Consider the 2D Euler incompressible equations

\begin{equation} \label{euler} \partial_t \vec{u} + (\vec{u} \cdot\nabla) \vec{u} = -\nabla p, \quad \nabla \cdot \vec{u} = 0, \end{equation}

where \(\vec{u} = (u_1, u_2)^T\) is the velocity field and \(p\) is the pressure. We study the fluid in the domain \(\Omega=\mathbb{T}_{2\pi}\times \mathbb{R}.\) By the incompressible condition \(\nabla \cdot \vec{u} = 0\), we can define the stream function \(\psi\), whose level curves are the stream lines of the fluid, by \(\vec{u} = \nabla ^\bot \psi = (\psi_y, -\psi_x)\).

\eqref{euler} is a great equation!