A counterexample concerning the pressure in the Navier-Stokes equations, as t → 0⁺

John Groves Heywood and Owen Walsh

Vol. 164 (1994), No. 2, 351–359

DOI: 10.2140/pjm.1994.164.351

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 u_t should be the same in a priori estimates for the solutions as in the equations themselves.

Mathematical Subject Classification 2000

Primary: 35Q30

Secondary: 35B40, 76D05

Milestones

Received: 5 November 1991

Published: 1 June 1994