Large Time Decay Estimates for the Muskat Equation

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 $\Vert f{\Vert }_s(t)={\int }_{{\mathbb{R}}^2}|\xi {|}^s|\hat{f}(\xi )|d\xi$ 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 $l\ge 3$ such that $\Vert f_0{\Vert }_1<k_0$ for a constant $k_0\approx 1/5$, we prove uniform in time bounds of $\Vert f{\Vert }_s(t)$ for $-d<s<l-1$ and assuming $\Vert f_0{\Vert }_{\nu }<\infty$ we prove time decay estimates of the form $\Vert f{\Vert }_s(t)\lesssim (1+t{)}^{-s+\nu }$ for $0\le s\le l-1$ and $-d\le \nu <s$. These large time decay rates are the same as the optimal rate for the linear Muskat equation. We also prove analogous results in 2D.

21 pages, made minor changes to the abstract, notation, strategy of proof and reference sections

References in corpus (1)

• A maximum principle for the Muskat problem for fluids with different densities

Cited by in corpus (9)

• Global mild solutions of the Landau and non-cutoff Boltzmann equations
• On The Muskat Problem With Viscosity Jump: Global In Time Results
• Global well-posedness for the 2D Muskat problem with slope less than 1
• Paralinearization of the Muskat equation and application to the Cauchy problem
• On the Cauchy problem for the Muskat equation with non-Lipschitz initial data
• Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces
• Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects
• Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data
• The Second Iterate of the Muskat Equation in Supercritical Spaces