽  
Hideki Murakawa ◽  
Markus Schmidtchen
Keyword(s):  
Finite Volume ◽  
Flow Structure ◽  
Gradient Flow ◽  
Diffusion Equations ◽  
Finite Volume Scheme ◽  
Time Step ◽  
Discrete Energy ◽  
Volume Scheme ◽  
Fully Discrete ◽  
Local Aggregation

We study an implicit finite-volume scheme for nonlinear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in [R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient flow structure, arXiv:1811.11502 ]. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

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.

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