Pressureless explicit solutions to incompressible Euler

Question

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

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

In other words, I am looking for explicit or semi explicit solutions $u:\mathbb{R}\times \mathbb{T}^d\rightarrow \mathbb{R}$, where $\Bbb T^d$, $d\ge 2$ is the $d$-dimensional torus, to the Euler system of PDEs $$ \begin{cases} \partial_t u+u\cdot \nabla u=0,\\ \operatorname{div} u =0, \end{cases} $$ with $\partial_t u$ not everywhere zero.

Answer 1

Partial answer. If you assume in addition that the solution is classical ('smooth'), then the Jacobian $A(t)$ of the velocity field along a trajectory ('characteristics') satisfies ${\rm Tr}A\equiv0$ (incompressibility) and the Riccati equation $$\frac{dA}{dt}+A^2=0_n.$$ Therefore $${\rm Tr}(A^2)=-({\rm Tr}A)'=0$$ and ${\rm Tr}(A^k)=0$ by induction, for every $k$. One deduces that $\nabla_xu$ is everywhere nilpotent. In particular $$A(t)=\sum_{k=1}^{n}(-t)^{k-1}A_0^k,\qquad A_0:=A(0).$$

Now your question rephrases to whether there exists a periodic vector field $v$ over ${\mathbb R}^n$, whose Jacobian is everywhere nilpotent, non-trivial in the sense that $(u\cdot\nabla)u\not\equiv0$.