Global regularity for 2D Muskat equations with finite slope
Global regularity for 2D Muskat equations with finite slope
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DOI：
10.1016/j.anihpc.2016.09.001
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发表时间：
2017-07-01
影响因子：
1.9
通讯作者：
Vicol, Vlad
中科院分区：
文献类型：
--
作者：
Constantin, Peter;Gancedo, Francisco;Vicol, Vlad
本文研究了不可压缩多孔介质中两个常密度流体界面的二维Muskat方程，其速度由达西定律给出。我们建立，只要两种流体之间的界面的斜率保持有界和一致连续，解决方案仍然是定期的。证明利用非局部非线性抛物方程的性质，通过一系列的非局部算子的非线性下界。这些被用来推断，只要界面的斜率保持一致有界，曲率保持有界。然后，非线性界允许我们获得W-2，W-p，1 < p类中任意大初始数据的局部存在性
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 W-2,W-p, 1 < p