A priori estimates in terms of the maximum norm for the solutions of the Navier–Stokes equations

Abstract

In this paper, we consider the Cauchy problem for the incompressible Navier–Stokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field. For illustrative purposes, we first derive corresponding a priori estimates for certain parabolic systems. Because of the pressure term, the case of the Navier–Stokes equations is more difficult, however.