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Steklov Mathematical Institute Seminar
May 21, 2015 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

On the Kelvin's 1880 problem and exact solutions of the Navier-Stokes equations

O. I. Bogoyavlenskij
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O. I. Bogoyavlenskij
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Abstract: Exact solutions to the steady equations of hydrodynamics are derived for that a classification of knots formed by the closed vorticity lines is obtained (Kelvin's 1880 problem).
Using the Alexander polynomial (that is a topological invariant of any knot in $\mathbb{R}^3$) it is shown which vortex torus knots are realized for the constructed exact solutions and which ones are not realized by the closed vorticity lines.
Exact solutions to the Navier-Stokes equations are obtained describing dynamics of a viscous incompressible fluid in $\mathbb{R}^3$. The presented solutions depend on an arbitrary vector field tangent to the 2-dimensional sphere $\mathbb{S}^2 \subset \mathbb{R}^3$ and on an arbitrary measure on the sphere $\mathbb{S}^2$. It is shown that dynamics of fluid for these solutions is not turbulent either in the Eulerian or in Lagrangian senses in spite of the corresponding Reynolds numbers can be arbitrarily large.

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