Piecewise Constant Subsolutions for the Muskat Problem

Communications in Mathematical Physics

Volumn 363, Issue 3, 2018, Pages 1051-1080

  Förster, Clemens ^a   Székelyhidi, László ^a  

^a UNIVERSITY OF LEIPZIG   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 85053251852     PISSN: 00103616     EISSN: 14320916     Source Type: Journal    
DOI: 10.1007/s00220-018-3245-2     Document Type: Article

References (29)

1. Alt H.W., Luckhaus S., Visintin A.: On nonstationary flow through porous media. Annali Mat. Pura Ed Appl. 136, 306–316 (1984)
2. Ambrose D.M.: Well-posedness of two-phase Hele-Shaw flow without surface tension. Eur. J. Appl. Math. 15, 597–607 (2004)
3. Bardos C., Székelyhidi L. Jr., Wiedemann E.: Non-uniqueness for the Euler equations: the effect of the boundary. Dedicated to the Memory of Professor Mark Vishik. Uspekhi Mat. Nauk 69:2(2), 3–22 (2014)
4. Bear J.: Dynamics of Fluids in Porous Media. Elsevier, New York (1972)
5. Castro, A., Córdoba, D., Faraco, D.: Mixing solutions for the Muskat problem (2016) arXiv:1605.04822 [math.AP]
6. Castro Á., Córdoba D., Fefferman C., Gancedo F., López-Fernández M.: Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. (2) 175, 909–948 (2012)
7. 2 initial data. Adv. Math. 286, 32–104 (2016)
8. Constantin P., Córdoba D., Gancedo F., Rodriguez-Piazza L., Strain R.M.: On the Muskat problem: global in time results in 2D and 3D. Am. J. Math. 138, 1455–1494 (2016)
9. Constantin P., Gancedo F., Vicol V., Shvydkoy R.: Global regularity for 2D Muskat equation with finite slope. Ann. Inst. H. Poincaré Anal. Non Linéaire 34, 1041–1074 (2017)
10. Córdoba D., Faraco D., Gancedo F.: Lack of uniqueness for weak solutions of the incompressible porous media equation. Arch. Ration. Mech. Anal. 200(3), 725–746 (2011)
11. Córdoba D., Gancedo F.: Contour dynamics of incompressible 3-D fluids in a porous medium with different densities. Commun. Math. Phys. 273(2), 445–471 (2007)
12. De Lellis C., Székelyhidi L. Jr.: The Euler equations as a differential inclusion. Ann. Math. (2) 170(3), 1417–1436 (2009)
13. De Lellis C., Székelyhidi L. Jr.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195(1), 225–260 (2008)
14. De Lellis C., Székelyhidi L. Jr.: The h-principle and the equations of fluid dynamics. Bull. Am. Math. Soc. (N.S.) 49(3), 347–375 (2012)
15. Chiodaroli E., De Lellis C., Kreml O.: Global ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Math. 68(7), 1157–1190 (2015)
16. Otto F.: Evolution of microstructure in unstable porous media flow: a relaxational approach. Commun. Pure Appl. Math. 52(7), 873–915 (1999)
17. Otto, F.: Evolution of microstructure: an example. In: Fiedler, B. (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 501–522. Springer, Berlin (2001)
18. Otto F.: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equ. 26, 101–174 (2001)
19. Majda A.J., Bertozzi A.L.: Vorticity and Incompressible Flow. volume 27 of Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
20. Muskat M.: The Flow of Homogeneous Fluids Through Porous Media. Springer, New York (1937)
21. Saffman P.G., Sir Geoffrey Taylor F.R.S.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245(1242), 312–329 (1958)
22. Scheffer V.: An inviscid flow with compact support in space–time. J. Geom. Anal. 3, 343–401 (1993)
23. Shnirelman A.I.: On the nonuniqueness of weak solution of the Euler equation. Commun. Pure Appl. Math. 50(12), 1261–1286 (1997)
24. Siegel M., Caflisch R.E., Howison S.: Global existence, singular solutions, and ill-posedness for the Muskat problem. Commun. Pure Appl. Math. 57, 1374–1411 (2004)
25. Székelyhidi, Jr. L.: From Isometric Embeddings to Turbulence. MPI-MIS Lecture notes 41 (2012)
26. Székelyhidi L. Jr.: Weak solutions to the incompressible Euler equations with vortex sheet initial data. C. R. Acad. Sci. Paris Ser. I Math. 349(19-20), 1063–1066 (2011)
27. Székelyhidi L. Jr.: Relaxation of the incompressible Porous Media Equation. Ann. Sci. Éc. Norm. Supér. 45(3), 491–509 (2012)
28. Wiedemann E.: Existence of weak solutions for the incompressible Euler equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 28(5), 727–730 (2011)
29. Wooding R.A., Morel-Seytoux H.J.: Multiphase fluid flow through porous media. Ann. Rev. Fluid Mech. 8, 233–274 (1976)