These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probability measure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear that Lorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

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.[-]

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.[-]

In this joint work with Athanasios Tzavaras (KAUST) and Corrado Lattanzio (L'Aquila) we develop a relative entropy framework for Hamiltonian flows that in particular covers the Euler-Korteweg system, a well-known diffuse interface model for compressible multiphase flows. We put a particular emphasis on extending the relative entropy framework to the case of non-monotone pressure laws which make the energy functional non-convex.The relative entropy computation directly implies weak (entropic)-strong uniqueness, but we will also outline how it can be used in other contexts. Firstly, we describe how it can be used to rigorously show that in the large friction limit solutions of Euler-Korteweg converge to solutions of the Cahn-Hilliard equation. Secondly, we explain how the relative entropy can be used for obtaining a posteriori error estimates for numerical approximation schemes.[-]

These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probability measure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear that Lorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probabilitymeasure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear thatLorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

We consider the problem of lagrangian controllability for two models of fluids. The lagrangian controllability consists in the possibility of prescribing the motion of a set of particle from one place to another in a given time. The two models under view are the Euler equation for incompressible inviscid fluids, and the quasistatic Stokes equation for incompressible viscous fluids. These results were obtained in collaboration with Thierry Horsin (Conservatoire National des Arts et Métiers, Paris)[-]

It is well known since the pioneering work of Scheffer and Shnirelman that weak solutions of the incompressible Euler equations exhibit a wild behaviour, which is very different from that of classical solutions. Nevertheless, weak solutions in three space dimensions have been studied in connection with a long-standing conjecture of Lars Onsager from 1949 concerning anomalous dissipation and, more generally, because of their possible relevance to the K41 theory of turbulence.
In recent joint work with Camillo De Lellis we established a connection between the theory of weak solutions of the Euler equations and the Nash-Kuiper theorem on rough isometric immersions. Through this connection we interpret the wild behaviour of weak solutions of Euler as an instance of Gromov's h-principle.
In this lecture we explain this connection and outline recent progress towards Onsager's conjecture.[-]