The Stability of the Solutions for a Quasilinear Degenerate Parabolic Equation

TL;DR

This paper investigates the stability of solutions to a degenerate quasilinear parabolic equation from boundary layer theory, establishing stability results under partial or no boundary conditions using BV estimates and Kruzkov's method.

Contribution

It introduces a novel partial boundary condition matching the degenerate equation and proves solution stability without boundary conditions, advancing understanding of such equations.

Findings

Existence of entropy solutions via BV estimates.

Stability established with partial boundary conditions.

Stability proved even without boundary conditions.

Abstract

The equation arising from Prandtl boundary layer theory is considered. The existence of the entropy solution can be proved by BV estimate method. The interesting problem is that, since a may be degenerate on the boundary, the usual boundary value condition may be overdetermined. Accordingly, only dependent on a partial boundary value condition, the stability of solutions can be expected. This expectation is turned to reality by Kruzkov's bi-variables method, a reasonable partial boundary value condition matching up with the equation is found first time. Moreover, the stability can be proved even without any boundary value condition.