Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singularity Formation in the Incompressible Porous Medium Equation without Boundary Mass

Kevin H. Dembski

TL;DR

The paper addresses finite-time blow-up for the 2D inviscid porous medium equation in wedge domains with density vanishing on the boundary, a setting where transport regularization would be expected to prevent singularity formation. It reduces the problem to scale-invariant 1D profiles, constructs a singular blow-up profile via a nonlocal ODE, and analyzes stability through a careful decomposition of the linearized operator into a coercive part and a finite-rank perturbation. A weighted Hardy-type framework is developed to control the stability analysis, and a stable manifold argument yields decaying perturbations that permit a truncation to compactly supported, finite-energy solutions that still blow up in finite time. The result demonstrates blow-up in a scenario where boundary mass is absent, highlighting the intrinsic blow-up mechanism of the IPM dynamics and advancing understanding of regularization by transport in incompressible flows.

Abstract

We prove finite-time singularity formation for Lipschitz continuous solutions of the inviscid porous medium equation which vanish on the boundary of the domain. As the density vanishes on the boundary of the domain, the full regularizing effect of transport is present and must be overcome. The solutions are smooth away from the origin and the density can be made compactly supported.

Singularity Formation in the Incompressible Porous Medium Equation without Boundary Mass

TL;DR

The paper addresses finite-time blow-up for the 2D inviscid porous medium equation in wedge domains with density vanishing on the boundary, a setting where transport regularization would be expected to prevent singularity formation. It reduces the problem to scale-invariant 1D profiles, constructs a singular blow-up profile via a nonlocal ODE, and analyzes stability through a careful decomposition of the linearized operator into a coercive part and a finite-rank perturbation. A weighted Hardy-type framework is developed to control the stability analysis, and a stable manifold argument yields decaying perturbations that permit a truncation to compactly supported, finite-energy solutions that still blow up in finite time. The result demonstrates blow-up in a scenario where boundary mass is absent, highlighting the intrinsic blow-up mechanism of the IPM dynamics and advancing understanding of regularization by transport in incompressible flows.

Abstract

We prove finite-time singularity formation for Lipschitz continuous solutions of the inviscid porous medium equation which vanish on the boundary of the domain. As the density vanishes on the boundary of the domain, the full regularizing effect of transport is present and must be overcome. The solutions are smooth away from the origin and the density can be made compactly supported.

Paper Structure

This paper contains 33 sections, 44 theorems, 311 equations.

Table of Contents

  1. Introduction
  2. The 2D Inviscid Porous Medium Equation
  3. Main Result
  4. Previous Results
  5. Regularization by Transport
  6. Scale Invariance and Corner Domains
  7. Discussion of The Proof
  8. Organization
  9. Notation and Definitions
  10. Local Well-Posedness in Scale-Invariant Spaces
  11. Construction of the Profile
  12. Local Solution of the Profile Equation
  13. Monotonicity Lemma
  14. Continuation of the Profile
  15. Shooting Method
  16. ...and 18 more sections

Key Result

Theorem 1.2

For any $k \geq 0$ and $0 < \alpha < 1$, there exist $\nabla u_0, \nabla \rho_0 \in \mathring{C}^{k,\alpha}(\Omega)$ with compact support such that the unique local in time solution to eq:IPM satisfies Here, $\Omega = \{(r, \theta) : -\beta \pi < \theta < \beta \pi\}$ for some $\beta < \pi/2$ where $(r, \theta)$ denote the standard polar coordinates on $\mathbb{R}^2$. Moreover, the initial densit

Theorems & Definitions (82)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Lemma 1.4
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • ...and 72 more