A counter-example concerning the pressure in the navier-stokes equations, as t → 0+

Abstract

We show the existence of solutions of the Navier-Stokes equations for which the Dirichlet norm, ∥∇u(t)∥L2(Ω), of the velocity is continuous as t = 0, while the normalized L2 norm, ∥p(t)∥L2(Ω)/R, of the pressure is not. This runs counter to the naive expectation that the relative orders of the spatial derivatives of u, p and ut should he the same in a priori estimates for the solutions as in the equations themselves. © 1994 by Pacific Journal of Mathematics.