the Navier-Stokes Millennium Prize problem
Here is a question that sounds like it should already be settled, but is not, and carries a one-million-dollar prize: if you start three-dimensional fluid flow off with perfectly smooth, finite-energy initial conditions and let the Navier-Stokes equations run forever, do the solutions stay smooth and finite for all time — or can they spontaneously blow up, developing infinite velocity at some point in finite time? Nobody knows. This is one of the seven Clay Mathematics Institute Millennium Prize Problems, posed in 2000.
Precisely, the problem asks one to either prove or disprove global existence and smoothness for the 3D incompressible Navier-Stokes equations on all of space (or a periodic box) given smooth, rapidly decaying initial data and no forcing. What is known sharpens the gap: in two dimensions, the answer is yes — solutions are smooth forever (Ladyzhenskaya). In three dimensions, weak solutions exist for all time (Leray, 1934) but might not be unique or smooth; and smooth solutions are known to exist and be unique only for a short time, or for all time if the initial data is small enough. The suspected danger is vortex stretching, the purely 3D mechanism that can in principle pile vorticity up without bound. Beale-Kato-Majda showed a blow-up, if it happens, must be accompanied by the time-integral of the maximum vorticity becoming infinite.
Why does a question we sidestep daily with simulations matter so much? Because a 'no' (a finite-time singularity) would mean the equations themselves break down as a model of reality, and a 'yes' would require genuinely new mathematics for nonlinear PDE that we do not currently possess. It is the sharpest honest statement of how the most-used equations in physics are still not fully understood. The Euler equations (no viscosity) version of the same question is also open and arguably harder.
An analogy in lower dimensions: the simplified model u_t + u u_x = nu u_xx (Burgers' equation) shows that nonlinear advection tends to steepen gradients toward a blow-up, while viscosity nu u_xx fights to smooth them out. Burgers stays smooth because viscosity wins in 1D. The open question is whether, in 3D Navier-Stokes, vortex stretching can ever outrun viscous damping.
Open since 2000: do smooth 3D solutions stay smooth forever, or can they blow up?
The prize is for the clean idealized problem (incompressible, smooth, decaying data); it does not claim Navier-Stokes is wrong as physics. And 'we cannot prove regularity' is not the same as 'blow-up has been observed' — no physical or numerical blow-up has been demonstrated either.