Voir la notice de l'article provenant de la source Numdam
We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class , . We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.
Keywords: Porous medium, Darcy's law, Muskat problem, Maximum principle
@article{AIHPC_2017__34_4_1041_0,
author = {Constantin, Peter and Gancedo, Francisco and Shvydkoy, Roman and Vicol, Vlad},
title = {Global regularity for {2D} {Muskat} equations with finite slope},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {1041--1074},
publisher = {Elsevier},
volume = {34},
number = {4},
year = {2017},
doi = {10.1016/j.anihpc.2016.09.001},
zbl = {1365.76304},
mrnumber = {3661870},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.09.001/}
}
TY - JOUR AU - Constantin, Peter AU - Gancedo, Francisco AU - Shvydkoy, Roman AU - Vicol, Vlad TI - Global regularity for 2D Muskat equations with finite slope JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 1041 EP - 1074 VL - 34 IS - 4 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.09.001/ DO - 10.1016/j.anihpc.2016.09.001 LA - en ID - AIHPC_2017__34_4_1041_0 ER -
%0 Journal Article %A Constantin, Peter %A Gancedo, Francisco %A Shvydkoy, Roman %A Vicol, Vlad %T Global regularity for 2D Muskat equations with finite slope %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 1041-1074 %V 34 %N 4 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.09.001/ %R 10.1016/j.anihpc.2016.09.001 %G en %F AIHPC_2017__34_4_1041_0
Constantin, Peter; Gancedo, Francisco; Shvydkoy, Roman; Vicol, Vlad. Global regularity for 2D Muskat equations with finite slope. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 1041-1074. doi: 10.1016/j.anihpc.2016.09.001
Cité par Sources :