Stable plane Euler flows with concentrated and sign-changing vorticity
Abstract
We construct a family of steady solutions to the two-dimensional incompressible Euler equation in a general bounded domain, such that the vorticity is supported in two well-separated regions of small diameter and converges to a pair of point vortices with opposite signs. Compared with previous results, we do not need to assume the existence of an isolated local minimum point of the Kirchhoff-Routh function. Moreover, due to their variational nature, the solutions obtained are Lyapunov stable in Lp norm of the vorticity. The proofs are achieved by maximizing the kinetic energy over an appropriate family of rearrangement classes of sign-changing functions and studying the limiting behavior of the maximizers.
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