Regularization of point vortices for the Euler equation in dimension two, part II

Abstract

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

This paper is the continuation of the paper (arXiv:1208.3002v2), 35 pages

Cited by in corpus (5)

• 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
• Multiple Vortices for the Shallow Water Equation
• Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations