Weak singularity dynamics in a nonlinear viscous medium

TL;DR

This paper investigates the dynamics of weak singularities in a nonlinear viscous medium modeled by degenerate parabolic equations, deriving conditions for their existence and evolution without relying on traditional comparison methods.

Contribution

It introduces a novel approach to analyze weak singularity dynamics in nonlinear viscous media, providing differential equations for singularity support evolution.

Findings

Necessary conditions for weak singular solutions are derived.

Differential equations describing singularity dynamics are established.

The approach does not depend on comparison theorems.

Abstract

We consider a system of nonlinear equations which can be reduced to a degenerate parabolic equation. In the case $x\in\bR^2$ we obtained necessary conditions for the existence of a weakly singular solution of heat wave type ($\codim\sing\supp=1$) and of vortex type ($\codim\sing\supp=2$). These conditions have the form of a sequence of differential equations and allow one to calculate the dynamics of the singularity support. In contrast to the methods used traditionally for degenerate parabolic equations, our approach is not based on comparison theorems.