Global in Time Estimates for Multi-phase Muskat Problem

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.