Lecture 1: Kolmogorov and Onsager's picture of 3D turbulence 

We will discuss some foundational aspects of three-dimensional incompressible fluid turbulence, including guiding experimental observations, Kolmogorov's 1941 theory on the structure of a turbulent flow, Onsager's 1949 conjecture on anomalous dissipation and weak Euler solutions, and Landau's Kazan remark concerning intermittency. Mathematical examples and constructions that exhibit features of turbulent behavior will be discussed. 

Lecture 2:  Transition to turbulence in two-dimensions 

We will discuss the formation of small and large scales in two-dimensional fluids (both viscous and inviscid). In the inviscid setting, we will discuss the mixing process which creates infinitely fine scales of motion at long times and serves as the dynamical mechanism for the direct enstrophy cascade. For viscous fluids, we will discuss the stability and instability of Kolmogorov flow on two-dimensional flat tori (Meshalkin-Sinai) and a related example of non-uniqueness of smooth steady states of the Navier-Stokes equations (Yudovich). Destabilization of this laminar regime relates to the transition to turbulence.

Lecture 3: Irreversibility of 2D perfect fluids near equilibrium 

We will discuss aspects of the long-term dynamics of 2d perfect fluids.  As an application of a certain stability of twisting for general Hamiltonian flows, we will show generic loss of smoothness near stable steady states, the existence of many wandering points, aging of the Lagrangian flow, along with other examples of complex behavior such as indefinite perimeter growth for special vortex patches.

The Vlasov-Poisson equations provide a statistical description of the classical N-body problem as $N\to\infty$: a distribution function on phase space is transported in a self-generated, Poisson potential field. In the physically relevant setting of phase space $\R^3\times\R^3$, the well-posedness theory for this model is relatively well understood, but a precise description of the long-time behavior of solutions remains elusive. In this minicourse we elaborate on the question of asymptotic dynamics in two related settings: near vacuum and near a point charge.

Lecture 1: Introduction and background, review of well-posedness theory. Dynamics near vacuum: preliminaries.

Lecture 2: Asymptotic stability of vacuum. Existence of wave operators and a scattering map.

Lecture 3: Dynamics with a point charge. Stability in the repulsive setting.

The Euler equations with a maximal density constraint (hard congestion model) can be approximated by the system of gas dynamics with a singular pressure law (soft congestion model). In this talk, I will present a mathematical justification of this singular limit in the setting of BV solutions, which is appropriate at least for two reasons: classical solutions break down in finite time and the analysis of the interface dynamics (even for the limit solution) is naturally performed within a Wave-Front Tracking algorithm. Our initial data are small BV perturbations of reference solutions, which are (possibly interacting) large shock waves, playing the role of the free/congested interface (in fact, this is a free boundary problem). The method is based on the use of weighted Glimm functionals and on the introduction of appropriate rescaling of the singular pressure. This is a work in collaboration with F. Ancona (Univ. of Padova) and C. Perrin (CNRS, Aix Marseille Univ.).

In this talk, we consider the uniqueness problem for Leray solutions of the three-dimensional Navier-Stokes equations.

In collaboration with D. Albritton and M. Colombo, we recently discovered non-unique Leray solutions in $R^3$ driven by a self-similar body force.

After a short overview of this construction, we shift our focus to the uniqueness question in bounded and periodic domains.

Specifically, we show that our non-unique self-similar solutions can be "glued" in both bounded and periodic domains. This result presents the first instance of non-uniqueness for Leray solutions in a bounded domain, showcasing the intrinsic locality and robustness of the self-similar non-uniqueness.

This talk will be devoted to the analysis of the Stokes-transport system in the domain T x (0,1), with no-slip conditions for the velocity on the boundaries of the domain. We prove that linearly stratified density profiles are orbitally stable: small perturbations (in H^6) of such profiles remain small (in H^4) for all times, and the total density converges for long times towards a rearrangement of the initial density. We also prove convergence rates in H^s for s<4. Eventually, we show that boundary layers are formed in the vicinity of z=0 and z=1 for large times, and limit the rate of decay of the solution. Decomposing the solution as the sum of a boundary layer and an interior part, we prove that the interior part enjoys a higher decay. This is a joint work with Julien Guillod and Antoine Leblond.

We will discuss mechanisms for singularity formation in incompressible fluid equations in finite and infinite time.

In this talk, we will present two kinetic models that are used to describe the evolution of charged particles in plasmas: the Vlasov-Poisson system and the Vlasov-Poisson system with massless electrons. These systems model respectively the evolution of electrons, and ions in a plasma. We will discuss the well-posedness of these systems, the stability of solutions, and their behavior under singular limits. Finally, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations.

