Equimeasurable functions and their application to vortex dynamics in lakes
(2019)
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- Authors
- Supervisors
- Van Schaftingen, Jean
- Abstract
- We study a variation of the two-dimensional Euler's equations known as the lake model, where the topography is non homogeneous. In this model, singular vortices are expected to follow lines of constant depth. This behaviour contrasts with what is known for the 2D Euler's equations, where the vortex core is known to move according to the geometry of the domain. We prove that the lake model is robust enough to yield singular-vortex solutions, and we give a mathematical justification of the vortex core evolution law. We also revisit folklore results and key estimates in terms on relevant physical conserved quantities. Although the first order behaviour of vortex dynamics differs from the 2D Euler model, our work also fills a conceptual gap between the two models, by showing that the Euler's 2D equations appear at second order in lake equations. The techniques of proof to attack the singular vortex problem require adaptation of symmetrisation techniques and tools from calculus of variation. In these topics, we also solve an open problem concerning the almost-sure convergence of shape symmetrisation along Markov processes, and we investigate an interesting question about the strict convergence of vector valued measures, whose study turned out to give rise to new interplays between measure-theory and convex analysis.
- Affiliations
Citations
Dekeyser, J. (2019). Equimeasurable functions and their application to vortex dynamics in lakes. https://hdl.handle.net/2078.5/63870