On the global existence for the Muskat problem

Constantin, 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 -