Finite time blow-up in a 1D model of the incompressible porous media equation

TL;DR

The paper derives a one-dimensional boundary-trace model for the 2D IPM equation, yielding a nonlocal transport equation $\partial_t \rho + g H_a\rho\, \partial_x \rho = 0$ on $\mathbb{T}$ with $H_a$ a convolution operator whose kernel $K_a(y)$ interpolates between the zero operator and the Hilbert transform. It proves local well-posedness in $C^\infty(\mathbb{T})$ for fixed $a,g>0$ and establishes a Beale-Kato-Majda-type blow-up criterion, then demonstrates finite-time blow-up for a class of smooth, even, nonnegative initial data satisfying $\rho_0(0)=0$ and $\rho'_0\ge 0$ on $[0,\pi)$. The blow-up argument adapts integral-inequality techniques from the CCF model, using monotonicity and an $H_a$-bound to derive a differential inequality that forces the $\partial_x$-norm to blow up in finite time. This work connects the 1D model to the CCF framework (via $a\to\infty$) and supports the view that CCF can serve as a reasonable 1D surrogate for IPM boundary dynamics, offering a rigorous route to understanding boundary-layer singularity formation in IPM-type systems.

Abstract

We derive a PDE that models the behavior of a boundary layer solution to the incompressible porous media (IPM) equation posed on the 2D periodic half-plane. This 1D IPM model is a transport equation with a non-local velocity similar to the well-known Córdoba-Córdoba-Fontelos (CCF) equation. We discuss how this modification of the CCF equation can be regarded as a reasonable model for solutions to the IPM equation. Working in the class of bounded smooth periodic data, we then show local well-posedness for the 1D IPM model as well as finite time blow-up for a class of initial data.

Figures (1)

• Figure 1: Example boundary trace of initial data suspected to lead to finite time blow-up in (\ref{['eq_intro_IPM']}). The boundary trace is even, attains 0 at the origin, and is increasing from $x_1 = 0$ to $x_1 = \pi$ (cf. \ref{['thm_intro_FTBU']}).

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3