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

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 $2<q\le \mathrm{\infty }$, 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 $L^{\mathrm{\infty }}$, an explicit convergence rate is obtained.