Large Time Decay Estimates for the Muskat Equation

TL;DR

This paper establishes optimal large-time decay rates for solutions to the Muskat equation in 2D and 3D, extending previous existence results with precise decay estimates based on initial data norms.

Contribution

It provides the first proof of uniform bounds and decay rates for the nonlinear Muskat problem in both 2D and 3D, matching the linear case's optimal decay.

Findings

Proved uniform bounds for _s(t) in 3D for -d < s < l-1.

Established decay rates _s(t)  (1+t)^{-s+ u} for initial data with finite _ u.

Extended decay results to 2D Muskat problem.

Abstract

We prove time decay of solutions to the Muskat equation in 2D and in 3D. In \cite{JEMS} and \cite{CCGRPS}, the authors introduce the norms $\|f\|_{s}(t)= \int_{\mathbb{R}^{2}} |\xi|^{s}|\hat{f}(\xi)| \ d\xi$ in order to prove global existence of solutions to the Muskat problem. In this paper, for the 3D Muskat problem, given initial data $f_{0}\in H^{l}(\mathbb{R}^{2})$ for some $l\geq 3$ such that $\|f_{0}\|_{1} < k_{0}$ for a constant $k_{0} \approx 1/5$, we prove uniform in time bounds of $\|f\|_{s}(t)$ for $-d < s < l-1$ and assuming $\|f_{0}\|_{\nu} < \infty$ we prove time decay estimates of the form $\|f\|_{s}(t) \lesssim (1+t)^{-s+\nu}$ for $0 \leq s \leq l-1$ and $-d \leq \nu < s$. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D.