Localized Mixing Zone for Muskat Bubbles and Turned Interfaces

We construct mixing solutions to the incompressible porous media equation starting from Muskat type data in the partially unstable regime. In particular, we consider bubble and turned type interfaces with Sobolev regularity. As a by-product, we prove the continuation of the evolution of IPM after the Rayleigh-Taylor and smoothness breakdown exhibited in [18, 17]. At each time slice the space is split into three evolving domains: two non-mixing zones and a mixing zone which is localized in a neighborhood of the unstable region. In this way, we show the compatibility between the classical Muskat problem and the convex integration method.

[1]  The Riemann Problem for the Multidimensional Isentropic System of Gas Dynamics is Ill-Posed if It Contains a Shock , 2017, 1708.01063.

[2]  V. Vicol,et al.  Nonuniqueness of weak solutions to the Navier-Stokes equation , 2017, Annals of Mathematics.

[3]  L. Székelyhidi,et al.  Non-uniqueness for the Transport Equation with Sobolev Vector Fields , 2017, Annals of PDE.

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

[5]  C. Klingenberg,et al.  On the density of “wild” initial data for the compressible Euler system , 2018, 1812.11802.

[6]  Antonio C'ordoba,et al.  Uniqueness for SQG patch solutions , 2016, 1605.06663.

[7]  M. G.,et al.  VISCOUS FINGERING IN POROUS MEDIA , 2002 .

[8]  V. Vicol,et al.  Wild solutions of the Navier–Stokes equations whose singular sets in time have Hausdorff dimension strictly less than 1 , 2018, Journal of the European Mathematical Society.

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

[10]  Steve Shkoller,et al.  Nonuniqueness of Weak Solutions to the SQG Equation , 2016, Communications on Pure and Applied Mathematics.

[11]  A. Córdoba,et al.  A pointwise estimate for fractionary derivatives with applications to partial differential equations , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[12]  R. Caflisch,et al.  Global existence, singular solutions, and ill‐posedness for the Muskat problem , 2004 .

[13]  R. Shvydkoy.  Convex integration for a class of active scalar equations , 2010, 1010.4755.

[14]  Robert M. Strain,et al.  Global Regularity for Gravity Unstable Muskat Bubbles , 2019, 1902.02318.

[15]  F. Yi.  Global classical solution of Muskat free boundary problem , 2003 .

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

[17]  Aniel,et al.  Lack of uniqueness for weak solutions of the incompressible porous media equation , 2010 .

[18]  Relaxation of the Boussinesq system and applications to the Rayleigh–Taylor instability , 2020, Nonlinear Differential Equations and Applications NoDEA.

[19]  Elisabetta Chiodaroli,et al.  On the Energy Dissipation Rate of Solutions to the Compressible Isentropic Euler System , 2013, 1312.4715.

[20]  Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime , 2020, 2001.06361.

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

[22]  Camillo De Lellis,et al.  The Euler equations as a differential inclusion , 2007 .

[23]  Stefano Modena,et al.  Non Uniqueness of Power-Law Flows , 2020, Communications in Mathematical Physics.

[24]  'Angel Castro,et al.  Degraded mixing solutions for the Muskat problem , 2018, Calculus of Variations and Partial Differential Equations.

[25]  Camillo De Lellis,et al.  Ill-Posedness of Leray Solutions for the Hypodissipative Navier–Stokes Equations , 2017, Communications in Mathematical Physics.

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

[27]  S. Modena,et al.  Convex integration solutions to the transport equation with full dimensional concentration , 2019, 1902.08521.

[28]  F. Lin,et al.  On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data , 2016, 1603.03949.

[29]  F. Otto.  Evolution of microstructure in unstable porous media flow: A relaxational approach , 1999 .

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

[31]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[32]  V. Vicol,et al.  Hölder Continuous Solutions of Active Scalar Equations , 2014, 1405.7656.

[33]  Francisco Mengual,et al.  Dissipative Euler Flows for Vortex Sheet Initial Data without Distinguished Sign , 2020, Communications on Pure and Applied Mathematics.

[34]  L. Székelyhidi,et al.  Piecewise Constant Subsolutions for the Muskat Problem , 2017, Communications in Mathematical Physics.

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

[36]  D. C'ordoba,et al.  Mixing solutions for the Muskat problem , 2016, Inventiones mathematicae.

[37]  V. Vicol,et al.  Convex integration and phenomenologies in turbulence , 2019, EMS Surveys in Mathematical Sciences.

[38]  Vlad Vicol,et al.  Onsager's Conjecture for Admissible Weak Solutions , 2017, Communications on Pure and Applied Mathematics.

[39]  Camillo De Lellis,et al.  High dimensionality and h-principle in PDE , 2016, 1609.03180.

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

[41]  H. J. Morel-Seytoux,et al.  Multiphase Fluid Flow Through Porous Media , 1976 .

[42]  Camillo De Lellis,et al.  On Admissibility Criteria for Weak Solutions of the Euler Equations , 2007, 0712.3288.

[43]  Lauri Hitruhin,et al.  Lamination Convex Hull of Stationary Incompressible Porous Media Equations , 2021, SIAM J. Math. Anal..

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

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

[46]  Oscillatory Solutions to Hyperbolic Conservation Laws and Active Scalar Equations , 2013 .

[47]  L. Székelyhidi,et al.  Non-uniqueness for the Euler Equations up to Onsager’s Critical Exponent , 2020, 2004.00391.

[48]  Ondrej Kreml,et al.  Global Ill‐Posedness of the Isentropic System of Gas Dynamics , 2013, 1304.0123.

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

[50]  Philip Isett,et al.  A Proof of Onsager's Conjecture , 2016, 1608.08301.

[51]  F. Mengual.  h-principle for the 2-dimensional incompressible porous media equation with viscosity jump , 2020, Analysis & PDE.

[52]  D. Faraco,et al.  Proof of Taylor’s Conjecture on Magnetic Helicity Conservation , 2018, Communications in Mathematical Physics.

[53]  George M. Homsy,et al.  Fingering instabilities in vertical miscible displacement flows in porous media , 1995, Journal of Fluid Mechanics.

[54]  Inwon C. Kim,et al.  Weak Solutions to the Muskat Problem with Surface Tension Via Optimal Transport , 2019, Archive for Rational Mechanics and Analysis.

[55]  S. Modena,et al.  Non-renormalized solutions to the continuity equation , 2018, Calculus of Variations and Partial Differential Equations.

[56]  Gianluca Crippa,et al.  Non-Uniqueness and prescribed energy for the continuity equation , 2014, 1412.2478.

[57]  L. Sz'ekelyhidi,et al.  Mixing solutions for the Muskat problem with variable speed , 2020, Journal of Evolution Equations.

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

[59]  V. Vicol,et al.  Weak Solutions of Ideal MHD Which Do Not Conserve Magnetic Helicity , 2019, Annals of PDE.

[60]  Bernold Fiedler,et al.  Ergodic theory, analysis, and efficient simulation of dynamical systems , 2001 .

[61]  G. Taylor,et al.  The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[62]  Robert M. Strain,et al.  On the global existence for the Muskat problem , 2010, 1007.3744.

[63]  Emil Wiedemann,et al.  NonUnique Admissible Weak Solutions of the Compressible Euler Equations with Compact Support in Space , 2020, SIAM J. Math. Anal..

[64]  Evolution of Microstructure: an Example , 2001 .

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

[66]  L. Sz'ekelyhidi.  Relaxation of the incompressible porous media equation , 2011, 1102.2597.

[67]  D. Córdoba,et al.  Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities , 2007 .

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

[69]  L. Székelyhidi.  Weak solutions to the incompressible Euler equations with vortex sheet initial data , 2011 .

[70]  Omar Lazar,et al.  Global well-posedness for the 2D stable Muskat problem in $H^{3/2}$ , 2018, Annales scientifiques de l'École Normale Supérieure.

[71]  Gretar Tryggvason,et al.  Numerical experiments on Hele Shaw flow with a sharp interface , 1983, Journal of Fluid Mechanics.

[72]  Camillo De Lellis,et al.  The $h$-principle and the equations of fluid dynamics , 2011, 1111.2700.