Singularity Formation in the Incompressible Porous Medium Equation without Boundary Mass

2025/11/03 by Kevin H. Dembski, Dembski, Kevin H. · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2511.01827

openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

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.

Citations

• Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source
• Vorticity blowup in 2D compressible Euler equations
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• On the Effects of Advection and Vortex Stretching
• Symmetries and Critical Phenomena in Fluids
• Remarks on functions with bounded Laplacian
• Small scale creation for solutions of the incompressible two dimensional\n Euler equation

Cited by

• Dynamics of Ideal Fluid Flows
• Finite time blow-up for a multi-dimensional model of the Kiselev-Sarsam equation

Related

• Finite time singularities of smooth solutions for the 2D incompressible porous media (IPM) equation with a smooth source
• Global existence and blow-up of solutions to the porous medium equation with reaction and singular coefficients
• Stable self-similar singularity formation for infinite energy solutions of the incompressible porous medium equations
• Flatness implies smoothness for solutions of the porous medium equation
• Large time behavior for a porous medium equation in a nonhomogeneous\n medium with critical density