Termine

We show how to regularize vortex sheets by means of smooth, compactly supported vorticities that asymptotically evolve according to the Birkhoff-Rott vortex sheet dynamics. More precisely, consider a vortex sheet initial datum $\omega^0_{\mathrm{sing}}$, which is a signed Radon measure supported on a closed curve. We construct a family of initial vorticities $\omega^0_\varepsilon\in C^\infty_c(\mathbb{R}^2)$ converging to $\omega^0_{\mathrm{sing}}$ distributionally as ε $\to 0^+$, and show that the corresponding solutions $\omega_ε(x,t)$ to the 2D incompressible Euler equations converge to the measure defined by the Birkhoff--Rott system with initial datum $\omega^0_{\mathrm{sing}}$. Based on joint work with Alberto Enciso and Antonio Fernandez.

Xmas PDE Colloquium - coffee break after (14:45 - 15:15, SRZ Lounge)

Imagine agents moving according to simple rules that disfavor close crowding but favor a mutual intermediate range proximity. Systems of this type appear across the sciences, from molecular dynamics to microbiology and ecology. The competition between the underlying short-range repulsion and intermediate-range attraction can lead to a phase transition, where the preferred state changes from a "crystalline" equidistribution to the formation of clusters (or colonies) separated by vacuum regions. I will describe recent work that analyzes this transition (or bifurcation) in some detail, emphasizing a curious aspect that makes this transition "non-hysteretic" or "reversible" in an infinite system-size limit, thus allowing for easy switching from crystal to cluster -- and back. Results include predictions for sizes and shapes of vacuum regions, corrections due to noise, and expansions for finite system size corrections. This is based on joint work with Angela Stevens and on a summer REU project mentored by Olivia Clifton.

Xmas PDE Colloquium - coffee break before (14:45 - 15:15, SRZ Lounge)

Many linear and nonlinear partial differential equations (PDEs) arise from the minimization of an underlying energy functional. Examples are ubiquitous and arise, for instance, in the study of non-Newtonian fluids, minimal surfaces, or nonlinear mechanics. Whereas classical numerical methods and their analysis mostly focus entirely on the solution of the associated PDE/Euler-Lagrange equations, recent contributions have started to take the energetic structure into account. This is particularly relevant for engineering applications where the energetic behavior of minimizers can be more relevant than the solution itself. In addition, analysis in terms of energy often naturally fits the structure of the problem, which can be exploited to derive error bounds with explicit constants. We will introduce the method of flux-equilibration which yields highly effective a posteriori error bounds that can subsequently be used to steer adaptive mesh-refinement. After a general introduction to the concept, we highlight recent advances and generalizations to the nonlinear setting and present numerical results highlighting the efficiency of the method.

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