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

We consider the problem of finding a solution to the incompressible Euler equations $${\omega }_t+v\cdot \mathrm{\nabla }\omega =0\text{ in }{\mathbb{R}}^2\times (0,\mathrm{\infty }),v(x,t)=\frac{1}{2\pi }{\int }_{{\mathbb{R}}^2}\frac{(y-x{)}^{\perp }}{\mathrm{\mid }y-x{\mathrm{\mid }}^2}\omega (y,t)dy$$ that is close to a superposition of traveling vortices as $t\to \mathrm{\infty }$. 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 $$\omega (x,t)={\omega }_0(x-cte)-{\omega }_0(x+cte)+o(1)\text{ as }t\to \mathrm{\infty }$$ where $${\omega }_0(x)=\frac{1}{{\epsilon }^2}W\left(\frac{x-q}{\epsilon }\right)-\frac{1}{{\epsilon }^2}W\left(\frac{x+q}{\epsilon }\right)+o(1)\text{ as }\epsilon \to 0$$ and $W(y)$ is a certain fixed smooth profile, radially symmetric, positive in the unit disc zero outside.