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Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation

Yaowei Xie, Huan Yu

Abstract

In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space $W^{1,\infty}$ when the initial data are small perturbations of stable profile $g(x_2).$ Consequently, instability can be inferred. Notably, our results are valid for arbitrary vertically stratified density profiles $g(x_2)$ without imposing any restrictions on the sign of $g'(x_2).$ From a physical perspective, since gravity acts downward, density profiles satisfying $g'(x_2) < 0$ typically correspond to stable configurations, whereas those with $g '(x_2) > 0$ are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when $g'(x_2) < 0$ and $g'(x_2)\in W^{2,\infty}(\mathbb{R})$. To the best of our knowledge, this work provides the first rigorous demonstration of IPM instability for vertically nonlinear density profiles, marking a significant departure from conventional physical expectations.

Mild ill-posedness in $W^{1,\infty}$ for the incompressible porous media equation

Abstract

In this paper, we establish the mild ill-posedness of 2D IPM equation in the critical Sobolev space when the initial data are small perturbations of stable profile Consequently, instability can be inferred. Notably, our results are valid for arbitrary vertically stratified density profiles without imposing any restrictions on the sign of From a physical perspective, since gravity acts downward, density profiles satisfying typically correspond to stable configurations, whereas those with are generally expected to be unstable. Surprisingly, our analysis uncovers an unexpected instability even when and . To the best of our knowledge, this work provides the first rigorous demonstration of IPM instability for vertically nonlinear density profiles, marking a significant departure from conventional physical expectations.

Paper Structure

This paper contains 14 sections, 16 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Key analytical estimates on Besov spaces
  4. Technical lemmas of composite functions
  5. Construction of special functions
  6. Local existence and uniqueness
  7. Proof of Theorem \ref{['thm1']}
  8. Lower bound estimate for $\|\nabla \eta\|_{L^\infty}$
  9. Conclusion of the Proof
  10. Appendix
  11. Littlewood-Paley Decomposition
  12. Besov Spaces
  13. Nonhomogeneous Bony Decomposition
  14. Commutator Estimates

Key Result

Theorem 1.1

Let $g(x_2)$ be any horizontal stratified state satisfying $g'(x_2)\not\equiv0$ and $g'(x_2)\in W^{2,\infty}(\mathbb{R})$. For every sufficiently small $\varepsilon>0$, there exists a initial data $\eta_0({\bf x})\in C^\infty_c(\mathbb{R}^2)$ satisfying such that the unique local solution $\eta(\cdot,t)$ of system p-IPM with initial data $\eta_0$ satisfies Here, $\tilde{c}, \tilde{C}$ are unive

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2: GTM-fourier-grafakos
  • Lemma 2.3
  • proof
  • Lemma 2.4: bahouriFourierAnalysisNonlinear2011miao2019-littlewood
  • Lemma 2.5: bahouriFourierAnalysisNonlinear2011miao2019-littlewood
  • ...and 17 more