Extremal energies of Laplacian operator: Different configurations for steady vortices

TL;DR

This paper investigates optimal configurations for steady vortices in fluid flows by solving extremal problems related to the Laplacian operator, providing analytical solutions in low contrast regimes and algorithms for high contrast regimes.

Contribution

It analytically determines unique optimal solutions in low contrast regimes and develops algorithms for high contrast regimes, including convergence proofs and numerical conjectures.

Findings

Analytical solutions are derived for low contrast regimes.

Algorithms are developed for high contrast regimes with proven convergence.

Numerical experiments suggest conjectures about maximizer locations.

Abstract

In this paper, we study a maximization and a minimization problem associated with a Poisson boundary value problem. Optimal solutions in a set of rearrangements of a given function define stationary and stable flows of an ideal fluid in two dimensions. The main contribution of this paper is to determine the optimal solutions. At first, we derive the solutions analytically when the problems are in low contrast regime. Moreover, it is established that the solutions of both problems are unique. Secondly, for the high contrast regime, two optimization algorithms are developed. For the minimization problem, we prove that our algorithm converges to the global minimizer regardless of the initializer. The maximization algorithm is capable of deriving all local maximizers including the global one. Numerical experiments leads us to a conjecture about the location of the maximizers in the set of rearrangements of a function.