Global decay estimates of rate (1+t)^{-s/2-1/4} are established for the linearized multi-phase Muskat problem around flat stable interfaces, slower than the classical single-interface rate.
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The Muskat problem describes how fluids of different densities move through a porous medium under gravity. When there are multiple layers with fixed densities, small wiggles in the flat interfaces between layers tend to flatten out over time. The authors linearize the governing equations around a perfectly flat stable setup, study the eigenvalues of the resulting linear operator, and find that low-frequency modes decay like (1+t) to the power of minus s over 2 minus 1/4 in a Wiener norm. They then control the nonlinear interactions to show the decay persists for all time. This rate is slower than the decay known for the simpler two-fluid case.
Core claim
We establish global-in-time decay estimates for the multi-phase Muskat problem... The asymptotic behavior at low frequencies of eigenvalues yields the decay rate of (1+t)^{-s/2-1/4} for Wiener norm ||f||_s
Load-bearing premise
The linearization around a flat stable configuration remains valid globally in time and the low-frequency eigenvalue asymptotics control the full nonlinear evolution without additional resonances or instabilities.
read the original 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.
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The result rests on the existence of a stable flat equilibrium, the validity of linearization for global control, and the low-frequency spectral behavior of the linearized operator; no free parameters or invented entities are mentioned.
axioms (2)
domain assumptionThe multi-phase Muskat system admits a flat stable equilibrium configuration around which linearization is justified. Invoked in the first sentence of the abstract as the starting point for the analysis.
domain assumptionLow-frequency asymptotics of the eigenvalues of the linearized operator determine the decay rate of the full nonlinear problem. Stated directly as the source of the (1+t)^{-s/2-1/4} rate.
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