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# Documents  35Q31 | enregistrements trouvés : 13

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## The Onsager Theorem De Lellis, Camillo | CIRM H

Post-edited

Research talks;Partial Differential Equations;Mathematical Physics

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 ...

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## Recent progress in the theory of the Euler and Navier-Stokes equations.Based on the workshop, 'The Navier-Stokes equations in Venice'Venice # April 8-12, 2013 Robinson, James C. ; Rodrigo, José L. ; Sadowski, Witold ; Vidal-Lopez, Alejandro | Cambridge University Press 2016

Congrès

- xiii; 232 p.
ISBN 978-1-107-55497-9

London mathematical society lecture note series , 0430

Localisation : Collection 1er étage

équation d'Euler # équation de Navier-Stokes

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## Relative entropy for the Euler-Korteweg system with non-monotone pressure Giesselmann, Jan | CIRM H

Multi angle

Research talks

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. 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 ...

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## Multiple traveling waves of the Euler-Korteweg system Audiard, Corentin | CIRM H

Multi angle

Research talks

The Euler-Korteweg system corresponds to compressible, inviscid fluids with capillary forces. It can be used to model diffuse interfaces. Mathematically it reads as the Euler equations with a third order dispersive perturbation corresponding to the capillary tensor.

In dimension one there exists traveling waves with equal or different limit at infinity, respectively solitons and kinks. Their stability is ruled by a simple criterion a la Grillakis-Shatah-Strauss. This talk is devoted to the construction of multiple traveling waves, namely global solutions that converge as $t\rightarrow \infty$ to a profile made of several (stable) traveling waves. The waves constructed have both solitons and kinks. Multiple traveling waves play a peculiar role in the dynamics of dispersive equations, as they correspond to solutions that follow in some sense a purely nonlinear evolution.
The Euler-Korteweg system corresponds to compressible, inviscid fluids with capillary forces. It can be used to model diffuse interfaces. Mathematically it reads as the Euler equations with a third order dispersive perturbation corresponding to the capillary tensor.

In dimension one there exists traveling waves with equal or different limit at infinity, respectively solitons and kinks. Their stability is ruled by a simple criterion a la ...

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## Ill-posedness for Leray solutions of the ipodissipative Navier-Stokes equations De Lellis, Camillo | CIRM H

Multi angle

Research talks;Partial Differential Equations

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 ...

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## An asymptotic regime for the Vlasov-Poisson system Miot, Evelyne | CIRM H

Multi angle

Research talks;Partial Differential Equations;Mathematical Physics

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.

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## The H-Principle and Turbulence Székelyhidi, László | CIRM H

Multi angle

Research talks;Partial Differential Equations;Mathematical Physics

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 ...

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## Control of the motion of a set of particles Glass, Olivier | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations

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 ...

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## Stochastic solutions of 2D fluids​ Flandoli, Franco | CIRM H

Multi angle

Research talks;Partial Differential Equations;Probability and Statistics

We revise recent contributions to 2D Euler and Navier-Stokes equations with and without noise, but always in the case of stochastic solutions. The role of white noise initial conditions will be stressed and related to some questions about turbulence.

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## Global regularity for 2D water waves with surface tension Ionescu, Alexandru D. ; Pusateri, Fabio | American Mathematical Society 2018

Ouvrage

- v; 123 p.
ISBN 978-1-4704-3103-7

Memoirs of the American Mathematical Society , 1227

Localisation : Collection 1er étage

vague irrotationnelle # tension superficielle # estimation d'énergie # analyse dispersive # singularité # résonance temporelle # opérateur de Dirichlet-Neumann en dimension 2

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## Analysis of hydrodynamic models Constantin, Peter | Society For Industrial And Applied Mathematics 2017

Ouvrage

- ix; 63 p.
ISBN 978-1-611974-79-9

CBMS-NSF regional conference series in applied mathematics , 0090

Localisation : Collection 1er étage

équation différentielle partielle # hydrodynamique # équation incompressible d'Euler # approximation de Boussinesq # milieu poreux incompressible # modèle d'Oldroyd-B # existence locale # unicité # équation scalaire

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## Shock formation in small-data solutions to 3D quasilinear wave equations Speck, Jared | American Mathematical Society 2016

Ouvrage

- xxiii; 515 p.
ISBN 978-1-4704-2857-0

Mathematical surveys and monographs , 0214

Localisation : Collection 1er étage

équation d'onde # onde de choc # équation différentielle non linéaire # quasilinéarisation

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## Séminaire Bourbaki. Vol. 2014/2015: exposés 1089-1103 | Société Mathématique de France 2016

Ouvrage

- x; 497 p.
ISBN 978-2-85629-836-7

Astérisque , 0380

Localisation : Périodique 1er étage

combinatoire # théorie des catégories # théorie des topos supérieurs # théorie de la mesure géométrique # équation aux dérivées partielles # théorie spectrale # géométrie différentielle # théorie ergodique # théorie géométrique des groupes # géométrie algébrique # représentation galoisienne # point rationnel

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