Global in Time Estimates for Multi-phase Muskat Problem

TL;DR

This paper derives global decay estimates for the multi-phase Muskat problem with multiple constant densities, analyzing linearized operators and nonlinear bounds to determine decay rates over time.

Contribution

It provides the first rigorous decay estimates for the multi-phase Muskat problem with multiple densities, extending classical results to more complex configurations.

Findings

Decay rate of (1+t)^{-s/2-1/4} for Wiener norm in multi-phase case

Linearization around stable configurations helps analyze asymptotic behavior

Bounded nonlinear terms close the decay estimate argument

Abstract

We establish global-in-time decay estimates for the multi-phase Muskat problem in the case where the density takes exactly n+1 distinct constant values. We first linearize the system around a flat stable configuration, followed by the study of associated linearized operator. The asymptotic behavior at low frequencies of eigenvalues yields the decay rate of (1+t)^{-s/2-1/4} for Wiener norm \|f\|_s, which is slower than the classical case, where the decay rate is (1+t)^{-s+\nu}. Afterwards we bound the nonlinear term to close the argument.