Junior Research Group Lars Eric Hientzsch

The junior research group's research activities concern the mathematical analysis of nonlinear PDEs arising in the description of quantum and geophysical fluid flow, often related to the propagation and interaction of waves.
The research develops along two main directions

• (quantised) vortices and nonlinear waves,
• hydrodynamic stability and regularity properties of geophysical fluid flow.

These research lines share a common file rouge given by the investigation of coherent structures. Specifically, we are interested in the properties (quantised) vortices, vortex filaments and the characterisation of related asymptotic dynamics.

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. Its mathematical theory faces similar outstanding questions and difficulties as the well-known ones for the compressible Euler equations.
Given its quantum nature, QHD admits a formal analogy through the Madelung transform to an effective wave-function dynamics. The latter is characterised by nonlinear Schrödinger equations (NLS) and motivates our hybrid approach combining both: dispersive techniques for NLS and hydrodynamic ones for QHD.
A key theme is to elucidate peculiarities of the behavior of quantum fluid flow compared to classical hydrodynamics. Mathematically, this mainly translates to exploiting dispersive properties in establishing a solution theory and the study of quantized vortices. A robust understanding of the properties of quantised vortices is instrumental for a theory of (quantum) turbulence.
The research activity is in large part finalized to study coherent structures for which non-trivial boundary conditions at spatial infinity are considered.

We are thus led to investigate

• nonlinear Schrödinger equations with non-vanishing conditions at infinity such as Gross-Pitaevskii equations. The study of these equations is of independent interest.
• hydrodynamic systems such as QHD and related models, again mainly with non-trivial far-field behavior.

The analysis of these hydrodynamic models describing the evolution in term of the physical observables mass and current density allows one (compared to NLS) to include dissipative or viscous phenomena in a natural way. The quantum Navier-Stokes equations for instance constitute an example for a dissipative quantum fluid model.
Further, the considered models fall with the class of Korteweg or capillarity fluids. Our analysis gives insights and extends to the broader class of these capillarity fluids for which an analogy to a seminlinear NLS equation is no longer available.

Key questions we pursue are well-posedness issues, the analysis of relevant dispersive effects of compressible flow, quantitative and qualitative properties of solutions, singularity formation, singular limits and characterisation of relevant asymptotic dynamics, e.g. vortex dynamics.

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 such as the propagation of vortex rings in inviscid or viscous 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.
(Partial) answers are available if considered in specific geometric situations and dimension reduced settings. 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.

A parallel research concerns ill-posedness phenomena and instabilities in models for geophysical fluid mechanics such as the (stratified) Boussinesq equations or the (degenerate) lake with rough geometries and topographies.
For the former for instance, we identify a nonlinear ill-posedness mechanism at critical regularity by a careful analysis of the interaction of nonlinear waves. The Boussinesq equations describing a non-homogeneous or stratified fluid are one of the models commonly used in oceanography and atmospheric sciences.

Beyond being of independent interest, the lake equations and 2D Boussinesq equations share strong mathematical analogies with the 3D axisymmetric incompressible Euler equations (the former without, the latter with swirl) which further motivates our study.

Staff in the junior research group