The dynamics of liquids and gases can be modeled by the Navier-Stokes system of partial differential equations describing the balance of mass and momentum in the fluid flow. In recent years their has been an increasing interest in random influences on the fluid motion modeled via stochastic partial differential equations.

In this course we will study the existence of weak martingale solutions to the stochastic Navier-Stokes equations (both incompressible and compressible). These solutions are weak in the analytical sense (derivatives exists only in the sense of distributions) and weak in the stochastic sense (the underlying probability space is not a priori given but part of the problem).

Lecture I: Measure-valued and dissipative solutions.
The concepts of measure-valued solutions and dissipative solutions were respectively introduced in the late 80s early 90s by DiPerna-Majda and Lions for the Euler equations of incompressible fluids. Although they look very different from each other, I will explain their close relationship in relation with the "weak-strong" uniqueness principle.

Lectures II and III: Variational and sharp measure-valued solutions to the Euler equations.
Using the least action principle, one may try to solve the Euler equation not as a Cauchy problem but rather as a minimization problem. This opens the way to a rather precise "sharp" concept of measure-valued solutions. Its relevence for the Cauchy problem will be discussed.

Lecture IV: Magnetic relaxation of the Euler equations and dissipative solutions.
Following K. Moffatt one can look for stationary solutions to the Euler equations with prescribed topology by solving a degenerate parabolic equations coming from MHD theory. This leads to a suitable concept of generalized solutions blending variational features with Lions' concept of dissipative solutions.

The aim of this course is to present some of the classical and more recent techniques to control oscillations and measure the regularity of weak solutions to various models of compressible fluids and in particular: The barotropic compressible Navier-Stokes and some Navier-Stokes-Fourier systems.

We will start by introducing the regularity theory on the simpler case of linear advection equations which follows from the notion of renormalized solutions. We will then review how the estimates developed for compressible fluids, first by P.L. Lions and then by E. Feireisl for the compressible Navier-Stokes, and E. Feireisl and A. Novotny for the Navier-Stokes-Fourier system. We will finish by presenting some recent estimates introduced with D. Bresch.

Phase field or diffuse interface approaches are frequently used to model the complex dynamics of two-phase flow problems. The major advantage of this ansatz is the ability to describe critical phenomena like topological changes due to merging/splitting of single phases or the phases' interaction with walls. In the last years there have been substantial advancements in the theory and numerics for diffuse-interface models. The most important directions for compressible free flows will be discussed concerning well-posedness and asymptotic behaviour for various limit regimes including the classical sharp interface limit. Besides the purely analytical issues it is important that diffuse interface models allow a reliable and efficient numerical approximation. This imposes new constraints on the whole model ansatz. While there is by now a quite well-established theory for compressible free flow situations the theory for two-phase flows in porous media is still in its very beginnings. Basic new developments for least partially homogenized flow of multiple phases and components are presented.

Main speakers

1. Pierre-Emmanuel Jabin
2. Yann Brenier
3. Dominic Breit
4. Christian Rohde

Participating organizers

5. Josef Malek
6. Milan Pokorny
7. Miroslav Bulicek
8. Ondrej Kreml
9. Michael Ruzicka
10. Eduard Feireisl
11. Mirko Rokyta

Participants

12. Martin Lanzendorfer
13. Jianfeng Cheng
14. Qin Zhang
15. Marta Lewicka
16. Erika Maringova
17. Swati Sharma
18. Vaclav Macha
19. Pappu Kumar
20. Sebastian Schwarzacher
21. Franz Gmeineder
22. Michal Bathory
23. Martin Michalek
24. Mikhail Turbin
25. Aleksandr Boldyrev
26. Simon Axmann
27. Jan Brezina
28. Tabea Tscherpel
29. Anna Abbatiello
30. Kehinde Ayodele Ajeigbe
31. Turgut Ak
32. Karel Tuma
33. Kamila Lyczek
34. Bogdan Raita
35. Karol Hajduk
36. Tomasz Debiec
37. Jakub Slavik
38. Marija Galic
39. Petr Kaplicky
40. Piotr Miszczak
41. Sarka Necasova
42. Martin Rehor
43. Radim Hosek
44. Pratik Nayak
45. Matteo Caggio
46. Martin Kalousek
47. Jan Blechta
48. Aneta Wróblewska-Kamiñska
49. Ondrej Soucek
50. Bangwei She
51. Oldrich Ulrych
52. Jana Peskova
53. Hana Bilkova