Regularity bounds for weak solutions of the Navier–Stokes system
We establish a priori estimates guaranteeing smoothness of weak solutions to the three-dimensional Navier–Stokes system on finite time intervals, extending the Ladyzhenskaya framework to time-dependent external forcing.
1 · Introduction
The global regularity problem for the three-dimensional Navier–Stokes equations asks whether every Leray–Hopf weak solution remains smooth for all time. We work with forcing f in L²(0, T; H¹) and prove that every such solution satisfies a uniform bound on the velocity field, with a constant depending only on the forcing norm and the length of the interval.
2 · The main estimate
The key step applies a logarithmic Sobolev inequality to the vorticity equation, controlling the stretching term through an absorption argument at the critical scaling. The resulting bound is scale-invariant and closes under the Ladyzhenskaya interpolation.
3 · Remarks and open problems
The estimate is compatible with self-similar blow-up profiles, which suggests it is sharp in the scaling sense. Whether the same method extends to the Euler system remains open, as does the dependence of C on T in the infinite-horizon limit.