In this CRC colloquium we're happy to welcome:

**Robert Lasarzik (WIAS / FU Berlin)**

Energy-variational solutions for different viscoelastic fluid models

Nonlinear evolution equations are ubiquitous in the modelling of processes in science and technology. Since they model phenomena like shocks and turbulence, classical solutions cannot be expected and generalized solutions need to be considered. After recalling different generalized solution concepts, the concept of energy-variational solutions is introduced for a general class of evolution equations. Under certain convexity assumptions, the existence of such solutions can be shown constructively by an adapted minimizing movement scheme. Weak-strong uniqueness follows by a suitable relative energy inequality. Moreover, energy-variational solutions are compared to other generalized solution concepts for specific systems in fluid dynamics. Finally, the general result is applied to two different viscoelastic fluid models without stress diffusion, and a short comparison of different viscoelastic models hints at advantages and disadvantages of this energy-variational approach.

**Sabine Jansen (LMU München)**

Virial expansion, inversion theorems, density functionals

Many quantities of interest in equilibrium statistical mechanics cannot be computed explicitly. At low density or activity (fugacity), however, they admit power series expansions, for example the Mayer and virial expansions. I will present recent results on convergence for inhomogeneous systems where the variable is a density profile. The mathematical key is a novel inversion theorem for holomorphic functionals in infinite-dimensional Banach spaces. Based on joint work with Tobias Kuna and Dimitrios Tsagkarogiannis.

**Sara Danieri (Gran Sasso Science Institute, L’Aquila)**

Energy driven pattern formation: an exploration into the mechanisms of self-made order.

Patterns, intended as ordered regular structures, are ubiquitous in nature. At micro- and mesoscopic scale ordered structures spontaneously emerge in a surprisingly diverse set of physical and chemical systems and they usually form simple structures with some degree of regularity, such as ``bubbles'' or ``lamellar/striped patterns''. One of the leading mechanisms at the base of pattern formation at such scale is since decades recognized to be the competition between short-range attractive forces (favouring pure phases) and long-range repulsive interactions (favouring instead oscillations between different phases). Despite the numerous numerical and experimental studies on the subject, the mathematical mechanisms behind the formation of regular periodic structures is still in most physical cases poorly understood. Two main difficulties reside in the nonlocality of the interactions and in the phenomenon of symmetry breaking, namely the fact that the ground states of such systems have less degrees of symmetry than the energy they minimize. In this talk we will give an overview of the main open problems in the field and discuss some recent results obtained in collaboration with E. Runa in which symmetry breaking and pattern formation has been rigorously proved.