Global in Time Vortex Configurations for the $2$D Euler Equations

Abstract

We consider the problem of finding a solution to the incompressible Euler equations $$\ddot{I}{‰}_t+v\cdot \nabla \ddot{I}‰=0in{\mathbb{R}}^2\times (0,\infty ),v(x,t)=\frac{1}{2\ddot{I}€}{\int }_{{\mathbb{R}}^2}\frac{(y-x{)}^{\perp }}{|y-x{|}^2}\ddot{I}‰(y,t)dy$$ that is close to a superposition of traveling vortices as $t\to \infty$. We employ a constructive approach by gluing classical traveling waves: two vortex-antivortex pairs traveling at main order with constant speed in opposite directions. More precisely, we find an initial condition that leads to a 4-vortex solution of the form $$\ddot{I}‰(x,t)=\ddot{I}{‰}_0(x-cte)-\ddot{I}{‰}_0(x+cte)+o(1)ast\to \infty$$ where $$\ddot{I}{‰}_0(x)=\frac{1}{{\epsilon }^2}W\left(\frac{x-q}{\epsilon }\right)-\frac{1}{{\epsilon }^2}W\left(\frac{x+q}{\epsilon }\right)+o(1)as\epsilon \to 0$$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside.

Accepted in the Journal of the European Mathematical Society