A note on the dissipation for the general Muskat problem
Authors:
Susanna V. Haziot and Benoît Pausader
Journal:
Quart. Appl. Math. 81 (2023), 367-373
MSC (2020):
Primary 35R35; Secondary 35Q35
DOI:
https://doi.org/10.1090/qam/1646
Published electronically:
February 14, 2023
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References |
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Additional Information
Abstract: We consider the dissipation of the Muskat problem and we give an elementary proof of a surprising inequality of Constantin-Cordoba-Gancedo-Strain [J. Eur. Math. Soc. (JEMS) 15 (2013), pp. 201–227 and Amer. J. Math. 138 (2016), pp. 1455–1494] which holds in greater generality.
References
- Thomas Alazard, Nicolas Meunier, and Didier Smets, Lyapunov functions, identities and the Cauchy problem for the Hele-Shaw equation, Comm. Math. Phys. 377 (2020), no. 2, 1421–1459. MR 4115021, DOI 10.1007/s00220-020-03761-w
- T. Alazard and Q.-H. Nguyen, Endpoint sobolev theory for the Muskat equation, Arxiv preprint, arXiv:2010.06915, 2020.
- Thomas Alazard and Quoc-Hung Nguyen, On the Cauchy problem for the Muskat equation. II: Critical initial data, Ann. PDE 7 (2021), no. 1, Paper No. 7, 25. MR 4242131, DOI 10.1007/s40818-021-00099-x
- 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 10.1007/s00205-013-0616-x
- 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 10.4171/JEMS/360
- Peter Constantin, Diego Córdoba, Francisco Gancedo, Luis Rodríguez-Piazza, and Robert M. Strain, On the Muskat problem: global in time results in 2D and 3D, Amer. J. Math. 138 (2016), no. 6, 1455–1494. MR 3595492, DOI 10.1353/ajm.2016.0044
- Fan Deng, Zhen Lei, and Fanghua Lin, On the two-dimensional Muskat problem with monotone large initial data, Comm. Pure Appl. Math. 70 (2017), no. 6, 1115–1145. MR 3639321, DOI 10.1002/cpa.21669
- Patrick T. Flynn and Huy Q. Nguyen, The vanishing surface tension limit of the Muskat problem, Comm. Math. Phys. 382 (2021), no. 2, 1205–1241. MR 4227171, DOI 10.1007/s00220-021-03980-9
- Eduardo García-Juárez, Javier Gómez-Serrano, Huy Q. Nguyen, and Benoît Pausader, Self-similar solutions for the Muskat equation, Adv. Math. 399 (2022), Paper No. 108294, 30. MR 4385135, DOI 10.1016/j.aim.2022.108294
- Giovanni Leoni and Ian Tice, Traces for homogeneous Sobolev spaces in infinite strip-like domains, J. Funct. Anal. 277 (2019), no. 7, 2288–2380. MR 3989147, DOI 10.1016/j.jfa.2019.01.005
- H. Q. Nguyen, Coercivity of the Dirichlet-to-Neumann operator and applications to the Muskat problem, Preprint, arXiv:2206.02321.
- Huy Q. Nguyen and Benoît Pausader, A paradifferential approach for well-posedness of the Muskat problem, Arch. Ration. Mech. Anal. 237 (2020), no. 1, 35–100. MR 4090462, DOI 10.1007/s00205-020-01494-7
References
- Thomas Alazard, Nicolas Meunier, and Didier Smets, Lyapunov functions, identities and the Cauchy problem for the Hele-Shaw equation, Comm. Math. Phys. 377 (2020), no. 2, 1421–1459. MR 4115021, DOI 10.1007/s00220-020-03761-w
- T. Alazard and Q.-H. Nguyen, Endpoint sobolev theory for the Muskat equation, Arxiv preprint, arXiv:2010.06915, 2020.
- Thomas Alazard and Quoc-Hung Nguyen, On the Cauchy problem for the Muskat equation. II: Critical initial data, Ann. PDE 7 (2021), no. 1, Paper No. 7, 25. MR 4242131, DOI 10.1007/s40818-021-00099-x
- 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 10.1007/s00205-013-0616-x
- 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 10.4171/JEMS/360
- Peter Constantin, Diego Córdoba, Francisco Gancedo, Luis Rodríguez-Piazza, and Robert M. Strain, On the Muskat problem: global in time results in 2D and 3D, Amer. J. Math. 138 (2016), no. 6, 1455–1494. MR 3595492, DOI 10.1353/ajm.2016.0044
- Fan Deng, Zhen Lei, and Fanghua Lin, On the two-dimensional Muskat problem with monotone large initial data, Comm. Pure Appl. Math. 70 (2017), no. 6, 1115–1145. MR 3639321, DOI 10.1002/cpa.21669
- Patrick T. Flynn and Huy Q. Nguyen, The vanishing surface tension limit of the Muskat problem, Comm. Math. Phys. 382 (2021), no. 2, 1205–1241. MR 4227171, DOI 10.1007/s00220-021-03980-9
- Eduardo García-Juárez, Javier Gómez-Serrano, Huy Q. Nguyen, and Benoît Pausader, Self-similar solutions for the Muskat equation, Adv. Math. 399 (2022), Paper No. 108294, 30. MR 4385135, DOI 10.1016/j.aim.2022.108294
- Giovanni Leoni and Ian Tice, Traces for homogeneous Sobolev spaces in infinite strip-like domains, J. Funct. Anal. 277 (2019), no. 7, 2288–2380. MR 3989147, DOI 10.1016/j.jfa.2019.01.005
- H. Q. Nguyen, Coercivity of the Dirichlet-to-Neumann operator and applications to the Muskat problem, Preprint, arXiv:2206.02321.
- Huy Q. Nguyen and Benoît Pausader, A paradifferential approach for well-posedness of the Muskat problem, Arch. Ration. Mech. Anal. 237 (2020), no. 1, 35–100. MR 4090462, DOI 10.1007/s00205-020-01494-7
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Additional Information
Susanna V. Haziot
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912
MR Author ID:
1235461
ORCID:
0000-0001-5156-8809
Email:
susanna_haziot@brown.edu
Benoît Pausader
Affiliation:
Department of Mathematics, Brown University, Providence, RI 02912
MR Author ID:
822827
ORCID:
0000-0002-1106-2122
Email:
benoit_pausader@brown.edu
Keywords:
Muskat,
Hele-Shaw,
free boundary problems
Received by editor(s):
October 17, 2022
Published electronically:
February 14, 2023
Additional Notes:
This material is based upon work supported by the National Science Foundation under Grant No. DMS-1929284 while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the program “Hamiltonian Methods in Dispersive and Wave Evolution Equations”. The work of the first author was funded by the National Science Foundation through the award DMS-2102961. The work of the second author was partially supported by a Simons fellowship and by NSF Grant No. DMS-2154162.
Dedicated:
This paper is dedicated to Professor C. Dafermos, with love and admiration.
Article copyright:
© Copyright 2023
Brown University
