Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation

By D. C'ordoba, Daniel Faraco, Francisco Gancedo. 2009

69 citations recorded by the source corpus.

In this work we consider weak solutions of the incompressible two-dimensional porous media (IPM) equation. By using the approach of De Lellis–Székelyhidi, we prove non-uniqueness for solutions in L∞ in space and time.

Subfields: Applied Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/baa514189febacff59398abffe9c80f5515758fe

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In Applied Mathematics

In Mechanics of Materials

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.”
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.”
Endpoint Sobolev Theory for the Muskat Equation (2020, background): “Let us also mention that the existence and (possible) non-uniqueness of some weak-solutions in the unstable regime has also been thoroughly studied (see [9,16,32,37,49,53]).”
Onsager's Conjecture for Admissible Weak Solutions (2017, background): “applied very efficiently to produce bounded weak solutions that violate the energy conservation in several forms (see also [9, 11, 12,16, 19, 31,37, 38] for related work on the compressible Euler equations, non-odd active scalar equations, and the stationary Euler equations).”