Siddhant Agrawal (ICMAT-CSIC) signs, together with Neel Patel (University of Maine) and Sijue Wu (University of Michigan), the article ‘Rigidity of acute angled corners for one phase Muskat interfaces’, published in Advances in Mathematics.

The one phase Muskat equation is an equation which models the interface between a porous medium and air, where the dynamics is driven by Darcy’s law. The equation is also equivalent to the Hele Shaw equation, with injection happening at infinity. This is a free boundary problem, as the fluid domain changes as a function of time. After writing the equation in an appropriate coordinate system, one can see that the equation is a nonlinear nonlocal parabolic equation. Previously this equation was shown to have unique global in time weak solutions for Lipschitz initial interfaces, and it was shown that for initial interfaces with slope less than 1, the interface instantaneously smoothens out.

In this paper, we study the problem when the initial interface has an acute angled corner or a cusp. We show that in this case, the interface does not smoothen out instantaneously, but rather the corner remains a corner with the same angle, at least for a short period of time. This is surprising as one generally expects smoothing to happen, due to the fact that the equation is parabolic in nature. This result shows that in this regime, the nonlinearity in the equation dominates the linear behavior.

We prove this result, by first writing the equation in Riemann mapping coordinates and then proving suitable weighted energy estimates. The fact that the weighted energy remains finite for a short time is then used to show that the corner remains a corner and does not smoothen out.

Reference: Siddhant Agrawal, Neel Patel, Sijue Wu, Rigidity of acute angled corners for one phase Muskat interfaces, Advances in Mathematics, Volume 412, 2023, 108801, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2022.108801.