Summer/Winter Schools

VSM Summer School 2025
September 14 to September 20, 2025 in Dienten am Hochkönig
The School takes place at the Universitätssport- und Seminarzentrum Dientnerhof.
The program will start on Monday, September 15 in the morning and finish on Friday, September 19 in the afternoon.
Participants are expected to arrive on Sunday, September 14 and to stay until Saturday, September 20. Registration will open in time and all student members of the VSM are welcome.

The confirmed speakers are:
David Damanik, Rice University – Furstenberg’s Theorem and the Anderson Model in the Non-Stationary Setting
After reviewing the classical Furstenberg theorem and its application to localization results for the one-dimensional Anderson model, we discuss recent extensions to the non-stationary case due to Anton Gorodetski and Victor Kleptsyn, as well as joint work with them on models with local correlations.

Ewelina Zatorska, University of Warwick – Mathematical Analysis of Multi-Phase Flow Models
This lecture series focuses on the mathematical analysis of multi-phase flows, with particular emphasis on compressible two-fluid systems in the isentropic and isothermal regimes. Such models are of considerable physical importance, appearing in a wide range of applications including geophysical and astrophysical flows, oil recovery, spray combustion, and coolant transport in nuclear reactors.

In contrast to compressible mixture models—where constituents interact at the microscopic level and are inseparably mixed—multi-phase flow models describe the macroscopic evolution of distinct, non-interacting compressible fluids that coexist in the same spatial domain. We will consider only macroscopic models, and discuss their derivation and connection to the single-phase compressible fluid models.

A central theme of the series will be the extension of well-posedness theory from single-phase to multi-phase systems. We will examine which results—such as the existence of local-in-time strong solutions and global-in-time weak solutions—can be generalized to two-fluid models. This part of the course will be based on a selection of recent results concerning both viscous and inviscid two-fluid systems with algebraic pressure closures. Some results also extend to models with PDE-based closures, such as the Baer–Nunziato system. If time permits, we will address the construction of relative entropy functionals and highlight the analytical advantages of PDE closures in this context.

In the final part of the series, we will explore the connection between compressible two-phase models and so-called constrained models, which have so far only been derived at a formal level. I will outline their main mathematical features and discuss emerging links with hydrodynamic models of interacting agents, a topic of growing interest in both applied mathematics and mathematical physics.

This course is designed for graduate students and researchers in mathematical fluid mechanics. Frequent references will be made to nowadays well-known results for the compressible Navier–Stokes equations. The problem sessions will introduce participants to analytical techniques originally developed for this system, which serve as a valuable foundation for the study of partial differential equations more broadly.

References:
1. D. Bresch, B. Desjardins, J.-M. Ghidaglia, E. Grenier, M. Hilliairet, Multifluid models including compressible fluids, in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, eds. Y. Giga, A. Novotný, Springer, Cham, pp. 1–52 (2018).
2. D. Bresch, P.B. Mucha, E. Zatorska, Finite-energy solutions for compressible two-fluid Stokes system, Archive for Rational Mechanics and Analysis, 232(2), 987–1029 (2019).
3. F. Bouchut, Y. Brenier, J. Cortes, J.-F. Ripoll, A hierarchy of models for two-phase flows, Journal of Nonlinear Science, 10, 639–660 (2000).
4. M. Ishii, T. Hibiki, Thermo-Fluid Dynamics of Two-Phase Flow, Springer, New York, (2006).
5. B.J. Jin, A. Novotný, Weak-strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids, Journal of Differential Equations, 268, 204–238 (2019).
6. Y. Li, E. Zatorska, On weak solutions to the compressible inviscid two-fluid model, Journal of Differential Equations, 299, 33–50 (2021).
7. Y. Li, E. Zatorska, Remarks on weak-strong uniqueness for two-fluid model, in: Geometric Potential Analysis, De Gruyter (2022).
8. Y. Li, Y. Sun, E. Zatorska, Large time behaviour for a compressible two-fluid model with algebraic pressure closure and large initial data, Nonlinearity, 33(8), 4075–4094 (2020).
9. A. Novotný, M. Pokorný, Weak solutions for some compressible multicomponent fluid models, Archive for Rational Mechanics and Analysis, 235, 355–403 (2020).
10. T. Piasecki, E. Zatorska, Maximal regularity for compressible two-fluid system, Journal of Mathematical Fluid Mechanics, 24(9), Article 96 (2022).

The third speaker will be announced in due time.

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VSM Summer School 2024
September 15 to September 21, 2024 in Dienten am Hochkönig

Grégory Miermont, École Normale Supérieure de Lyon
Combinatorial and probabilistic aspects of maps

Laure Saint-Raymond, Institut des Hautes Études Scientifiques (IHES)
An introduction to kinetic theory

Bernd Sturmfels, Max Planck Institute for Mathematics in the Sciences Leipzig
Computational Algebraic Geometry
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