Claudia C. Raithel

Personal Homepage

Welcome to my webpage!   I am an applied mathematician working in partial differential equations. Currently, I am a postdoc at the TU Dresden with Prof. Stefan Neukamm, before I was a postdoc at the TU Wien with Prof. Ansgar Jüngel. I completed my PhD in 2019 under the supervision of Prof. Felix Otto at the Max-Planck-Institute for Mathematics in the Sciences in Leipzig. My main research interests are stochastic homogenization of elliptic and parabolic PDEs -in particular, in the presence of defects or boundary phenomenon-; in particle systems -in particular, in fluctuating hydrodynamics-; and, more recently, also in cross-diffusion systems. Please see below for more details!

Contact Details

Email: claudia.raithel@​tu-dresden.de

Address: Bürogebäude Z21, 225.1
                      Zellescher Weg 25
                      01217 Dresden

Research

Stochastic homogenization of linear elliptic PDEs:   The central qualitative result in the homoenization of linear elliptic PDEs is classical and states that, under the assumptions of stationarity and ergodicity, as one zooms-out, a linear elliptic PDE with random coefficients may be approximated (in some sense, almost-surely) by a homogenized PDE of the same form with constant coefficients. In the last years robust theories have been developed to, under various quantifications of the ergodicity assumption, obtain quantitative results describing the homogenization process: e.g., convergence rates or estimates for the stochastic fluctuations. In my research, I have mainly been interested in settings in which the classical stationarity assumption is broken (e.g., when there is a boundary or an interface) or when the macroscopic situation is itself in some sense singalur (e.g., in a domain with corners).
[1], [2], [4], [6], [9]

Interacting particle systems:   The dynamics of a system of interacting particles is described via a system of SDEs, each containing a drift and interaction term. Due to the computational complexity required to simulate such a many-particle system, it is desirable to derive effective models. In particular, one shows that in the many-particle limit the distribution of the empirical measure solves an effective PDE. With the effective mean-field behavior recovered, one is also interested in descriptions of the fluctuations of the stochastic particle system around this effective behavior. In the theory of fluctuating hydrodynamics these fluctations are described via a super-critical singular SPDE called the Dean-Kawasaki equation.
[5], [8]

Cross-diffusion systems with entropy structure: These are (possibly degenerate) quasilinear parabolic systems which are coupled in the diffusion term. Due to the cross-terms, the rigorous analysis of these systems is difficult. It helps to consider a subclass of cross-diffusion systems: ones which are formally given as the gradient flow of an entropy functional. Such cross-diffusion systems appear very naturally in a variety of contexts; examples incude the Shigesada-Kawasaki-Teramoto (SKT) model for population dynamics and the Maxwell-Stefan model, which can be used to model non-Fickian diffusion. Even within this special class of cross-diffusion systems, not much is known concerning the regularity of weak solutions. We have been able to derive the first widely applicable (to, e.g., solutions of the Maxwell-Stefan system and bounded solutions of the SKT model) partial regularity result in this context.
[7]

The contributions [5] and [8] are also concerned with cross-diffusion systems, but without entropy structure.

Singular SPDEs: [3], [8]

Publications

Click on the number and it will take you to the journal article or preprint.

[9] P. Bella, J. Fischer, M. Josien, and C. Raithel. Boundary layer estimates in stochastic homogenization. arXiv preprint: 2403.12911, 2024.

[8] F. Cornalba, J. Fischer, J. Ingmanns, and C. Raithel. Density fluctuations in weakly interacting particle systems via the Dean-Kawasaki equation. arXiv preprint: 2303.00429, 2023 (submitted).

[7] M. Braukhoff, C. Raithel, and N. Zamponi. Partial Hölder regularity for solutions of a class of cross-diffusion systems with entropy structure. J. Math. Pures Appl., 166: 30-69, 2022.

[6] M. Josien, C. Raithel, and M. Schäffner. Stochastic homogenization and geometric singularities: a study on corners. arXiv preprint: 2201.09938, 2022. (accepted in SIAM J. Math Anal.)

[5] E. Daus, M. Ptashnyk, and C. Raithel. Derivation of a fractional cross-diffusion system as the limit of a stochastic many-particle system driven by Lévy Noise. J. Differential Equations, 309: 386--426, 2022.

[4] M. Josien and C. Raithel. Quantitative homogenization for the case of an interface between two heterogeneous media. SIAM J. Math Anal., 53: 813--854, 2020.

[3] C. Raithel and J. Sauer. The initial value problem for singular SPDEs via rough paths. arXiv preprint: 2001.00490, 2020.

[2] J. Fischer and C. Raithel. Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space. SIAM J. Math. Anal. 49 (1): 82-114, 2017.

[1] C. Raithel. A large-scale regularity theory for random elliptic operators on the half-space with homogeneous Neumann boundary data. arXiv preprint: 1703.04328, 2017.

Teaching

TU Dresden (from 2023): Exercise Sessions for the Lectures:

Analysis 1 for Teachers (Winter 2023)

TU Wien (2019-2023): Exercise Sessions for the Lectures:

Modelling with Partial Differential Equations (Winter 2019)

Calculus of Variations (Summer 2020)

Partial Differential Equations (Winter 2020; Winter 2022)

Ordinary Differential Equations (Summer 2021; Summer 2022)

Analysis 1 (Winter 2021)