The Euler equation of incompressible flows \[ \left\{ \begin{aligned} \nabla \cdot \mathbf{v} &= 0\\ \mathbf{v}_t + \mathbf{v} \cdot \nabla \mathbf{v} &= -\nabla p, \end{aligned} \right. \] has in two dimensions singular solutions, where the vorticity is the sum of Dirac’s masses. The position of the vortices is governed by a Hamiltonian system. The solutions are solutions in the sense of distributions.

With Didier Smets, we have shown that some solutions could be approximated by stationnary classical solutions. We construct the velocity field as \(\nabla \psi^\perp\), where \[ \left\{ \begin{aligned} -\varepsilon^2 \Delta \psi & = \psi_+^p & & \text{dans \(\Omega\)},\\ \psi&=\psi_0-\frac{\kappa}{2\pi} \ln \frac{1}{\varepsilon} & & \text{sur \(\partial \Omega\)}. \end{aligned} \right. \] and \(\varepsilon \to 0\). We treat the case of a single stationnary vortex and a pair of stationnary vortices in a bounded domain and also the case of a pair of vortices in translation in \(\mathbb{R}^2\).

With S. De Valeriola, we have studied the corresponding problem for vortex rings in axisymmetric three-dimensional flows and for the shallow water equation. In that case, the velocity diverges as \(\log \varepsilon\) at the boundary.

In a word with Justin Dekyser, we have showed that vortices for the lake equations move asymptotically along the level lines for the depth (isobaths).