no. 4
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.

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 W2,p, 1<p≤∞. 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.

DOI : 10.1016/j.anihpc.2016.09.001
Classification : 76S05, 35Q35
Mots clés : Porous medium, Darcy's law, Muskat problem, Maximum principle 
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     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},
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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. http://www.numdam.org/articles/10.1016/j.anihpc.2016.09.001/

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