Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem

Thomas Alazard; Quoc Hung Nguyen

doi:10.1016/j.aim.2022.108278

Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem

Thomas Alazard
, Quoc Hung Nguyen

Research output: Contribution to journal › Article › peer-review

Abstract

We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first well-posedness result for arbitrary large data in the critical space H˙²(R²)∩W^1,∞(R²). Moreover, we prove the existence of solutions for initial data which are not Lipschitz.

Original language	English
Article number	108278
Journal	Advances in Mathematics
Volume	399
DOIs	https://doi.org/10.1016/j.aim.2022.108278
Publication status	Published - 16 Apr 2022
Externally published	Yes

Keywords

Muskat equation

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10.1016/j.aim.2022.108278

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title = "Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem",

abstract = "We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first well-posedness result for arbitrary large data in the critical space H˙2(R2)∩W1,∞(R2). Moreover, we prove the existence of solutions for initial data which are not Lipschitz.",

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AU - Nguyen, Quoc Hung

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N2 - We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first well-posedness result for arbitrary large data in the critical space H˙2(R2)∩W1,∞(R2). Moreover, we prove the existence of solutions for initial data which are not Lipschitz.

AB - We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first well-posedness result for arbitrary large data in the critical space H˙2(R2)∩W1,∞(R2). Moreover, we prove the existence of solutions for initial data which are not Lipschitz.

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DO - 10.1016/j.aim.2022.108278

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SN - 0001-8708

VL - 399

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108278

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Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem

Abstract

Keywords

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