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

Liangchen Zou

TL;DR

This work addresses global well-posedness for the dissipative 2D IPM equation near a special anisotropic class $\rho=f(x_2,t)$, where $f$ evolves under the 1D fractional heat equation. The authors employ a decomposition $\rho=\rho_0+\rho_1$, with $\rho_0$ the 1D heat-diffused profile and $\rho_1$ the perturbation, and derive detailed a priori estimates and energy bounds that couple $\rho_1$ to $\rho_0$ through the velocity field $u$ given by $u_1=-R_1R_2\rho$, $u_2=R_1^2\rho$. Under two regimes for the dissipation exponent $0<\alpha<1$ and $1\le\alpha<2$ with corresponding regularity on $f$, the paper proves the existence and uniqueness of global solutions in $C([0,\infty);H^s(\mathbb{R}^2))$ for small perturbations $g\in H^s(\mathbb{R}^2)$, thereby extending stability results for dissipative IPM near a structured, lower-dimensional manifold. The results rely on precise control of nonlinear terms via commutator-type estimates and decay properties of the 1D and 2D fractional heat semigroups, and establish a robust framework for understanding anisotropic dissipation in porous media models.

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 $ρ(x_1,x_2,t)=f(x_2,t)$, 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.

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

TL;DR

This work addresses global well-posedness for the dissipative 2D IPM equation near a special anisotropic class , where evolves under the 1D fractional heat equation. The authors employ a decomposition , with the 1D heat-diffused profile and the perturbation, and derive detailed a priori estimates and energy bounds that couple to through the velocity field given by , . Under two regimes for the dissipation exponent and with corresponding regularity on , the paper proves the existence and uniqueness of global solutions in for small perturbations , thereby extending stability results for dissipative IPM near a structured, lower-dimensional manifold. The results rely on precise control of nonlinear terms via commutator-type estimates and decay properties of the 1D and 2D fractional heat semigroups, and establish a robust framework for understanding anisotropic dissipation in porous media models.

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 , 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 satisfies certain regularity assumptions.
Paper Structure (3 sections, 3 theorems, 57 equations)

This paper contains 3 sections, 3 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Decomposition and a priori estimate
  3. Proof of Theorem 1.1

Key Result

Theorem 1.1

Suppose one of the following holds: (i) $0<\alpha<1$, $s=2-\alpha$, $\nabla f\in L^{\infty}(\mathbb{R)}$, $\tilde{\Lambda}^{1+s-\frac{\alpha}{2}} f\in L^{p}(\mathbb{R})$ with $p\in[2,\infty)$, and $\nabla f\in L^q(\mathbb{R})$ for a given $q\in\left[1,\frac{1}{\alpha}\right)$. (ii) $1\leq\alpha<2$, has a unique global solution in $C\left([0,\infty);H^{s}(\mathbb{R}^2)\right)$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof