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 \( e_y \).