Scheffer has shown that there is a nontrivial weak solution $u(x,t)\in L^2(\mathbb R^2\times\mathbb R)$ to the incompressible Euler equations in 2D
$$\left\{\begin{eqnarray} \frac{\partial u}{\partial t}+\nabla\cdot(u\otimes u) +\nabla p=0 \\ \nabla\cdot u=0\qquad\qquad\qquad\qquad\qquad\ \end{eqnarray}\right.$$ such that $u(x,t)\equiv 0$ for $|x|^2+|t|^2>1$. In other words, the solution is identically zero for $t<-1$, then "something happens" and the solution becomes non-zero, and for all $t>1$ the solution vanishes again. In the real world, this would look like if the water suddenly started to move in a cup that stands firmly on a table.
See V. Scheffer, "An inviscid flow with compact support in space-time", Journal of Geometric Analysis, vol. 3 (1993), pp. 343-401.
