Singular limits of the quasi-linear Kolmogorov-type equation with a source term

TL;DR

This paper investigates the behavior of solutions to a nonlinear ultra-parabolic equation with a source term, establishing existence and stability, and analyzing the limit as the source term becomes impulsive, collapsing to a delta function.

Contribution

It provides a rigorous analysis of the singular limit of the Kolmogorov-type equation with a collapsing source term, using kinetic and compactness methods.

Findings

Existence and uniqueness of solutions are established.

The limit as the source term becomes a delta function is rigorously justified.

The method of kinetic equations and compensated compactness are employed.

Abstract

Existence, uniqueness and stability of kinetic and entropy solutions to the boundary value problem for the Kolmogorov-type genuinely nonlinear ultra-parabolic equation with a smooth source term is established. After this, we consider the case when the source term contains a small positive parameter and collapses to the Dirac delta-function, as this parameter tends to zero. In this case, the limiting passage from the original equation with the smooth source to the impulsive ultra-parabolic equation is fulfilled and rigorously justified. The proofs rely on the method of kinetic equation and on the compensated compactness techniques for genuinely nonlinear equations.