# Research of Carina Geldhauser

## For links to the publications, see Publications

N.B. This site is currently not maintained. For a more up-to-date list of my preprints, please check **both** the servers of Calculus of Variations and Geometric Measure Theory at Pisa, Open Review and arxiv.org

I studied the existence and behavior of solutions to nonlinear PDEs in several different settings:

.

### Interacting Particle Systems

Interacting particle systems model complex phenomena in natural and social sciences. These phenomena involve a large number of interrelated components, which are modeled as particles confined to a lattice. I study so-called interacting diffusion models, i.e. I consider continuous on-site variables. Therefore my models take the form of a system of coupled stochastic differential equations. My goal is to describe the macroscopic behavior of the interacting diffusion as a nonlinear stochastic partial differential equation.

### Gradient flows of non-convex potentials

Gradient flows describe the evolution of a system as the steepest descent of an energy potential. This means that our system is minimizing its energy over time. Non-convex potentials, appearing for example in phase transitions or image processing, give rise to forward-backward parabolic PDEs. I try to determine the regime of initial data under which we can prove existence of solutions to such PDEs. Moreover, I study the behavior and properties of solutions to forward-backward parabolic PDEs.

### Methods of Statistical Mechanics in Turbulence

A very prominent feature of turbulent flows, which appear in fluid dynamics, meteorology and engineering (e.g. in combustion phenomena), is the spontaneous appearance of large-scale, long-lived vortices, e.g. Jupiter's Great Red Spot. Though the distributions of vorticity in the actual flow of normal fluids are continuous, in many cases a set of discrete vortices provides a reasonable approximation. I study these point vortex models with methods of statistical mechanics.

#### Research updates

*On the right are some new related to my research, i.e. new preprints, talks, upcoming trips etc.*

*This page not so frequently updated. For a more up-to-date list of my preprints, please check both the servers of Calculus of Variations and Geometric Measure Theory at Pisa and arxiv.org - Some of my papers are only on one of the two servers, e.g. because arxiv doesn't deal well with compiling images.*

For links to the publications, see the header Publications

### July 2021

A PhD position will soon be opened in my group. Please contact me for expressions of interest.

### June 2021

My project "Stochstic Models of Turbulence" is funded with 1 200 000 SEK (= 120.000 EUR) by the Crafoord foundation.

### April 2021

My project "Mathematical Models for Material Science" was accepted for funding with a volume of 1 200 000 SEK (= 120.000 EUR).

### Jan 2021

My article Limit theorems and fluctuations for point vortices of generalized Euler equations, with Marco Romito, arxiv preprint was accepted for publication in Journal of Statistical Physics..

### July 2020

My article Point vortices for inviscid generalized surface quasi-geostrophic models with Marco Romito (Pisa) is finally online on DCDS Ser B, Volume 25, Issue 7..

### Feb 2020

My new preprint Travelling waves for discrete stochastic bistable equations with Christian Kuehn from TU Munich is now available on arxiv.org.

### June 2019

My review article with Marco Romito (Pisa) is now open access on AIMS' topical section on Matehamtical Analysis in Fluid Dynamics: The point vortex model for the Euler equation

### June 2019

My review article with Marco Romito (Pisa) is now open access on AIMS' topical section on Matehamtical Analysis in Fluid Dynamics: The point vortex model for the Euler equation

### Jan 2019

Talk at the collaborative research center Energy transfers in Atmosphere and Ocean, Hamburg

### Dec 2018

Together with Marco Romito (Pisa), we investigated further point vortices for generalized surface quasigeostrophic models, see arxiv.org

### Oct 2018

My latest work Limit theorems and fluctuations for point vortices of generalized Euler equations in collaboration with Marco Romito (Pisa)is now on arxiv.org