# Research of Carina Geldhauser

## For links to the publications, see Publications

My research deals with phenomena arising in physics and image processing. To solve the questions arising there, I use tools from applied analysis, partial differential equations, and probability theory. My goal is to capture the qualitative and quantitative behavior of solutions of nonlinear PDEs, which can be perturbed by noise.

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

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### 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