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Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation on the three fundamental domains

Min Jun Jo, Junha Kim

Abstract

We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on $\bbR^2$ with respect to the buoyancy frequency $N$. Our target density of stratification is the sum of the large background linear profile with its slope $N$ and the small perturbation that could be both non-linear and non-monotone. Quantification in $N$ will be performed not only on how large the initial density disturbance is allowed to be but also on how much the target densities can deviate from the purely linear density stratification without losing their stability. For the purely linear density stratification, our method robustly applies to the three fundamental domains $\bbR^2,$ $\bbT^2,$ and $\bbT\times[-1,1]$, improving both the previous result by Elgindi (On the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation, Archive for Rational Mechanics and Analysis, 225(2), 573-599, 2017) on $\bbR^2$ and $\bbT^2$, and the study by Castro-Córdoba-Lear (Global existence of quasi-stratified solutions for the confined IPM equation. Archive for Rational Mechanics and Analysis, 232(1), 437-471, 2019) on $\bbT\times[-1,1]$. The obtained temporal decay rates to the stratified density on $\bbR^2$ and to the newly found asymptotic density profiles on $\bbT^2$ and $\bbT\times[-1,1]$ are all sharp, fully realizing the level of the linearized system. We require the initial disturbance to be small in $H^m$ for any integer $m\geq 4$, which we even relax to any positive number $m>3$ via a suitable anisotropic commutator estimate.

Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation on the three fundamental domains

Abstract

We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on with respect to the buoyancy frequency . Our target density of stratification is the sum of the large background linear profile with its slope and the small perturbation that could be both non-linear and non-monotone. Quantification in will be performed not only on how large the initial density disturbance is allowed to be but also on how much the target densities can deviate from the purely linear density stratification without losing their stability. For the purely linear density stratification, our method robustly applies to the three fundamental domains and , improving both the previous result by Elgindi (On the asymptotic stability of stationary solutions of the inviscid incompressible porous medium equation, Archive for Rational Mechanics and Analysis, 225(2), 573-599, 2017) on and , and the study by Castro-Córdoba-Lear (Global existence of quasi-stratified solutions for the confined IPM equation. Archive for Rational Mechanics and Analysis, 232(1), 437-471, 2019) on . The obtained temporal decay rates to the stratified density on and to the newly found asymptotic density profiles on and are all sharp, fully realizing the level of the linearized system. We require the initial disturbance to be small in for any integer , which we even relax to any positive number via a suitable anisotropic commutator estimate.
Paper Structure (33 sections, 33 theorems, 302 equations)

This paper contains 33 sections, 33 theorems, 302 equations.

Table of Contents

  1. Introduction
  2. Hydrodynamic stability
  3. The inviscid IPM equation
  4. The goal of this study
  5. Darcy's law
  6. Density stratification as stablizing mechanism
  7. Stationary solutions to the IPM equation
  8. Previous results
  9. Summary of main results
  10. Main results
  11. Notations
  12. Statements
  13. Strategy for proofs
  14. Preliminaries
  15. Existence of local-in-time solutions
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $\rho_s(x_2)=x_2.$ There exists $\delta_0$ such that if $\|\theta_0\|_{H^{m}\cap W^{4,1}( \mathbb R^2)} \leq \delta \leq \delta_0$ with $m\geq 20$, then the solution $\theta$ to eq:SIPM_g on $\mathbb R^2$ satisfies where $u=R_1 R^{\perp} \theta$. For the case of $\mathbb T^2$, there exists $\delta_1>0$ such that if $\|\theta_0\|_{H^m(\mathbb T^2)} \leq \delta \leq \delta_1$ with $m\geq 20$, t

Theorems & Definitions (62)

  • Theorem 1.1: T. Elgindi, 2017
  • Theorem 1.2: A. Castro, D. Córdoba, and D. Lear, 2019
  • Theorem 1.3: R. Bianchini, T. Crin-Barat, M. Paicu, 2024
  • Theorem A: Quantitative asymptotic stability of quasi-linear densities
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Theorem B
  • ...and 52 more