Joint BCAM-UPV/EHU Analysis and PDE seminar: The small island problem for the degenerate lake equations

Abstract
The lake equations are introduced as a 2D model for the vertically averaged horizontal component of a 3D incompressible fluid. A lake is described by a 2D domain, the surface of the lake, and a non-negative topography function. The 2D velocity is subject to an anelastic constraint depending on the topography. The equations are degenerate if the topography is allowed to vanish. Indeed, vorticity and velocity are then related through non-uniformly elliptic equations.

In this talk, we prove two stability results for the lake equations for singular geometries and degenerated topographies. First, motivated by natural phenomena such as flooding or erosion we consider a sequence of lakes with an island of vanishing topography that shrinks to a point. In the limit, a vortex-wave type solution is obtained. We highlight key differences between the problem under consideration and the respective problem for the incompressible 2D Euler equations (flat topography). The degeneracy of the topography crucially alters the behavior of solutions.

Second, we address the stability for a sequence of lakes without island for which an island appears in the limit, e.g. due to a decreasing level of water in the lake.

This is joint work with C. Lacave and E. Miot.

Link to the session: https://zoom.us/j/97395120947?pwd=MnRpd3NUcUJidit6bEFJUEx2TUZqUT09

More info at https://sites.google.com/view/apdebilbao/home