Ovsyannikov vortex: exact solution of classical and relativistic hydrodynamic equations

Ovsyannikov vortex is a exact solution of continuum mechanics equations which is partially symmetric with respect to the group of rotations $SO(3)$ in the space $\mathbb R^3(\overrightarrow x)\times\mathbb R^3(\overrightarrow u)$. Physical interpretation of this solution is vortex gas or fluid flowing from spherical surface with transverse velocity component. This solution was discovered and first studied by Ovsyannikov [1]. Later, it was studied in a series of papers with additional symmetry for the equations of gas dynamics [2–4, 6], magnetic hydrodynamics [5], relativistic hydrodynamics [7]. The report provides an overview of previously obtained results and new concerning the mathematical properties of this solution and its physical interpretation.
REFERENCES
1. Ovsyannikov L. V. Special vortex. J. Appl. Mech. Tech. Phys. 1995. V. 36, N 3. P. 45–52.
2. Chupakhin A.P. Invariant submodels of a special vortex. J. Appl. Math. Mech. 2003. V. 67, N 3. P. 390–405.
3. Cherevko A.A., Chupakhin A.P. Stationary Ovsyannikov vortex. Preprint No1–05. Novosibirsk: Lavrentyev Institute of Hydrodynamics SB RAS 2005.
4. Pavlenko A.S. Projective submodel of the Ovsyannikov vortex. J. Appl. Mech. Tech. Phys. 2005. V. 46, N 4. P. 3–16.
5. Golovin S.V. Singular vortex in magnetohydrodynamics. J. Phys. A: Math. Gen. 2005. Vol. 38. P. 4501–4516.
6. Cherevko A.A., Chupakhin A.P. About automodel Ovsyannikov vortex. Proceedings of the Steklov Institute of Mathematics. 2012. V. 278. P. 276–287.
7. Chupakhin A.P., Yanchenko A.A. Special vortex in relativistic hydrodynamics. J. Phys.: Conference Series. 2017. Vol. 894, Art. 012114.