Relaxation of the incompressible porous media equation

By L. Sz'ekelyhidi. 2011

101 citations recorded by the source corpus.

It was shown recently by Cordoba, Faraco and Gancedo that the 2D porous media equation admits weak solutions with compact support in time. The proof, based on the convex integration framework, uses ideas from the theory of laminates, in particular T4 configurations. In this note we calculate the explicit relaxation of IPM, thus avoiding T4 configurations. We then use this to construct weak solutions to the unstable interface problem (the Muskat problem), as a byproduct shedding new light on the gradient flow approach introduced by Otto.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8426c43cfdd9ab0166d93822ae329078ecee35e5

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Paralinearization of the Muskat Equation and Application to the Cauchy Problem (2019, methodology): “We refer also to [22, 47] where uniqueness issues have been studied using the convex integration scheme.”
Global Ill‐Posedness of the Isentropic System of Gas Dynamics (2013, background): “This work of Székelyhidi was in turn motivated by the so-called Muskat problem (see [6, 13, 21, 23] for a rather detailed survey).”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, result): “1) is a non-odd function of frequency), extending prior results in [36,79,82] for the incompressible porous media equation.”
Dissipative continuous Euler flows (2012, background): “2) In all applications so far of convex integration to fluid mechanics [1, 5, 23, 26, 27, 28, 8] a key starting point is a plane-wave analysis, in order to identify compatible plane-waves that will be used in the subsequent construction.”