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# Documents  35D30 | enregistrements trouvés : 9

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## Pathwise regularisation by noise in PDEs Gubinelli, Massimiliano | CIRM H

Post-edited

Research talks;Partial Differential Equations;Probability and Statistics

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) 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 ...

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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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## On the analysis of a class of thermodynamically compatible viscoelastic fluids with stress diffusion Málek, Josef | CIRM H

Multi angle

Research talks;Partial Differential Equations;Mathematical Physics

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. 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 ...

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## Mean field type control with congestion Laurière, Mathieu | CIRM H

Multi angle

Research talks;Control Theory and Optimization;Partial Differential Equations;Numerical Analysis and Scientific Computing

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.
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 Hamilto...

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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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## Séminaire Bourbaki. Volume 2016/2017:exposés 1120-1135 Bourbaki, Nicolas | Société Mathématique de France 2019

Ouvrage

- x; 584 p.
ISBN 978-2-85629-897-8

Astérisque , 0407

Localisation : Périodique 1er étage

problème de Cauchy en relativité générale # équations de contraintes relativistes # nœud # nœud trivial # entrelacs # mouvement de Reidemeister # surface normale # présentation par arcs # positivité au sens de Nakano # métrique de Bergman # conjecture d'ouverture # métriques de Kähler-Einstein # théorème de Bando-Mabuchi # rationalité # décomposition de Chow de la diagonale # mesures gaussiennes # corrélation positive # lois gamma multivariées # isomorphismes de graphes # isomorphismes de chaînes # configurations cohérentes # designs [sic] combinatoires # opérade # théorie des déformations # quantification par déformation # valeurs zêta multiples # le complexe de graphes # inégalités géométriques # isopérimétrie # courbure de Ricci # entropie # transport optimal # analyse non lisse # géométrie synthétique # D-modules holonomes # ind-faisceaux # cohomologie modérée # Équation de Yang-Baxter # groupes quantiques # opérateurs d'entrelacement # géométrie symplectique # cohomologie équivariante # enveloppes stables # algèbres affines quantiques # catégorification # algèbres amassées # systèmes intégrables quantiques # laplacien géométrique hypoelliptique # intégrales orbitales # formule de trace de Selberg # fonctions de zêta dynamiques # théorie de l'indice et théorèmes du point fixe # torsion analytique # équations hypoelliptiques # théorie de Hodge # structures de contact # h-principe # topologie symplectique # graphes expanseurs # variétés hyperboliques # nœuds # empilements de sphères # réseau E8 # réseau de Leech # formes quasi-modulaires # théorème de la valeur moyenne de Vinogradov # découplage # congruence efficace # restriction # Kakeya # méthode du cercle de Hardy-Littlewood # problème de Waring # fluides compressibles # viscosité dégénérée # construction de solutions faibles globales problème de Cauchy en relativité générale # équations de contraintes relativistes # nœud # nœud trivial # entrelacs # mouvement de Reidemeister # surface normale # présentation par arcs # positivité au sens de Nakano # métrique de Bergman # conjecture d'ouverture # métriques de Kähler-Einstein # théorème de Bando-Mabuchi # rationalité # décomposition de Chow de la diagonale # mesures gaussiennes # corrélation positive # lois gamma multivariées # ...

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## On the theory of weak turbulence for the nonlinear Schrödinger equation Escobedo, M. ; Velazquez, J. J. L. | American Mathematical Society 2015

Ouvrage

- v; 107 p.
ISBN 978-1-4704-1434-4

Memoirs of the american mathematical society , 0238

Localisation : Collection 1er étage

équation de Schrödinger # système non linéaire # mécanique ondulatoire # turbulence faible # explosion en temps fini # condensation # pulsation # transfert d'énergie

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## Degenerate complex Monge-Ampère equations Guedj, Vincent ; Zeriahi, Ahmed | European Mathematical Society 2017

Ouvrage

- xxiv; 472 p.
ISBN 978-3-03719-167-5

Tracts in mathematics , 0026

Localisation : Ouvrage RdC (GUED)

équation de Monge-Ampère # domaine d'holomorphie # théorie pluripotentielle # fonction sous-harmonique # opérateur complexe de Monge-Ampère # capacité généralisée # solution faible # estimation a priori # métrique canonique de Kähler # variété singulière

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## Elliptic PDEs, measures and capacities:from the Poisson equation to nonlinear Thomas-Fermi problems Ponce, Augusto C. | European Mathematical Society 2016

Ouvrage

- x; 453 p.
ISBN 978-3-03719-140-8

Tracts in mathematics , 0023

Localisation : Ouvrage RdC (PONC)

équation différentielle elliptique # théorie de la mesure géométrique # méthode du balayage # mesure de Borel # équation de Chern-Simons # potentiel continu # mesure diffuse # problème de Dirichlet # équation de Euler-Lagrange # solution extrême # inégalité fractionnelle de Sobolev # lemme de Frostman # mesure de Hausdorff # inégalité de Kato # Laplacien # ensemble de Lebesgue # espace de Marcinkiewicz # principe maximum # problème de minimisation # intégration de Morrey # problème d'obstacle # méthode de Perron # équation de Poisson # théorie du potentiel # représentant précis # mesure réduite # théorie de la régularité # théorème de représentation de Riesz # opérateur de Schrödinger # équation semi-linéaire # capacité de Sobolev # sous-harmonique # super-harmonique # équation de Thomas-Fermi # lemme de Weyl équation différentielle elliptique # théorie de la mesure géométrique # méthode du balayage # mesure de Borel # équation de Chern-Simons # potentiel continu # mesure diffuse # problème de Dirichlet # équation de Euler-Lagrange # solution extrême # inégalité fractionnelle de Sobolev # lemme de Frostman # mesure de Hausdorff # inégalité de Kato # Laplacien # ensemble de Lebesgue # espace de Marcinkiewicz # principe maximum # problème de m...

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