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Endpoint Sobolev Theory for the Muskat Equation

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Abstract

This paper is devoted to the study of solutions with critical regularity for the two-dimensional Muskat equation. We prove that the Cauchy problem is well-posed on the endpoint Sobolev space of \(L^2\) functions with three-half derivative in \(L^2\). This result is optimal with respect to the scaling of the equation. One well-known difficulty is that one cannot define a flow map such that the lifespan is bounded from below on bounded subsets of this critical Sobolev space. To overcome this, we estimate the solutions for a norm which depends on the initial data themselves, using the weighted fractional Laplacians introduced in our previous works. Our proof is the first in which a null-type structure is identified for the Muskat equation, allowing to compensate for the degeneracy of the parabolic behavior for large slopes.

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Notes

  1. Since it is not immediately obvious that the right-hand side is well defined, we refer to the discussion in paragraph Sect. 4.1 where we recall that the Muskat equation can be written under the form \(\partial _t f+\left| D\right| f={\mathcal {T}}(f)f\) where \({\mathcal {T}}(f)f\) is given by a well-defined integral.

  2. Let us mention that we will make extensive of this symmetry to handle the contributions of terms involving \(\Delta _{-\alpha }\), without explicitly recalling this argument.

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Acknowledgements

The authors want to thank Benoît Pausader for his stimulating comments as well as for discussions about the critical problem for the Muskat equation. T.A. acknowledges the SingFlows project (Grant ANR-18-CE40-0027) of the French National Research Agency (ANR). Q.H.N. is supported by the Academy of Mathematics and Systems Science, Chinese Academy of Sciences startup fund, and the National Natural Science Foundation of China (No. 12050410257 and No. 12288201) and the National Key R &D Program of China under grant 2021YFA1000800.

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Authors and Affiliations

  1. CNRS, Centre Borelli UMR9010, ENS Paris-Saclay, Université Paris-Saclay, Avenue des Sciences, F-91190, Gif-sur-Yvette, France

    Thomas Alazard

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Quoc-Hung Nguyen

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Alazard, T., Nguyen, QH. Endpoint Sobolev Theory for the Muskat Equation. Commun. Math. Phys. 397, 1043–1102 (2023). https://doi.org/10.1007/s00220-022-04514-7

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