Non-splat singularity for the one-phase Muskat problem

Córdoba, Diego; Pernas-Castaño, Tania

doi:10.1090/tran6688

Non-splat singularity for the one-phase Muskat problem

Authors: Diego Córdoba and Tania Pernas-Castaño
Journal: Trans. Amer. Math. Soc. 369 (2017), 711-754
MSC (2010): Primary 35Q35
DOI: https://doi.org/10.1090/tran6688
Published electronically: April 14, 2016
MathSciNet review: 3557791
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For the water wave equations, the existence of splat singularities has been shown, i.e., the interface self-intersects along an arc in finite time. The aim of this paper is to show the absence of splat singularities for the incompressible fluid dynamics in porous media.

References

Ángel Castro, Diego Córdoba, Charles Fefferman, and Francisco Gancedo, Breakdown of smoothness for the Muskat problem, Arch. Ration. Mech. Anal. 208 (2013), no. 3, 805–909. MR 3048596, DOI https://doi.org/10.1007/s00205-013-0616-x

Ángel Castro, Diego Córdoba, Charles Fefferman, and Francisco Gancedo, Splash singularities for the one-phase Muskat problem in stable regimes. arXiv:1311.7653v1, 2013.

Angel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, and Javier Gómez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, Ann. of Math. (2) 178 (2013), no. 3, 1061–1134. MR 3092476, DOI https://doi.org/10.4007/annals.2013.178.3.6

Ángel Castro, Diego Córdoba, Charles Fefferman, Francisco Gancedo, and María López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2) 175 (2012), no. 2, 909–948. MR 2993754, DOI https://doi.org/10.4007/annals.2012.175.2.9

C. H. Arthur Cheng, Rafael Granero-Belinchón, and Steve Shkoller, Well-posedness of the Muskat problem with ${H}^{2}$ initial data. arXiv:14127737v1, 2014.

Peter Constantin, Diego Córdoba, Francisco Gancedo, and Robert M. Strain, On the global existence for the Muskat problem, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 201–227. MR 2998834, DOI https://doi.org/10.4171/JEMS/360

Antonio Córdoba and Diego Córdoba, A pointwise estimate for fractionary derivatives with applications to partial differential equations, Proc. Natl. Acad. Sci. USA 100 (2003), no. 26, 15316–15317. MR 2032097, DOI https://doi.org/10.1073/pnas.2036515100

Antonio Córdoba, Diego Córdoba, and Francisco Gancedo, Interface evolution: the Hele-Shaw and Muskat problems, Ann. of Math. (2) 173 (2011), no. 1, 477–542. MR 2753607, DOI https://doi.org/10.4007/annals.2011.173.1.10

Antonio Córdoba, Diego Córdoba, and Francisco Gancedo, Porous media: the Muskat problem in three dimensions, Anal. PDE 6 (2013), no. 2, 447–497. MR 3071395, DOI https://doi.org/10.2140/apde.2013.6.447

Diego Córdoba and Francisco Gancedo, Contour dynamics of incompressible 3-D fluids in a porous medium with different densities, Comm. Math. Phys. 273 (2007), no. 2, 445–471. MR 2318314, DOI https://doi.org/10.1007/s00220-007-0246-y

Daniel Coutand and Steve Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Comm. Math. Phys. 325 (2014), no. 1, 143–183. MR 3147437, DOI https://doi.org/10.1007/s00220-013-1855-2

Daniel Coutand and Steve Shkoller, On the impossibility of finite-time splash singularities for vortex sheets. arXiv:1407.1479v1, 2014.

Joachim Escher and Bogdan-Vasile Matioc, On the parabolicity of the Muskat problem: well-posedness, fingering, and stability results, Z. Anal. Anwend. 30 (2011), no. 2, 193–218. MR 2793001, DOI https://doi.org/10.4171/ZAA/1431

Charles Fefferman, Alexandru D. Ionescu, and Victor Lie, On the absence of “splash” singularities in the case of two-fluids interfaces. arXiv:1312.2917v2, 2013.

Francisco Gancedo and Robert M. Strain, Absence of splash singularities for surface quasi-geostrophic sharp fronts and the Muskat problem, Proc. Natl. Acad. Sci. USA 111 (2014), no. 2, 635–639. MR 3181769, DOI https://doi.org/10.1073/pnas.1320554111

Javier Gómez-Serrano and Rafael Granero-Belinchón, On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof, Nonlinearity 27 (2014), no. 6, 1471–1498. MR 3215843, DOI https://doi.org/10.1088/0951-7715/27/6/1471

Rafael Granero-Belinchón, Global existence for the confined Muskat problem, SIAM J. Math. Anal. 46 (2014), no. 2, 1651–1680. MR 3196945, DOI https://doi.org/10.1137/130912529

Tosio Kato and Gustavo Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), no. 7, 891–907. MR 951744, DOI https://doi.org/10.1002/cpa.3160410704

Morris Muskat, Two fluid systems in porous media. The encroachment of water into an oil sand, J. Appl. Phys. (5):250–264, 1934.

Lord Rayleigh, On The Instability Of Jets, Proc. Lond. Math. Soc. 10 (1878/79), 4–13. MR 1576197, DOI https://doi.org/10.1112/plms/s1-10.1.4

P. G. Saffman and Geoffrey Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London Ser. A 245 (1958), 312–329. (2 plates). MR 97227, DOI https://doi.org/10.1098/rspa.1958.0085

Michael Siegel, Russel E. Caflisch, and Sam Howison, Global existence, singular solutions, and ill-posedness for the Muskat problem, Comm. Pure Appl. Math. 57 (2004), no. 10, 1374–1411. MR 2070208, DOI https://doi.org/10.1002/cpa.20040