Relaxation approximation and asymptotic stability of stratified solutions to the IPM equation

TL;DR

This paper establishes the nonlinear asymptotic stability of stratified solutions to the IPM equation with less restrictive initial data, using improved techniques and detailed analysis of the equations' anisotropic properties.

Contribution

It provides a simplified proof of global well-posedness for the Boussinesq system, proves convergence to IPM, and demonstrates asymptotic stability with weaker initial data assumptions.

Findings

Proved asymptotic stability for initial data in $ ext{dot}H^{1- au} igcap ext{dot}H^s$ with $s>3$.

Established uniform energy estimates using anisotropic Littlewood-Paley decomposition.

Showed integrable time decay of vertical velocity for weaker initial data.

Abstract

We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in $\dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and for any $0 < \tau <1$. Such result improves the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to $H^{20}(\mathbb{R}^2)$. More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in $H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s > 3$ and $0 < \tau <1$. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity $\|u_2(t)\|_{L^\infty (\mathbb{R}^2)}$ for initial data only in $\dot H^{1-\tau}(\mathbb{R}^2) \cap \dot H^s(\mathbb{R}^2)$ with $s >3$.