Asymptotics of solutions of a parabolic equation near singular points

TL;DR

This paper investigates the asymptotic behavior of solutions to a quasi-linear parabolic equation near singular points, with implications for physical systems like fluid dynamics and turbulence.

Contribution

It provides new analysis of solution asymptotics near various singularities in parabolic equations with small parameters, relevant to physical and probabilistic models.

Findings

Asymptotic descriptions near jump discontinuities and shock collisions

Analysis of gradient catastrophe and transition of weak discontinuities

Results applicable to acoustic waves, turbulence, and nonlinear diffusion

Abstract

Results of investigation of the asymptotic behavior of solutions to the Cauchy problems for a quasi-linear parabolic equation with a small parameter at a higher derivative near singular points of limit solutions are presented. Interest to the problem under consideration is explained by its applications to a wide class of physical systems and probabilistic processes such as acoustic waves in fluid and gas, hydrodynamical turbulence and nonlinear diffusion. The following cases are considered: a singularity generated by a jump discontinuity of the initial function, collision of two shock waves, gradient catastrophe, transition of a weak discontinuity into a shock wave, a singularity generated by a large initial gradient.