Rigidity of acute angled corners for one phase Muskat interfaces

We consider the one-phase Muskat problem modeling the dynamics of the free boundary of a single fluid in porous media. We prove local well-posedness for fluid interfaces that are general curves and can have singularities. In particular, the free boundary can have acute angle corners or cusps. Moreover, we show that isolated corners/cusps on the interface must be rigid, meaning the angle of the corner is preserved for a finite time, there is no rotation at the tip, the particle at the tip remains at the tip and the velocity of that particle at the tip points vertically downward.

[1]  Nastasia Grubic,et al.  Local wellposedness for the free boundary incompressible Euler equations with interfaces that exhibit cusps and corners of nonconstant angle , 2021, 2107.09751.

[2]  M. Muskat.  Two Fluid Systems in Porous Media. The Encroachment of Water into an Oil Sand , 1934 .

[3]  D. Ambrose.  Well-posedness of two-phase Hele–Shaw flow without surface tension , 2004, European Journal of Applied Mathematics.

[4]  J. Vázquez,et al.  Persistence of corners in free boundaries in Hele-Shaw flow , 1995, European Journal of Applied Mathematics.

[5]  C. Fefferman,et al.  Breakdown of Smoothness for the Muskat Problem , 2012, 1201.2525.

[6]  Sijue Wu.  Wellposedness of the 2D full water wave equation in a regime that allows for non-$$C^1$$C1 interfaces , 2018, Inventiones mathematicae.

[7]  Inwon C. Kim,et al.  Waiting time phenomena of the Hele-Shaw and the Stefan problem , 2006 .

[8]  Siddhant Agrawal,et al.  Angled crested type water waves with surface tension II: Zero surface tension limit , 2020, 2009.13469.

[9]  K. Chen,et al.  The Muskat problem with $C^1$ data , 2021, 2103.09732.

[10]  Charles Fefferman,et al.  Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves , 2011, 1102.1902.

[11]  J. Escher,et al.  On the parabolicity of the Muskat problem: Well-posedness, fingering, and stability results , 2010, 1005.2512.

[12]  R. H. Kinsey,et al.  A Priori Estimates for Two-Dimensional Water Waves with Angled Crests , 2014, 1406.7573.

[13]  酒井 良.  Small modifications of quadrature domains , 2010 .

[14]  Stephen Cameron.  Global well-posedness for the two-dimensional Muskat problem with slope less than 1 , 2017, Analysis & PDE.

[15]  P. Constantin,et al.  Global regularity for 2D Muskat equations with finite slope , 2015, 1507.01386.

[16]  Quoc-Hung Nguyen,et al.  On the Cauchy problem for the Muskat equation with non-Lipschitz initial data , 2020, 2009.04343.

[17]  C. Fefferman,et al.  Splash Singularities for the One-Phase Muskat Problem in Stable Regimes , 2013, 1311.7653.

[18]  T. Alazard,et al.  Paralinearization of the Muskat Equation and Application to the Cauchy Problem , 2019, Archive for Rational Mechanics and Analysis.

[19]  Inwon C. Kim,et al.  Regularity for the one-phase Hele-Shaw problem from a Lipschitz initial surface , 2007 .

[20]  Hongjie Dong,et al.  Global well‐posedness for the one‐phase Muskat problem , 2021, Communications on Pure and Applied Mathematics.

[21]  Robert M. Strain,et al.  On the Muskat problem with viscosity jump: Global in time results , 2017, Advances in Mathematics.

[22]  Nataliya Vasylyeva,et al.  The Two-Phase Hele-Shaw Problem with a Nonregular Initial Interface and Without Surface Tension , 2014 .

[23]  Quoc-Hung Nguyen,et al.  Endpoint Sobolev Theory for the Muskat Equation , 2020, Communications in Mathematical Physics.

[24]  N. Meunier,et al.  Lyapunov Functions, Identities and the Cauchy Problem for the Hele–Shaw Equation , 2019, Communications in Mathematical Physics.

[25]  Robert M. Strain,et al.  On the Muskat problem: Global in time results in 2D and 3D , 2013, 1310.0953.

[26]  B. Matioc,et al.  The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results , 2016, Analysis & PDE.

[27]  S. Dragomir.  Some Gronwall Type Inequalities and Applications , 2003 .

[28]  A. Córdoba,et al.  Interface evolution: the Hele-Shaw and Muskat problems , 2008, 0806.2258.

[29]  Antonio Córdoba,et al.  Porous media: The Muskat problem in three dimensions , 2013 .

[30]  Benoit Pausader,et al.  Self-similar solutions for the Muskat equation , 2021, Advances in Mathematics.

[31]  S. Shkoller,et al.  Well-posedness of the Muskat problem with H2 initial data , 2014, 1412.7737.

[32]  B. Pausader,et al.  A Paradifferential Approach for Well-Posedness of the Muskat Problem , 2019, Archive for Rational Mechanics and Analysis.