Seminar Analysis: Christian Zillinger (Universität Bonn)

On linear inviscid damping, boundary effects and blow-up

The Euler equations of fluid dynamics are time-reversible equations and possess many conserved quantities, including the kinetic energy and entropy. Furthermore, as shown by Arnold, they even have the structure of an infinite-dimensional Hamiltonian system. Despite these facts, in experiments one observes a damping phenomenon for small velocity perturbations to monotone shear flows, where the perturbations decay with algebraic rates. In this talk, I discuss the underlying phase-mixing mechanism of linear inviscid damping, its mathematical challenges and how to establish decay with optimal rates for a general class of monotone shear flows. Here, a particular focus will be on the setting of a channel with impermeable walls, where boundary effects asymptotically result in the formation of singularities. 

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