Vikas Krishnamurthy

Liouville Chain Solutions of the Euler Equation

We describe a large class of solutions of the two-dimensional steady incompressible Euler equation in which point vortices are embedded in a non-constant background vorticity field such that the whole arrangement is stationary. This background vorticity is proportional to the exponential of the stream function and leads to a Liouville-type partial differential equation. We exploit the known solution structure of a class of solutions of this Liouville-type equation to construct solutions, which we call Liouville chains, of the Euler equation mentioned above. Liouville chains can be constructed as iterated solutions starting from a simple purely point vortex equilibrium (without a background field). Each iteration forms one link in the chain and can terminate after one step, a finite number of steps, or go on indefinitely. The solutions are given in terms of an arbitrary positive real parameter, and as this parameter goes to zero or infinity, we find that the background vorticity concentrates into point vortices, and in the limit the solutions go over into purely point vortex equilibria. The solutions also contain arbitrary complex-valued parameters which arise as integration constants in the iteration. We describe several examples illustrating all these properties.