The vortex method is a common numerical and theoretical approach used to implement the motion of an ideal flow, in which the vorticity is approximated by a sum of point vortices, so that the Euler equations read as a system of ordinary differential equations. Such a method is well justified in the full plane, thanks to the explicit representation formulas of Biot and Savart. In an exterior domain, we also replace the impermeable boundary by a collection of point vortices generating the circulation around the obstacle. The density of these point vortices is chosen in order that the flow remains tangent at midpoints between adjacent vortices and that the total vorticity around the obstacle is conserved.
In this work, we provide a rigorous justification of this method for any smooth exterior domain, one of the main mathematical difficulties being that the Biot--Savart kernel defines a singular integral operator when restricted to a curve (here, the boundary of the domain). We also introduce an alternative method---the fluid charge method---which, as we argue, is better conditioned and therefore leads to significant numerical improvements.
Uniqueness for two dimensional incompressible ideal flow on singular domains
在[Gérard-Varet and Lacave，The 2D Euler equation on singular domains，submitted]中建立了奇异域中二维不可压缩Euler方程解的存在性。目前的工作是关于这样的解决方案的唯一性时，域是外部或内部的单连通集的角，虽然速度爆炸附近的这些角落。在具有两个端点的曲线的外部，在[Lacave，Two Dimensional Incompressor Ideal Flow Around a Thin Obstacle Tending to a Curve，Ann. IHP，Anl \textbf{26}（2009），1121-1148]中示出，该解具有一些有趣的性质，如被视为特殊涡面。因此，我们证明了唯一性，而一般涡面的问题是开放的。
Vortex-sheet desingularization for three-dimensional ideal fluids
Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究委员会数学科学研究所）；Departamento de Matemáticas, Universidad Autónoma de Madrid（马德里自治大学）；Max Planck Institute for Mathematics in the Sciences（马普科学数学研究所）
Two Dimensional Incompressible Ideal Flow Around a Small Curve
本文研究了二维不可压缩Euler方程在曲线外收缩到一点时解的渐近性态。这项工作联系了两个先前的结果：[Iftimie，Lopes Filho和Nussenzveig Lopes，Two Dimensional Incompressor Ideal Flow Around a Small Obstacle，Comm. PDE，28（2003），349-379]和[Lacave，Two Dimensional Incompressor Ideal Flow Around a Thin Obstacle Tending to a Curve，Ann. IHP，Anl，26（2009），1121-1148]。这项工作的第二个目标是完成上一篇文章，定义障碍物收缩为曲线的方式。特别地，我们给出了区域收敛的几何性质，以使极限流是欧拉方程的解。
Uniformly Rotating Vortex Patches with Arbitrarily Many Genuine Holes
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, Xiamen University（厦门大学数学科学学院）