Abstract

In the 1960s, Lynden-Bell, studying the dynamics of galaxies around steady states of the gravitational Vlasov-Poisson equation, described a phenomenon he called "violent relaxation," a convergence to equilibrium through phase mixing analogous in some respects to Landau damping in plasma physics. In this talk, I will discuss recent work on this gravitational Landau damping for the linearised Vlasov-Poisson equation and, in particular, the critical role of regularity of the steady states in distinguishing damping from oscillatory behaviour in the perturbations. This is based on joint work with Mahir Hadzic, Gerhard Rein, and Christopher Straub.

In this talk, I will discuss a joint work with Scott Armstrong in which we construct a class of incompressible vector fields that have many of the properties observed in a fully turbulent velocity field, and for which the associated scalar advection-diffusion equation generically displays anomalous diffusion. We also propose an analytical framework in which to study anomalous diffusion, via a backward cascade of renormalized eddy viscosities. Our proof is by "fractal" homogenization, that is, we perform a cascade of homogenizations across arbitrarily many length scales.

The Keller-Segel system is a model for chemotactic aggregation and is a nonlocal non-linear reaction diffusion equation. I will discuss existence of solutions with type II finite time blow-up in the 3-dimensional, axially-symmetric case. An intermediate step is a construction by gluing techniques of finite time blow-up solutions in the 2-dimensional case. This is a joint work with Juan Dávila, Manuel del Pino and Monica Musso.

We investigate the long-time asymptotics of a coagulation model for a system of interacting particles characterized by their volume and surface area. We describe the coagulation process as a combination between collision and fusion of particles. The solutions of the system can exhibit different behaviors depending on the chosen fusion rate. We prove existence of self-similar profiles when the fusion rates are selected such that the particles maintain a shape that does not differ much from spherical. On the other hand, for other fusion mechanisms and some suitable initial data, we show that the particle distribution describes a system of ramified-like particles. Lastly, I will talk about how we are able to recover the standard coagulation equation in the case of fast fusion. This is joint work with J. J. L. Velázquez.

We obtain a new a priori regularity result for the isentropic Euler equations with gamma = 3. The improvement in regularity comes in the form of traces allowing us to rule out the existence of certain pathologies resembling the Gibbs phenomenon. Consequently, we expand the known uniqueness class for the equation. The main result relies upon a new epsilon regularity criterion for the system, which is shown by a combination of dispersive and elliptic techniques. 

I will discuss joint work with Nahmod, Pavlović, Rosenzweig, and Staffilani on the rigorous derivation of the Hamiltonian structure of the Vlasov equation (see arxiv.org/abs/2206.07589). It is known that the Vlasov equation may be rigorously derived as the effective dynamics of a large system of classical particles via BBGKY or empirical measures. In our work, we rigorously show how the Lie-Poisson structures of the various finite and infinite particle systems in addition to the Hamiltonian structure of the Vlasov equation are related via Poisson morphisms and mean-field limits, proving what we describe as a "geometric mean-field limit".

We study a class of second order strongly degenerate kinetic operators in the framework of special relativity. More precisely, the operator we consider here is a possible suitable relativistic generalization of the kinetic Fokker-Planck operator. We first describe it as a Hörmander operator which is invariant with respect to Lorentz transformations. We then prove a Lorentz-invariant Harnack type inequality, and we derive sharp asymptotic lower bounds for positive solutions to the equation. As a consequence, we obtain lower bounds for the density of the relativistic stochastic process associated to our operator.

In this talk, we present recent results on anomalous dissipation, mathematical definition inspired by Kolmogorov's zeroth law of turbulence, and on the non-selection of solutions via vanishing viscosity, i.e. sending the viscosity parameter $\nu \to 0$.

More precisely, for every $\varepsilon > 0$, we provide  smooth solutions to the forced Navier-Stokes equations, which have the uniform regularity $\sup_{\nu > 0} \| u_\nu \|_{L^3_t C^{1/3 - \varepsilon}_x \cap L^\infty_t L^2_x} < \infty$ and exhibit anomalous dissipation.

Within the same framework, we also present the lack of selection among solutions to the forced 3D Euler equations via vanishing viscosity, using body forces with suitable regularity that depend on $\nu > 0$. These results are based on the investigation of $2 + \frac{1}{2}$-dimensional NS equations, and we delve into the proof of similar results in the simplified model of the advection-diffusion equation.

These are joint works with E. Bruè, M. Colombo, G. Crippa, and C. De Lellis.