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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 ...

35R60 ; 35Q53 ; 35D30 ; 60H15

Post-edited  The Onsager Theorem
De Lellis, Camillo (Auteur de la Conférence) | CIRM (Editeur )

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

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 ...

76A10 ; 80A10 ; 35D30 ; 35Q35

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