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Local uniqueness for vortex patch problem in incompressible planar steady flow. (English. French summary) Zbl 1436.35153
The authors consider the steady planar flow of an ideal fluid in a bounded region \(\Omega \) of \(\mathbb{R}^{2}\) and especially the flow whose vorticity \(\omega \) is a constant \(\lambda \) in a region \(\Omega _{\lambda }\) which has \(k\) simply connected components \(\Omega _{\lambda ,j}\) and \(d(x,x_{0,j})\rightarrow 0\) uniformly for all \(x\in \Omega _{\lambda ,j}\) as \(\lambda \rightarrow +\infty \) for some points \(x_{0,j}\in \overline{\Omega }\), \(j=1,\ldots ,k\), while \(\omega =0\) elsewhere. The authors prove that the stream function \(\psi \) of the flow is a solution to the problem \(-\Delta \psi =\lambda \sum_{j=1}^{k}1_{\Omega _{\lambda ,j}}\) in \(\Omega \), with the boundary condition \(\psi =0\) on \(\partial \Omega \). Here the vorticity set \(\Omega _{\lambda ,j}\) is unknown. Observing that \(\Omega _{\lambda ,j}=B_{\delta }(x_{0,j})\cap \{\psi >\widetilde{k}_{i,j}\}\), where \(x_{0,j}\) is the point such that \(d(x,x_{0,j})\rightarrow 0\) for all \(x\in \Omega_{\lambda ,j}\) as \(\lambda \rightarrow +\infty \), and \(\delta >0\) is a fixed small constant, the authors end with the elliptic problem \(-\Delta \psi =\lambda \sum_{j=1}^{k}1_{B_{\delta }(x_{0,j})}1_{\{\psi >\widetilde{k}_{i,j}\}}\) for some large \(\widetilde{k}_{i,j}\) satisfying the following prescribed vortex strength condition \(\lambda \left\vert \Omega _{\lambda,j}\right\vert =\kappa _{j}>0\). The purpose of the paper is to determine necessary conditions on the location of \(x_{0,j}\) such that the last elliptic problem is solvable, and the uniqueness of solutions to this elliptic problem which satisfy the vortex strength condition when it is solvable. From the Green function \(G\) for \(-\Delta \) in \(\Omega \) with zero boundary condition, the authors deduce the Robin function \(\varphi (x)=H(x,x) \), where \(H\) is the regular part of \(G\) and the Kirchhoff-Routh function \(\mathcal{W}(x_{1},\ldots ,x_{k})=-\sum_{i\neq j}^{k}\kappa _{i}\kappa_{j}G(x_{i},x_{j})+\sum_{i=1}^{k}\kappa _{i}^{2}\varphi (x_{i})\), for every integer \(k>0\). The authors first prove that for \(k\) given positive numbers \(\kappa _{j}\), \(j=1,\ldots ,k\), if \(\psi _{\lambda }\) is a solution of the elliptic problem which satisfies the vortex strength condition such that each component of vorticity set \(\Omega _{\lambda ,j}\), \(j=1,\ldots ,k\), shrinks to \(x_{0,j}\in \overline{\Omega }\), as \(\lambda \rightarrow +\infty\), then \(x_{0,j}\in \Omega \), \(j=1,\ldots ,k\), \(x_{0,j}=x_{0,i}\) for \(j=i\) and \(x_{0}=(x_{0,1},\ldots ,x_{0,k})\) is a critical point of \(\mathcal{W}\). The first main result proves that if \(x_{0}\in \Omega ^{k}\) is an isolated critical point of \(\mathcal{W}(x)\), which is non-degenerate, then for large \(\lambda >0\), the elliptic problem together with the vortex strength condition have a unique solution. The second main result proves that if \(\Omega \) is convex, the vortex patch problem with prescribed vorticity strength has a unique solution and \(k=1\) if \(\lambda >0\) is large. For the proofs, the authors mainly apply a blow-up procedure using the diameter \(D_{\lambda ,j}\) of the unknown set \(\Omega _{\lambda ,j}\) and they use Pohozaev’s identity and Harnack’s inequality.

35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35Q05 Euler-Poisson-Darboux equations
Full Text: DOI
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