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Uniqueness of the Leray-Hopf solution for a dyadic model

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Periodic and Ergodic Spectral Problems

We consider the system of nonlinear differential equations

label{1} egin{cases} dot u_n(t) + la u_n(t) - la{e n} u_{n-1}(t)2 + la{e(n+1)} u_n(t) u_{n+1}(t) = 0,\ u_n(0) = a_n, n in mathbb{N}, quad la > 1, e > 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 $e le 2$, whereas the “strong” solution does not exist if $e > 3$.

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

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

This talk is part of the Isaac Newton Institute Seminar Series series.

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