The a priori bound on (d/dt)||u||_q^q for 3D Navier-Stokes is sharp up to a prefactor, as shown by Riemannian conjugate-gradient maximizers for several q>3.
On maximum enstrophy growth in a hy- drodynamic system
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On the sharpness of bounds on the rate of growth of Lebesgue norms of the velocity in Navier-Stokes flows
The a priori bound on (d/dt)||u||_q^q for 3D Navier-Stokes is sharp up to a prefactor, as shown by Riemannian conjugate-gradient maximizers for several q>3.