The Muskat problem with a large slope

TL;DR

This paper proves local well-posedness for the Muskat problem with large initial slopes by introducing a new quantity that captures local monotonicity, expanding the class of initial data for which solutions exist.

Contribution

The paper introduces a novel quantity eta__0' that captures local slope properties, enabling well-posedness results for initial data with large slopes.

Findings

Established local well-posedness in any dimension.

Identified a new class of initial data with large slopes.

Demonstrated existence of classical solutions under new conditions.

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.