Vortices and Nonlinear Waves

Recent guests of the junior research group

Quantized vortices and nonlinear waves

A quantum fluid is a system of interacting particles that exhibits effects of quantum statistics also at a macroscopic scale such as for superfluidity, Bose-Einstein condensation and electron transport in semiconductors. The prototype model is the so called Quantum Hydrodynamic system (QHD) describing an inviscid compressible fluid flow featuring a dispersive tensor accounting for the quantum effects. From a mathematical perspective, the QHD system reads as a compressible Euler system augmented by a dispersive stress tensor figuring as quantum correction.
With a hybrid approach relying on both dispersive PDEs and fluid mechanical models and techniques, we investigate the Cauchy theory for these systems and relevant physical phenoemna such as the occurence of quantized vortices.
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Hydrodynamic (in-)stability of geophysical fluid flow

A fundamental question for the understanding of hydrodynamic stability consists in rigorously describing the dynamics of vortices and vortex filaments in incompressible fluids. The vortex filament conjecture states that initial concentration around such a curve persists and the curve evolves through the binormal curvature flow. In its generality, the conjecture remains unsolved and the evolution and interaction of vortex filaments in incompressible fluids remains a puzzling problem. Vortex solutions are of low regularity and as such fall in general outside the usual well-posedness framework.
We derive and analyse the relevant asymptotic dynamics in contexts with spatial heterogeneities such as the (degenerate) lake equations describing a 2D fluid with prescribed but varying topography and the 3D axisymmetric Euler eeqeuations.
A parallel research concerns ill-posedness phenomena and instabilities in models for geophysical fluid mechanics such as the (stratified) Boussinesq equations caputered by a careful analysis of the interaction of nonlinear waves.
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