Program

A quantum fluid is a system of interacting particles that shows the effects of quantum statistics also on a macroscopic scale. Typical examples comprise superfluid Helium II, or Bose-Einstein condensates. From a mathematical perspective, these systems are usually described by a compressible, inviscid, barotropic fluid, subject to a stress tensor depending on the mass density and its derivatives.

During this short course, some rigorous results concerning the analysis of solutions to such models will be presented. In particular, we will discuss the mathematical framework that allow the study of global in time weak solutions with vacuum and, consequently, the treatment of quantized vortices. This class of solutions is particularly relevant because of its deep physical motivations. Within this framework, we will analyze various singular limits corresponding to asymptotic regimes, that are studied in some applications. If time permits, some open problems and directions of research will be mentioned.

Based on the mechanical viewpoint that living tissues present a fluid-like behaviour, PDE models inspired by fluid dynamics are nowadays well established as one of the main mathematical tools for the macroscopic description of tissue growth. Depending on the type of tissue, these models link the pressure to the velocity field using either Brinkman’s law (visco-elastic models) or Darcy’s law (porous-medium equations (PME)). Moreover, the stiffness of the pressure law plays a crucial role in distinguishing density-based (compressible) models from free boundary (incompressible) problems where saturation of the density holds.

This course aims to analyse how to relate different mechanical models of living tissues through singular limits. In particular, we will address the inviscid limit towards the PME and the incompressible limit from the PME to free boundary problems of Hele-Shaw type for equations including convective effects as well as for systems of coupled nonlinear equations.

Multicomponent reactive flow models derived from the kinetic theory are considered. The mathematical structure of the resulting systems of equations is closely related to the underlying kinetic framework. We first address multicomponent diffusion, complex reaction networks, entropic structure, local existence and asymptotic stability of constant equilibrium states. We further address relaxation issues for reactive fluids as well as for two temperature models and discuss the apparition of the bulk viscosity coefficient. We finally consider nonideal fluids and their cohesive properties that lead to single species as well as multicomponent diffuse interfaces models.

Abstract: A variety of pattern-forming phenomena, ranging from the growth of bacterial colonies to snowflake formation, share similar underlying physical mechanisms and mathematical structure. Dense-branching or dendritic morphologies are among the most common forms of microstructural patterning in systems driven out of equilibrium. Understanding the formation kinetics and the interplay of system parameters can lead to understanding of growth and form in nature, as well as improved control and efficiency in a variety of physical, biological, and engineering systems. Prediction and control of the emergent patterns are difficult due to the nonlocality and nonlinearity of the system.

In this course, we focus on viscous fingering as a paradigm for such phenomena. We will study theoretically and numerically the dynamics and control of viscous fingering patterns and, when possible, present experimental validation of model predictions. Extensions of the results to other pattern-forming systems will be discussed.