Year: 2012
Type: preprint
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 $p>1$, $Ω\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $Ω$ 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 $κ^+_i=κ>0\,(i=1,...,m)$ and $κ^-_j=-κ\,(j=1,...,n)$, there is a stationary classical solution approximating stationary $m+n$ points vortex solution of incompressible Euler equations with total vorticity $(m-n)κ$.
Source: arXiv (Cornell University)
Authors: Cao, Daomin, Liu, Zhongyuan, Wei, Juncheng
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2342965283