On the convergence of Euler-Stokes splitting of the Navier-Stokes equations

Abstract

We consider Euler--Stokes splitting approximation of the Navier--Stokes equations with no--slip boundary condition. This consists in alternate solving of the Euler equations with tangential boundary condition and Stokes equations with no-slip boundary condition on small time intervals of the same length $k$. In a previous paper, J.T. Beale and C. Greengard proved the convergence of this approximation scheme in $L^p$ norm as $k$ tends to zero, for smooth solutions of the Navier--Stokes equations. Here we show how a certain simplification in their arguments improves their main result in the following way: the convergence holds without any additional regularity assumption on the solution of the Navier--Stokes equations.