# On the global existence for the Muskat problem

Peter Constantin; Diego Córdoba; Francisco Gancedo; Robert M. Strain

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 1, page 201-227
- ISSN: 1435-9855

## Access Full Article

top## Abstract

top## How to cite

topConstantin, Peter, et al. "On the global existence for the Muskat problem." Journal of the European Mathematical Society 015.1 (2013): 201-227. <http://eudml.org/doc/277179>.

@article{Constantin2013,

abstract = {The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\mathbb \{R\})$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\left\Vert f\right\Vert _1\le 1/5$. Previous results of this sort used a small constant $\epsilon <<1$ which was not explicit. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy $\left\Vert f_0\right\Vert _\{L^\infty \}<\infty $ and $\left\Vert \partial _xf_0\right\Vert _\{L^\infty \}<1$. We take advantage of the fact that the bound $\left\Vert \partial _xf_0\right\Vert _\{L^\infty \}<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.},

author = {Constantin, Peter, Córdoba, Diego, Gancedo, Francisco, Strain, Robert M.},

journal = {Journal of the European Mathematical Society},

keywords = {porous media; incompressible flows; fluid interface; global existence; dynamics of the interface; porous media; incompressible flows; fluid interface; dynamics of the interface},

language = {eng},

number = {1},

pages = {201-227},

publisher = {European Mathematical Society Publishing House},

title = {On the global existence for the Muskat problem},

url = {http://eudml.org/doc/277179},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Constantin, Peter

AU - Córdoba, Diego

AU - Gancedo, Francisco

AU - Strain, Robert M.

TI - On the global existence for the Muskat problem

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 1

SP - 201

EP - 227

AB - The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an $L^2(\mathbb {R})$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance $\left\Vert f\right\Vert _1\le 1/5$. Previous results of this sort used a small constant $\epsilon <<1$ which was not explicit. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy $\left\Vert f_0\right\Vert _{L^\infty }<\infty $ and $\left\Vert \partial _xf_0\right\Vert _{L^\infty }<1$. We take advantage of the fact that the bound $\left\Vert \partial _xf_0\right\Vert _{L^\infty }<1$ is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.

LA - eng

KW - porous media; incompressible flows; fluid interface; global existence; dynamics of the interface; porous media; incompressible flows; fluid interface; dynamics of the interface

UR - http://eudml.org/doc/277179

ER -

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.