# Global well-posedness for the 2D Muskat problem with slope less than 1

## TL;DR

This paper establishes the global existence and decay of smooth solutions for the 2D Muskat problem under a slope condition less than 1, extending techniques from quasi-geostrophic equations.

## Contribution

It proves global well-posedness for the 2D Muskat problem in the stable regime with slope product less than 1, using a novel modulus of continuity approach.

## Key findings

- Solutions decay to flatness as time progresses.
- Unique solutions exist for initial data with $C^{1, ext{epsilon}}$ regularity.
- The slope condition ensures global regularity and stability.

## Abstract

We prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the product of the maximal and minimal slopes is strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon}$. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Nazarov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation.