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Uniqueness of the Leray-Hopf solution for a dyadic model

Presented by: 
N Filonov [Russian Academy of Sciences St Peterburg]
Wednesday 3rd June 2015 - 15:10 to 16:10
INI Seminar Room 2
We consider the system of nonlinear differential equations

\label{1} \begin{cases} \dot u_n(t) + \la^{2n} u_n(t) - \la^{\be n} u_{n-1}(t)^2 + \la^{\be(n+1)} u_n(t) u_{n+1}(t) = 0,\\ u_n(0) = a_n, n \in \mathbb{N}, \quad \la > 1, \be > 0.

In this talk we explain why this system is a model for the Navier-Stokes equations of hydrodynamics. The natural question is to find a such functional space, where one could prove the existence and the uniqueness of solution. In 2008, A.~Cheskidov proved that the system (0.1) has a unique "strong" solution if $\be \le 2$, whereas the "strong" solution does not exist if $\be > 3$.

(Note, that the 3D-Navier-Stokes equations correspond to the value $\be = 5/2$.)

We show that for sufficiently "good" initial data the system (0.1)has a unique Leray-Hopf solution for all $\be > 0$.

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons