**Authors**: Ángel Castro (ICMAT-CSIC); Diego Córdoba (ICMAT-CSIC) and Daniel Faraco (ICMAT-UAM)

**Source**: *Inventiones Mathematicae* vol. 226, pages 251–348

**Date of publication**: 5 May 2021

**Review**:

In fluid mechanics, one is typically interested in describing the evolution of a scalar transported by a fluid, where the fluid and the velocity and the fluid are coupled. In most of the classical texts, in physics and mathematics, one computes the linearized problem around some stationary solution and declares the problem stable or unstable depending on the sign of the spectrum. In the linearly stable case, deep research amounts to show that the non-linear equation is indeed well posed and in the best scenario there is a global in time solution. In the unstable situation, classical tools in analysis and PDE fail and either one finds explicit solutions in particular situations, performs experiments or run simulations.

In this work, the authors present a very robust method to construct weak solutions to unstable problems by combining the convex integration method of De Lellis and Székelyhidi with the contour dynamics evolution.

The paper is devoted to the Muskat problem, the evolution of a scalar through a porous media, were the effect of the porous and the gravitiy is represented by the well-known Darcy´s law. In experiments going back to Saffman and Taylor, and brought back to the mathematical community by F. Otto, one sees the presence of a so-called mixing zone where the two fluids mix stochastically.

Their method describes such a mixing zone as an envelope of a curve, whose evolution is dictated by an average of the, by now classical, Muskat operator. The double average regularizes the operator for positive time but blows up when times goes to zero. In order to circumvent that difficulty, they create a semi classical calculus for non-smooth operator understanding time as the small parameter playing the role of the Planck constant.

In simulations, one sees different pointwise solutions at every sample, a phenomenon that physicists have baptised as spontaneous stochasticity and the strong butterfly effect. However observable quantities are expected to behave classically. Thus deterministic models are consistent with stochastic ones.

The method described in the paper does not only yield the expected weath of non-unique weak solutions, but predicts the evolution of macroscopical quantities such as size and shape of the mixing zone. Non trivial elaboration of these ideas has been use to deal with vortex sheet type problems or partially unstable situations