## On stability of a regular vortex polygon in the circular domain.(English)Zbl 1097.76031

The paper deals with Kelvin problem for the case in which the vortex $$n$$-gon is located within a circular domain with the common center of symmetry. According to the well-known Lyapunov theorem, the equilibrium of a complete system is unstable when a linearized system is exponentially unstable. The power-law instability is insufficient to draw this conclusion; therefore, nonlinear terms should be involved in the analysis. The paper presents necessary and sufficient conditions for the stability and instability of a regular $$n$$-gon of point vortices located at the circle. For a vortex pentagon, the answer to the question concerning the instability remains unclear for the null set of a governing parameter $$p$$. The author also formulates two theorems concerning the stability in the Routh sense.

### MSC:

 76E30 Nonlinear effects in hydrodynamic stability 76B47 Vortex flows for incompressible inviscid fluids

### Keywords:

resonance; Lyapunov theorem; Routh stability
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