The summer school will include the following mini-courses (click on title to view outline):

• On the non-uniqueness of solutions of the Navier-Stokes equations by Maria Colombo (École polytechnique fédérale de Lausanne)

  In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. In this course we will introduce the problem and aim at exhibiting two distinct Leray solutions with zero initial velocity and identical body force. The starting point of our construction is Vishik's answer to another long-standing problem about the 2D Euler equations. Building on Vishik's work, we construct a `background' solution which is unstable for the 3D Navier-Stokes dynamics in similarity variables; the second solution from the same initial datum is a trajectory on the unstable manifold associated to the background solution, in accordance with the predictions of Jia and Sverak.

• Selfgravitating fluids: criticality and stellar collapse by Mahir Hadžić (University College London)
  We examine the question of rigorous singularity formation for selfgravitating compressible fluids, in both the Newtonian and the fully relativistic context. A first step in this direction is to understand the scaling symmetries of the Euler-Poisson system and its relativistic analogue - the Einstein-Euler system. We will introduce natural notions of criticality and formulate reasonable expectations about the existence of self-similar imploding flows. Lecture 2 will focus on the question of existence of self-similar solutions that blow up in finite time, first discovered numerically by Larson and Penston in 1969. These solutions are prototype models of stellar collapse. We will show how to construct them rigorously, thereby highlighting the main obstacle in our analysis - the presence of so-called sonic points in the flow. In the last lecture we will sketch the proof of existence of relativistic Larson-Penston solutions, i.e. self-similar imploding solutions to the Einstein-Euler system first discovered numerically by Ori and Piran in 1990. These also happen to give examples of so-called naked singularities in the context of general relativity theory. All the difficulties from the Newtonian case will be amplified and some entirely new challenges associated with the causal structure of the resulting spacetime will be highlighted.
• Introduction to the Vlasov-Poisson system by Mikaela Iacobelli (Eidgenössische Technische Hochschule Zürich) The mini-course has been cancelled
  In this mini-course, I will give an introduction to non-collisional kinetic theory. In particular, I will focus on the Vlasov-Poisson system and explain its main properties. Then, I will discuss some relevant mathematical questions (like the system's well-posedness and derivation), focusing on the uniqueness and stability issues.

as well as the following talks, given by young researchers:

• Spectral properties of anisotropic electromagnetic metamaterials by Francesco Ferraresso (Cardiff University)
  In this talk I will consider metamaterials having negative electric permittivity \(\epsilon\) and negative magnetic permeability \(\mu\) simultaneously , and more in general having complex electromagnetic parameters. There is a huge interest in the spectral properties of these materials both from an applied and a theoretical point of view, since the successful construction of a left-handed composite material in 2000.
  After a brief survey on the main applications of metamaterials and the related theoretical challenges, I will focus on a specific simplified system of PDEs describing anisotropic metamaterials: the lossy Drude-Lorentz model.
  I will present some recent spectral enclosures and spectral approximation results for this model in a possibly unbounded Lipschitz domain of \(\mathbb{R}^N\).
  In particular, approximation by domain truncation is shown to be spectrally exact outside a set of spectral pollution enclosed in a one-dimensional curve. Based on a joint work with M. Marletta.

• On the Kronig-Penney model in a constant electric field by Simon Larson (University of Gothenburg - Chalmers)

  I will discuss the nature of the spectrum of the one-dimensional Schrödinger operators \[ -\frac{d^2}{dx^2}-Fx + \sum_{n\in \mathbb{Z}} g_n\delta(x-an) \] with \(F,a > 0\) and two different choices of the coupling constants \(\{g_n\}_{n\in \mathbb{Z}}\). We shall see that the nature of the spectrum changes drastically depending on the choice of parameters. Based on joint work with Rupert Frank.

• On some existence results for degenerate cross-diffusion systems by Havva Yoldaş (University of Vienna)
  In this talk, we look at a cross-diffusion system consisting of two Fokker-Planck equations where the gradient of the density for each species acts as a potential for the other one. The model is initially proposed for describing the competition between two rival gang members. The first part of the talk is about the local existence result obtained by the application of the boundedness by entropy method. The system is also the gradient flow for the Wasserstein distance of a functional which is not lower semi-continuous, thus it is not well-posed. We then use a variational approach, more precisely, we compute the convexification of the integral and provide a global existence proof in a suitable sense for the gradient flow of the corresponding relaxed functional. The talk is based on two joint works: A. Barbaro, N. Rodriguez, H.Y. & N. Zamponi (CMS, 19(8): 2139-2175, 2021) and R. Ducasse, F. Santambrogio & H.Y. (arXiv:2111.13764, 2021).