Nonstationary flows of viscous and ideal fluids in R3

Abstract

The Cauchy problem for the nonstationary Navier-Stokes equation in R 3 is considered. It is shown that the solution exists on a time interval independent of the viscosity v and tends as v → 0 to the solution of the limiting equation, provided that the initial velocity field and the external force field are sufficiently smooth and small at infinity (in the sense that they belong to the Sobolev space over R 3 of order 3). Such a result is not altogether new but the proof, which depends on the theory of nonlinear evolution equation in Hilbert space, is simpler than the existing one due to Swann.