The Muskat problem with a large slope

Stephen Cameron; Ke Chen; Ruilin Hu; Quoc Hung Nguyen; Yiran Xu

doi:10.1016/j.jfa.2025.111257

The Muskat problem with a large slope

Stephen Cameron, Ke Chen, Ruilin Hu, Quoc Hung Nguyen, Yiran Xu

Department of Mathematical Sciences

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

Abstract

In this paper, we establish local well-posedness results for the Muskat equation in any dimension using modulus of continuity techniques. By introducing a novel quantity β_σ(f0′) which encapsulates local monotonicity and slope, we identify a new class of initial data within W^1,∞(R^d)[jls-end-space/]. This includes scenarios where the product of the maximal and minimal slopes is large, thereby guaranteeing the local existence of a classical solution.

Original language	English
Article number	111257
Journal	Journal of Functional Analysis
Volume	290
Issue number	4
Early online date	4 Nov 2025
DOIs	https://doi.org/10.1016/j.jfa.2025.111257
Publication status	Published - 15 Feb 2026

Data Availability Statement

No data was used for the research described in the article.

Keywords

Modulus of continuity
The Muskat problem
Well-posedness theory

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2025.111257

Cite this

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abstract = "In this paper, we establish local well-posedness results for the Muskat equation in any dimension using modulus of continuity techniques. By introducing a novel quantity βσ(f0′) which encapsulates local monotonicity and slope, we identify a new class of initial data within W1,∞(Rd)[jls-end-space/]. This includes scenarios where the product of the maximal and minimal slopes is large, thereby guaranteeing the local existence of a classical solution.",

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KW - The Muskat problem

KW - Well-posedness theory

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The Muskat problem with a large slope

Abstract

Data Availability Statement

Keywords

ASJC Scopus subject areas

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