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Regularization of point vortices pairs for the Euler equation in dimension two. (English) Zbl 1293.35223
Summary: 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 \[ \begin{cases} -\varepsilon^2\Delta u=\sum\limits_{i=1}^m\chi_{\Omega_i^+}\biggl(u-q-\frac{\kappa_i^+}{2\pi}\ln\frac{1}{\varepsilon}\biggl)_+^p\\ -\sum_{j=1}^n\chi_{\Omega_j^-}\biggl(q-\frac{\kappa_j^-}{2\pi}\ln\frac{1}{\varepsilon}-u\biggl)_+^p,\qquad & x\in\Omega,\\ u=0,\quad & x\in\partial\Omega,\end{cases} \] where \(p>1\), \(\Omega\subset\mathbb R^2\) is a bounded domain, \(\Omega_i^+\) and \(\Omega_j^-\) are mutually disjoint subdomains of \(\Omega\) and \(\chi_{\Omega_i^+}\) (resp. \(\chi_{\Omega_j^-}\)) are characteristic functions of \(\Omega_i^+\) (resp. \(\Omega_j^-\)), \(q\) is a harmonic function. We show that if \(\Omega\) is a simply-connected smooth domain, then for any given \(C^1\)-stable critical point of Kirchhoff-Routh function \(\mathcal W(x_1^+,\dots, x_m^+,x_1^-,\dots,x_n^-)\) with \(\kappa^+_i> 0\) \((i=1,\dots,m)\) and \(\kappa^-_j>0\) \((j=1,\dots,n)\), there is a stationary classical solution approximating stationary \(m+n\) points vortex solution of incompressible Euler equations with total vorticity \(\sum_{i=1}^m\kappa^+_i-\sum_{j=1}^n\kappa_j^-\). The case that \(n=0\) can be dealt with in the same way as well by taking each \(\Omega_j^-\) as an empty set and set \(\chi_{\Omega_j^-}\equiv 0\), \(\kappa^-_j=0\).

35Q31 Euler equations
76B47 Vortex flows for incompressible inviscid fluids
35B65 Smoothness and regularity of solutions to PDEs
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