JournalsjemsVol. 15, No. 1pp. 201–227

On the global existence for the Muskat problem

  • Peter Constantin

    Princeton University, United States
  • Diego Córdoba

    Universidad Autónoma de Madrid, Spain
  • Francisco Gancedo

    University of Chicago, USA
  • Robert M. Strain

    University of Pennsylvania, Philadelphia, USA
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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 L2(R)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 f11/5\|f\|_1 \le 1/5. Previous results of this sort used a small constant ϵ1\epsilon \ll1 which was not explicit. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy f0L<\|f_0\|_{L^\infty}<\infty and xf0L<1\|\partial_x f_0\|_{L^\infty}<1. We take advantage of the fact that the bound xf0L<1\|\partial_x f_0\|_{L^\infty}<1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.

Cite this article

Peter Constantin, Diego Córdoba, Francisco Gancedo, Robert M. Strain, On the global existence for the Muskat problem. J. Eur. Math. Soc. 15 (2013), no. 1, pp. 201–227

DOI 10.4171/JEMS/360