Global well-posedness for dissipative IPM with data close to a class of special solutions

TL;DR

This paper proves the global well-posedness of the 2-D dissipative IPM equation for initial data near special solutions, extending understanding of solution behavior in supercritical and subcritical regimes.

Contribution

It establishes the global existence and uniqueness of solutions close to special solutions for the dissipative IPM equation under certain regularity conditions.

Findings

Global well-posedness for initial data near special solutions

Decay properties of special solutions related to fractional heat equation

Applicability to both supercritical and subcritical cases

Abstract

In this paper, we consider the 2-D dissipative incompressible porous media (IPM) equation in both supercritical and subcritical cases. The dissipative IPM equation admits a class of special solutions of the form $\rho(x_1,x_2,t)=f(x_2,t)$, which decay in the mode of the 1-D fractional heat equation. Our main result is the global well-posedness for the dissipative IPM equation with initial data close to this class of special solutions provided that $f$ satisfies certain regularity assumptions.