Regularization of point vortices for the Euler equation in dimension two

Abstract

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 Δu=(u-q-\fracκ{2π}\ln\frac{1}{\ep})_+^p, \quad & x\inΩ, u=0, \quad & x\in\partialΩ, ] where $p>1$, $\Omega \subset {\mathbb{R}}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $\kappa >0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $m\kappa$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).

32pages

References in corpus (1)

• Desingularization of vortices for the Euler equation

Cited by in corpus (5)

• Gluing methods for vortex dynamics in Euler flows
• Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem
• Periodic solutions of the N-vortex Hamiltonian system in planar domains
• Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type
• The N-vortex Problem on a Riemann Sphere