

International scientific conference "Days of Classical Mechanics"
January 26, 2015 10:05–10:40, Moscow, Steklov Mathematical Institute of RAS, Gubkina, 8






KAM theory and the 3d Euler equation
S. B. Kuksin^{} 
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Abstract:
Consider the Euler equation on a closed 3d manifold (e.g. on the 3torus or on the 3sphere). Let $u(t,x)$ be its solution, and $\omega(t,x)$ – the vorticity of this solution. By the Kelvin theorem the vorticityvectorfields, calculated for different values of time, are conjugated by means of volumepreserving diffeomorphisms. Therefore the quantity $\kappa(\omega)$, equal to the volume of the set which is the union of all twodimensional invariant tori of $\omega$ is timeindependent. We use KAM and the Arnold theorem on the structure of steadystates of the 3d Euler to prove that $\kappa$ is continuous at $\omega$'s which are nondegenerate steadystates of the equation, and use this integral of motion to study qualitative properties of the dynamical system, defined by the equation in the space of sufficiently smooth vectorfields. Namely, to study its nonergodicity and study the problem “does the manifold of steadystates of the equation attracts (in a suitable sense) all trajectories which start from its vicinity?”.
This is a joint work with B. Khesin and D. PeraltaSalas.

