Famous PDEs, Euler's equations

Similarly, famous systems of non-linear partial differential equations are for example Euler's equations for incompressible, inviscid flow $$ \begin{equation*} \frac{\partial \mathbf{u}}{\partial t} +\mathbf{u}\nabla\mathbf{u}= -Dp; \hspace{1cm} \mathrm{div} \mathbf{u} = 0, \end{equation*} $$ with \( p \) being the pressure and $$ \begin{equation*} \nabla = \frac{\partial}{\partial x}e_x+\frac{\partial}{\partial y}e_y, \end{equation*} $$ in the two dimensions. The unit vectors are \( e_x \) and \( e_y \).