Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport
Jacobs, Matt; Kim, Inwon; Mészáros, Alpár R.
Dr Alpar Meszaros email@example.com
Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoḡlu–Otto and Laux–Otto), we introduce a novel framework to approximate solutions of the Muskat problem with surface tension. Our approach is based on interpreting the Muskat problem as a gradient flow in a product Wasserstein space. This perspective allows us to construct weak solutions via a minimizing movements scheme. Rather than working directly with the singular surface tension force, we instead relax the perimeter functional with the heat content energy approximation of Esedoḡlu–Otto. The heat content energy allows us to show the convergence of the associated minimizing movement scheme in the Wasserstein space, and makes the scheme far more tractable for numerical simulations. Under a typical energy convergence assumption, we show that our scheme converges to weak solutions of the Muskat problem with surface tension. We then conclude the paper with a discussion on some numerical experiments and on equilibrium configurations.
Jacobs, M., Kim, I., & Mészáros, A. R. (2020). Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport. Archive for Rational Mechanics and Analysis, 239(1), 389-430. https://doi.org/10.1007/s00205-020-01579-3
|Journal Article Type||Article|
|Acceptance Date||Sep 15, 2020|
|Online Publication Date||Oct 14, 2020|
|Deposit Date||Oct 22, 2020|
|Publicly Available Date||Oct 23, 2020|
|Journal||Archive for Rational Mechanics and Analysis|
|Peer Reviewed||Peer Reviewed|
Published Journal Article (Advance online version)
Publisher Licence URL
Advance online version This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
You might also like
A variational approach to first order kinetic Mean Field Games with local couplings
Degenerate nonlinear parabolic equations with discontinuous diffusion coefficients
The planning problem in mean field games as regularized mass transport