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 $(u\cdot\nabla)u\not\equiv0$. I guess it doesn't.

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 $(u\cdot\nabla)u\not\equiv0$.

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 Ricatti 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 $(u\cdot\nabla)u\not\equiv0$. I guess it doesn't.

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 $(u\cdot\nabla)u\not\equiv0$. I guess it doesn't.