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arXiv · 2411.01501

The Muskat problem with a large slope

Yiran Xu·Stephen Cameron·Ke Chen·Ruilin Hu·Quoc-Hung Nguyen
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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 \(\beta_\sigma(f_0')\) which encapsulates local monotonicity and slope, we identify a new class of initial data within \(W^{1,\infty}(\mathbb{R}^d)\). 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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Yiran Xu, Stephen Cameron, Ke Chen, Ruilin Hu, Quoc-Hung Nguyen. 2024-11-03. The Muskat problem with a large slope. https://arxiv.org/abs/2411.01501

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