Approximate Lagrangian controllability for the 2-D Euler equation. Application to the control of the shape of vortex patches

AbstractIn this paper, we consider the two-dimensional Euler equation in a bounded domain Ω, with a boundary control located on an arbitrary part of the boundary. We prove that, given two Jordan curves which are homotopic in Ω and which surround the same area, given an initial data and a positive time T, one can find a control such that the corresponding solution drives the first curve inside Ω arbitrarily close to the second one (in any Ck norm) at time T. We also prove that given two vortex patches satisfying the same conditions on their contour, one can approximately deform the first one into the second one