Global regularity for 2D Muskat equations with finite slope
By Peter Constantin, Francisco Gancedo, Roman Shvydkoy, Vlad Vicol. 2015
86 citations recorded by the source corpus.
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 $W^{2,p}$, $1<p\leq \infty$. 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.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/944ef9116d48c9b733b891f2059c17e0521d0565
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In Applied Mathematics
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Citations in the corpus
- Endpoint Sobolev Theory for the Muskat Equation (2020, background): “There are now many different proofs that the Cauchy problem is well-posed, locally in time, on the sub-critical Sobolev spaces Hs(R) with s > 3/2 (see [2,24,28,36,44,46]).”
- A Paradifferential Approach for Well-Posedness of the Muskat Problem (2019, methodology): “For the case of constant viscosity, using nonlinear lower bounds, the authors in [21] obtained local well-posedness for η ∈ W 2,p(R) with p ∈ (1,∞].”
- Global well-posedness for the two-dimensional Muskat problem with slope less than 1 (2017, unknown): “Then the original proof of [Constantin et al. 2017] goes through.”