scholarly journals Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport

2020 ◽  
Author(s):  
Matt Jacobs ◽  
Inwon Kim ◽  
Alpár R. Mészáros
Keyword(s):  
Surface Tension ◽  
Weak Solutions ◽  
Mean Curvature Flow ◽  
Optimal Transport ◽  
Heat Content ◽  
Threshold Dynamics ◽  
Muskat Problem ◽  
Equilibrium Configurations ◽  
Wasserstein Space ◽  
Energy Convergence

Abstract 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.

scholarly journals The Muskat problem with surface tension and equal viscosities in subcritical $$L_p$$-Sobolev spaces

2021 ◽  
Author(s):  
Anca-Voichita Matioc ◽  
Bogdan-Vasile Matioc
Keyword(s):  
Mathematical Model ◽  
Surface Tension ◽  
Global Existence ◽  
Sobolev Spaces ◽  
Existence Of Solutions ◽  
Well Posedness ◽  
Global Existence Of Solutions ◽  
Muskat Problem ◽  
The Mathematical Model ◽  
Quasilinear Parabolic

AbstractIn this paper we establish the well-posedness of the Muskat problem with surface tension and equal viscosities in the subcritical Sobolev spaces $$W^s_p(\mathbb {R})$$ W p s ( R ) , where $${p\in (1,2]}$$ p ∈ ( 1 , 2 ] and $${s\in (1+1/p,2)}$$ s ∈ ( 1 + 1 / p , 2 ) . This is achieved by showing that the mathematical model can be formulated as a quasilinear parabolic evolution problem in $$W^{\overline{s}-2}_p(\mathbb {R})$$ W p s ¯ - 2 ( R ) , where $${\overline{s}\in (1+1/p,s)}$$ s ¯ ∈ ( 1 + 1 / p , s ) . Moreover, we prove that the solutions become instantly smooth and we provide a criterion for the global existence of solutions.

scholarly journals Second-Order Models for Optimal Transport and Cubic Splines on the Wasserstein Space

2019 ◽  
Vol 19 (5) ◽  
pp. 1113-1143 ◽  
Author(s):  
Jean-David Benamou ◽  
Thomas O. Gallouët ◽  
François-Xavier Vialard
Keyword(s):  
Optimal Transport ◽  
Cubic Splines ◽  
Second Order ◽  
Wasserstein Space

Embryonic tissues are viscoelastic materials

2000 ◽  
Vol 78 (3) ◽  
pp. 243-251 ◽  
Author(s):  
D A Beysens ◽  
G Forgacs ◽  
J A Glazier
Keyword(s):  
Surface Tension ◽  
Metastatic Potential ◽  
Relevant Information ◽  
Specific Cell ◽  
Biologically Relevant ◽  
Embryonic Tissues ◽  
Differential Adhesion ◽  
Equilibrium Configurations ◽  
Experimental Findings ◽  
Differential Adhesion Hypothesis

Early embryonic development is characterized by spectacular morphogenetic processes such as sorting or spreading of tissues. Analogy between viscoelastic fluids and certain properties of embryonic tissues turned out to be useful in interpreting some aspects of these morphogenetic phenomena. In accordance with the differential adhesion hypothesis, the values of tissue-specific surface tensions have been shown to be consistent with the equilibrium configurations such tissues reach in the course of sorting and spreading. A method to measure tissue surface tension and viscoelastic properties is described. Notions like the Laplace's equation relating surface tension to radii of curvature, or the Kelvin model of viscoelasticity are used to analyze the results of these measurements. The fluid analogy is extended to time-dependent phenomena, in particular, to the analysis of cellular pattern evolution in the course of spreading. On the basis of recent experimental findings, we demonstrate that the kinetics of spreading and nucleation in binary fluids can be analyzed using the same formalism. We illustrate how our results can be used to obtain biologically relevant information on the strength of binding between specific cell adhesion molecules under near physiological conditions. We also suggest a diagnostic application of our method to monitor the metastatic potential of tumors. PACS No.: 03.65Ge

scholarly journals Mixing solutions for the Muskat problem with variable speed

2020 ◽  
Author(s):  
Florent Noisette ◽  
László Székelyhidi
Keyword(s):  
Weak Solutions ◽  
Variable Speed ◽  
Mixing Speed ◽  
Link Type ◽  
Muskat Problem ◽  
Quick Proof ◽  
Math Phys

AbstractWe provide a quick proof of the existence of mixing weak solutions for the Muskat problem with variable mixing speed. Our proof is considerably shorter and extends previous results in Castro et al. (Mixing solutions for the Muskat problem, 2016, arXiv:1605.04822) and Förster and Székelyhidi (Comm Math Phys 363(3):1051–1080, 2018).

The Muskat problem with surface tension and a nonregular initial interface

2011 ◽  
Vol 74 (17) ◽  
pp. 6074-6096 ◽  
Author(s):  
Borys V. Bazaliy ◽  
Nataliya Vasylyeva
Keyword(s):  
Surface Tension ◽  
Muskat Problem

The action of certain chemical substances on the zoospores of Pseudoperonospora Humuli (Miy. et Takah.) Wils

1929 ◽  
Vol 19 (1) ◽  
pp. 185-200 ◽  
Author(s):  
W. Goodwin ◽  
E. S. Salmon ◽  
W. M. Ware
Keyword(s):  
Surface Tension ◽  
Weak Solutions ◽  
Soft Soap ◽  
Fungicidal Action ◽  
Chemical Substances ◽  
Pseudoperonospora Humuli ◽  
Other Substances

1. Both Pseudoperonospora Humuli and Phytophihora infestans are extremely susceptible in the zoospore stage to the action of weak solutions of soap or saponin. The zoospores are caused to disintegrate suddenly, apparently by changes in surface tension, within 60 seconds, in solutions containing over 0·1 per cent, soft soap. Those of P. Humuli are more vulnerable than those of P. infestans.2. The fungicidal action of soap and saponin mixed with certain adherent substances was tested on hop plants.3. The power of adhesion and the fungicidal efficiency of the mixtures were tested by allowing single drops to dry on the surface of watch glasses and by then adding drops of water containing zoospores.4. Other substances, e.g. aluminium-lime mixture, glycerine, iodine, bromine, were also found to kill zoospores rapidly.

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