Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their detailed derivations.
Please give one example per answer, preferably with clear descriptions and pointers to literature.
Best Answer
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
$$\begin{cases} \frac{\partial u}{\partial t}+\nabla\cdot(u\otimes u) +\nabla p=0, \\[5pt] \nabla\cdot u=0 . \end{cases}$$ 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.