In many mechanical systems where energy is conserved, the phenomenon of resonance can occur, meaning that for certain time-periodic forces, the solution of the system becomes unbounded. Examples of partial differential equations describing such systems include the wave equation and equations of linearized elasticity (Lamé system). On the other hand, resonance does not occur in systems with strong dissipation, such as systems described by the heat equation. More precisely, in such a system, there exists a unique time-periodic solution for each time-periodic right-hand side. In this lecture, we will address the question "how much dissipation is necessary to prevent the occurrence of resonance?". We will analyze periodic solutions to the so-called heat-wave system, where the wave equation is coupled with the heat conduction equation via a common boundary. In this system, dissipation only exists in the heat component, and the system can be viewed as a simplified model of fluid-structure interaction. We will demonstrate that in certain geometric configurations, there exists a unique time-periodic solution for each time-periodic right-hand side, assuming sufficient regularity of the forcing term. A counterexample illustrates that this regularity requirement is stronger than in the case of the Cauchy problem. Finally, we will discuss the open question of whether the result is valid for arbitrary geometry or if there exists a geometry where resonance can occur.[-]

We discuss some examples of the "good" effects of "very bad", "irregular" functions. In particular we will look at non-linear differential (partial or ordinary) equations perturbed by noise. By defining a suitable notion of "irregular" noise we are able to show, in a quantitative way, that the more the noise is irregular the more the properties of the equation are better. Some examples includes: ODE perturbed by additive noise, linear stochastic transport equations and non-linear modulated dispersive PDEs. It is possible to show that the sample paths of Brownian motion or fractional Brownian motion and related processes have almost surely this kind of irregularity. (joint work with R. Catellier and K. Chouk)[-]

In the fifties John Nash astonished the geometers with his celebrated isometric embedding theorems. A folkloristic explanation of his first theorem is that you should be able to put any piece of paper in your pocket without crumpling or folding it, no matter how large it is.
Ten years ago László Székelyhidi and I discovered unexpected similarities with the behavior of some classical equations in fluid dynamics. Our remark sparked a series of discoveries and works which have gone in several directions. Among them the most notable is the recent proof of Phil Isett of a long-standing conjecture of Lars Onsager in the theory of turbulent flows. In a joint work with László, Tristan Buckmaster and Vlad Vicol we improve Isett's theorem to show the existence of dissipative solutions of the incompressible Euler equations below the Onsager's threshold.[-]

We first summarize the derivation of viscoelastic (rate-type) fluids with stress diffusion that generates the models that are compatible with the second law of thermodynamics and where no approximation/reduction takes place. The approach is based on the concept of natural configuration that splits the total response between the current and initial configuration into the purely elastic and dissipative part. Then we restrict ourselves to the class of fluids where elastic response is purely spherical. For such class of fluids we then provide a mathematical theory that, in particular, includes the long-time and large-data existence of weak solution for suitable initial and boundary value problems. This is a joint work with Miroslav Bulicek, Vit Prusa and Endre Suli.[-]

The theory of mean field type control (or control of MacKean-Vlasov) aims at describing the behaviour of a large number of agents using a common feedback control and interacting through some mean field term. The solution to this type of control problem can be seen as a collaborative optimum. We will present the system of partial differential equations (PDE) arising in this setting: a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation. They describe respectively the evolution of the distribution of the agents' states and the evolution of the value function. Since it comes from a control problem, this PDE system differs in general from the one arising in mean field games.
Recently, this kind of model has been applied to crowd dynamics. More precisely, in this talk we will be interested in modeling congestion effects: the agents move but try to avoid very crowded regions. One way to take into account such effects is to let the cost of displacement increase in the regions where the density of agents is large. The cost may depend on the density in a non-local or in a local way. We will present one class of models for each case and study the associated PDE systems. The first one has classical solutions whereas the second one has weak solutions. Numerical results based on the Newton algorithm and the Augmented Lagrangian method will be presented.
This is joint work with Yves Achdou.[-]

In a joint work with Maria Colombo and Luigi De Rosa we consider the Cauchy problem for the ipodissipative Navier-Stokes equations, where the classical Laplacian $-\Delta$ is substited by a fractional Laplacian $(-\Delta)^\alpha$. Although a classical Hopf approach via a Galerkin approximation shows that there is enough compactness to construct global weak solutions satisfying the energy inequality à la Leray, we show that such solutions are not unique when $\alpha$ is small enough and the initial data are not regular. Our proof is a simple adapation of the methods introduced by Laszlo Székelyhidi and myself for the Euler equations. The methods apply for $\alpha < \frac{1}{2}$, but in order to show that they produce Leray solutions some more care is needed and in particular we must take smaller exponents.[-]

Weak solutions of the incompressible Euler equations which are weak limits of vanishing viscosity Navier-Stokes solutions inherit, in two dimensions, conservation properties which are not available for general weak solutions. Research has focused on the behavior of energy, enstrophy and, more generally, the distribution function of vorticity, always in fluid domains with no boundary, with and without forcing. In this talk I will report on recent work in this direction.[-]