Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source

Diego Córdoba, Luis Martínez-Zoroa

TL;DR

This paper proves the existence of smooth, finite-energy solutions to the 2D IPM equation with a smooth, compactly supported source that blow up in finite time. The authors develop a multi-scale, p-layered construction, introducing velocity approximations and a first-order solution operator to iteratively build higher-frequency layers while rigorously controlling errors. The resulting infinite-layer solution remains smooth and compactly supported up to a finite time but exhibits unbounded growth in the C^1 norm as t approaches the blow-up time, with ∇u also becoming unbounded. This constitutes a novel blow-up result for IPM with finite energy and a smooth forcing, highlighting a carefully orchestrated, scale-separated mechanism to drive singular behavior while preserving the overall analytic structure.

Abstract

We establish the existence of smooth, finite-energy solutions to the 2D incompressible porous media equation (IPM), with a compactly supported uniformly smooth source, which develop singularities in finite time.

Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source

TL;DR

This paper proves the existence of smooth, finite-energy solutions to the 2D IPM equation with a smooth, compactly supported source that blow up in finite time. The authors develop a multi-scale, p-layered construction, introducing velocity approximations and a first-order solution operator to iteratively build higher-frequency layers while rigorously controlling errors. The resulting infinite-layer solution remains smooth and compactly supported up to a finite time but exhibits unbounded growth in the C^1 norm as t approaches the blow-up time, with ∇u also becoming unbounded. This constitutes a novel blow-up result for IPM with finite energy and a smooth forcing, highlighting a carefully orchestrated, scale-separated mechanism to drive singular behavior while preserving the overall analytic structure.

Abstract

We establish the existence of smooth, finite-energy solutions to the 2D incompressible porous media equation (IPM), with a compactly supported uniformly smooth source, which develop singularities in finite time.

Paper Structure

This paper contains 20 sections, 23 theorems, 442 equations.

Table of Contents

  1. Introduction
  2. Motivation and related work
  3. Local well-posedness for IPM.
  4. Ill-posedness at critical regularity for IPM.
  5. Stability results for IPM from a stable profile.
  6. Small scale formation for IPM.
  7. Finite time blow-up results of classical solutions for 3D incompressible Euler equations.
  8. Ideas of the proof
  9. The use of different scales
  10. Approximating the velocity
  11. Dealing with passive transport
  12. The first order solution operator
  13. Structure of the paper
  14. Notation
  15. Approximations for the velocity
  16. ...and 5 more sections

Key Result

Theorem 1

There exists smooth compactly supported initial conditions $\rho_{0}\in C^{\infty}_{c}$ and a smooth compactly supported source term $F(x,t)\in L^{\infty}_{t}C^{\infty}_{x}$ such that the only classical solution to the IPM equation $\rho(x,t)\in C^{\infty}_{c}$ for $t\in[0,1)$ exhibits finite time b

Theorems & Definitions (67)

  • Theorem : Finite time blowup for the IPM equation with a smooth source
  • Remark 1
  • Remark 2
  • Definition 2.1
  • Remark 3
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 57 more