In this project we plan to cover a wide scope of analysis of partial differential equations originating from the following applications: fluid mechanics, aerodynamics, geophysics, biomathematics, phase transitions and image processing. Our project is divided into four work packages:

- Compressible Navier-Stokes equations
- Crystal growth and image processing
- Regularity criteria
- Asymptotic analysis

## 1. Compressible Navier-Stokes equations

The mathematical description of compressible flows find applications in aerodynamics, geophysics or astro-physics. In spite of intensive research in the field, the mathematical theory of compressible flows is still far from being complete. Better understanding of the governing equations on the level of existence of solutions and their qualitative properties is crucial for the aforementioned applications. From the mathematical point of view the theory delivers a number of challenging problems. We plan to study quantitative and qualitative aspect of solutions to chosen particular models from this subject (most important are the Navier-Stokes system and Navier-Stokes-Fourier systems). Our work in this field will be aimed at answering the questions concerning existence and uniqueness of solutions, their stability (in particular for large velocity vectors), asymptotic behavior and structure of solutions, and finally their regularity.

## 2. Crystal growth and image processing

The crystal growth and image processing problem lead to: TV flows and its generalization; evolution of closed curves by singular mean curvature and Stefan type problems with Gibbs-Thomson relation involving singular mean curvature. Mathematically speaking, we are interested in showing well-posedness of the system and their asymptotic behavior. The central issue, however, is generation, evolution and persistence of facets.

Problem of regularity of weak solution to the non-stationary 3D Navier-Stokes equations is one of the most challenging problems of the modern PDE theory. We are planning to investigate local properties of weak solutions to the NS system. Our main goal is to find out new sufficient conditions of regularity of weak solutions to the Navier-Stokes equations in a neighborhood of a given point in space-time. We are also going to show the global existence of regular solutions being close to special regular global solutions like two-dimensional or axially-symmetric, with respect to non-homogenous and compressible Navier-Stokes equations. Our other aim is to analyze the behavior of solutions near potential blow up points.

Asymptotic analysis of solutions to models arising in mathematical fluid mechanics includes a substantial number of problems of very different nature, and a large variety of mathematical methods to be employed. There open questions on the side of modelling as well as purely mathematical problems. We focus on issues, where the prospective team members contributed significantly to the present state of art:

- Long-time behaviour of solutions and related problems, including stabilisation of solutions to equilibria, existence of global attractors, time periodic and time almost-periodic solutions.
- Scale analysis and singular limits, relations between different models of fluid dynamics.
- Effect of domain geometries on the boundary behaviour, results based on asymptotic analysis, scaling and study of boundary layers.
- Stability of special regular solutions to Navier-Stokes and MHD equations.