3 research outputs found
Breakdown of smoothness for the Muskat problem
In this paper we show that there exist analytic initial data in the stable
regime for the Muskat problem such that the solution turns to the unstable
regime and later breaks down i.e. no longer belongs to .Comment: 93 pages, 10 figures (6 added
Rigidity of a non-elliptic differential inclusion related to the Aviles-Giga conjecture
International audienceIn this paper we prove sharp regularity for a differential inclusion into a set that arises in connection with the Aviles-Giga functional. The set is not elliptic, and in that sense our main result goes beyond \v{S}ver\'{a}k's regularity theorem on elliptic differential inclusions. It can also be reformulated as a sharp regularity result for a critical nonlinear Beltrami equation. In terms of the Aviles-Giga energy, our main result implies that zero energy states coincide (modulo a canonical transformation) with solutions of the differential inclusion into . This opens new perspectives towards understanding energy concentration properties for Aviles-Giga: quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions. All these reformulations of our results are strong improvements of a recent work by the last two authors Lorent and Peng, where the link between the differential inclusion into and the Aviles-Giga functional was first observed and used. Our proof relies moreover on new observations concerning the algebraic structure of entropies