What are some examples of (semi-)explicit solutions of the incompressible Euler equations which satisfy the following

1. they are presureless
2. they are periodic in space
3. they have nontrivial time dependence (i.e. not steady state)

In other words $u:\mathbb{R}\times \mathbb{T}^d\rightarrow \mathbb{R}$ where $d\ge 2$ is the $d$-dimensional torus satisfies $$\partial_t u+u\cdot \nabla u=0,\ \text{div}\ u =0$$ with $\partial_t u$ not everywhere zero.