If a smooth solution of the 3D incompressible Navier-Stokes equations ceases to exist at a finite time T_f, several of its norms must diverge, and they must diverge at least as fast as an explicit algebraic rate in (T_f - t). Students learn how a bound on the velocity's maximum norm propagates to every derivative (so blow-up forces the maximum norm to become unbounded), how differential and integral inequalities satisfied by the Lq norms yield lower bounds such as ||u(t)||_Lq >= c_q (T_f - t)^{-(q-3)/2q} for q > 3, and how interpolation converts one such bound into others. The same machinery gives smallness conditions on the initial data that rule out blow-up entirely, shows that higher Lr norms blow up faster than lower ones, and yields the Beale-Kato-Majda criterion on the time integral of the maximum vorticity.
A review, with complete proofs, of what must happen to a smooth solution of the 3D incompressible Navier-Stokes Cauchy problem as time approaches a hypothetical finite blow-up time T_f, the first ins…