Vanishing viscosity limits for the degenerate lake equations with Navier boundary conditions

TL;DR

This paper studies the behavior of solutions to degenerate viscous lake equations as viscosity vanishes, proving convergence to inviscid solutions under Navier boundary conditions with explicit rates in certain cases.

Contribution

It establishes the global existence and uniqueness of solutions for degenerate viscous lake equations with Navier boundary conditions and proves their convergence to inviscid solutions, including explicit rates when vorticity is bounded.

Findings

Global solutions exist for initial vorticity in L^q, 2<q≤∞.

Solutions converge to inviscid lake equations as viscosity vanishes.

Explicit convergence rate obtained when vorticity is in L^∞.

Abstract

The paper is concerned with the vanishing viscosity limit of the two-dimensional degenerate viscous lake equations when the Navier slip conditions are prescribed on the impermeable boundary of a simply connected bounded regular domain. When the initial vorticity is in the Lebesgue space $L^q$ with $2<q\le\infty$, we show the degenerate viscous lake equations possess a unique global solution and the solution converges to a corresponding weak solution of the inviscid lake equations. In the special case when the vorticity is in $L^\infty$, an explicit convergence rate is obtained.