Open-problem statements, with their sources attached.

For three-dimensional incompressible flow, determine whether $\partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\Delta u$ with $\nabla\cdot u=0$ always has a global smooth solution, or exhibit finite-time breakdown.

For every compact simple gauge group $G$, construct a nontrivial quantum Yang-Mills theory on $\mathbb{R}^4$ whose spectrum has a positive mass gap $\Delta>0$.