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

机译:欧拉方程维数点涡的正则化   二,第二部分

摘要

In this paper, we continue to construct stationary classical solutions of theincompressible Euler equation approximating singular stationary solutions ofthis equation. This procedure now is carried out by constructing solutions to the followingelliptic problem \[ {cases} -\ep^2 \Deltau=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p-(q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep}-u)_+^p,\quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, {cases} \] 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 anygiven 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 $\kappa^-_j=-\kappa\,(j=1,...,n)$, thereis a stationary classical solution approximating stationary $m+n$ points vortexsolution of incompressible Euler equations with total vorticity $(m-n)\kappa$.
机译:在本文中,我们将继续构造不可压缩的Euler方程的平稳经典解,近似该方程的奇异平稳解。现在,通过构造以下椭圆问题的解决方案\ [{cases}-\ ep ^ 2 \ Deltau =(uq- \ frac {\ kappa} {2 \ pi} \ ln \ frac {1} {\ ep })_ + ^ p-(q- \ frac {\ kappa} {2 \ pi} \ ln \ frac {1} {\ ep} -u)_ + ^ p,\ quad&x \ in \ Omega,u = 0,\ quad&x \ in \ partial \ Omega,{cases \\]其中$ p> 1 $,$ \ Omega \ subset \ mathbb {R} ^ 2 $是有界域,$ q $是谐波功能。我们证明了,如果$ \ Omega $是一个简单连接的光滑域,那么对于任何给定的Kirchhoff-Routh函数的非退化临界点$ \ mathcal {W}(x_1 ^ +,...,x_m ^ +,x_1 ^ -,...,x_n ^-)$和$ \ kapp ^ + _ i = \ kappa> 0 \,(i = 1,...,m)$和$ \ kapp ^ -_ j =-\ kappa \, (j = 1,...,n)$,存在一个平稳的经典解,它近似不可压缩的Euler方程的平稳$ m + n $个点的涡旋解,总涡度为$(mn)\ kappa $。

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