Degenerate lake equations: classical solutions and vanishing viscosity limit

TL;DR

This paper establishes the existence of global classical solutions for degenerate inviscid lake equations and demonstrates the convergence of viscous solutions to these inviscid solutions as viscosity vanishes, validating the models physically.

Contribution

It provides the first proof of global classical solutions for degenerate lake equations and rigorously shows the vanishing viscosity limit with a convergence rate.

Findings

Existence of global classical solutions for degenerate inviscid lake equations.

Weak viscous solutions converge to inviscid solutions as viscosity approaches zero.

A convergence rate for the vanishing viscosity limit is derived.

Abstract

The objective of this paper is twofold. First, we show the existence of global classical solutions to the degenerate inviscid lake equations. This result is achieved after revising the elliptic regularity for a degenerate equation on the associated stream-function, and adapting the method used for construction of classical solutions to the incompressible Euler equations. Second, we show that the weak solutions of the viscous lake equations converge to classical solutions of the inviscid lake equations when the viscosity coefficient goes to zero, which constitutes an important physical validation of these models. The later result is achieved by the use of energy method as in the proofs of Kato-type theorems. This method also allows us to expose a convergence rate.