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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 ...
35Q31 ; 35D30 ; 76B03
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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 ...
35Q31 ; 35A01 ; 35D30
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- v; 79 p.
ISBN 978-0-8218-4914-9
Memoirs of the american mathematical society , 0991
Localisation : Collection 1er étage
fonction Q-valuée # énergie de Dirichlet # existence et régularité # espace métrique # application harmonique # théorie de la mesure géométrique
49Q20 ; 35J55 ; 54E40 ; 53A10
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- vi; 124 p.
ISBN 978-3-03719-044-9
Zurich lectures in advanced mathematics
Localisation : Ouvrage RdC (DELE)
théorie de la mesure # mesure géométrique # fonction à plusieurs variables # intégration # problème variationnel # théorème de Mastrand # critère de rectifiabilité
28A75 ; 26B15 ; 49Q15 ; 49Q20
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