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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 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 ...
35Q31 ; 35D30 ; 76B03

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We investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system in a regime studied by F. Golse and L. Saint-Raymond. First we establish the convergence towards the Euler equation under several assumptions on the energy and on the norms of the initial data. Then we provide a first analysis of the asymptotics for a Vlasov-Poisson system describing the interaction of a bounded density with a moving point charge.
82D10 ; 82C40 ; 35Q35 ; 35Q83 ; 35Q31

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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)
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 ...
35Q93 ; 35Q31 ; 76D55 ; 93B05

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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 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 ...
35Q31 ; 35A01 ; 35D30

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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.
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 ...
35Q31 ; 76B03