Asymptotic-behavior in incompressible Navier-Stokes equations

Abstract

We give a development up to the second order of strong solutions u of incompressible Navier-Stokes equations in R(n), n greater-than-or-equal-to 2 for several classes of initial data u0. The first term is the solution h(t) = G(t) * u0 of the heat equation taking the same initial data. A better approximation is provided by the divergence free solutions with initial data u0 of v(t) = DELTAv = -h(i) partial derivative(i) h - partial derivative(j) del E(n) * h(i) partial derivative(i) h(j) in R+ x R(n) where E(n) stands for the fundamental solution of -DELTA in R(n). For initial data satisfying some integrability conditions (and small enough, if n greater-than-or-equal-to 3) we obtain, for 1 less-than-or-equal-to q less-than-or-equal-to infinity, t(1/2)+(n/2) (1-(1/q))/delta(t)\\u(t)-v(t)\\q = t(1/2)+(n/2) (1-(1/q))/delta(t)\\u(t)-G(t)*u0+R(t)\\q --> 0 when t --> infinity, where delta(t) is equal to log t if n = 2 and to a constant if n greater-than-or-equal-to 3 and R (t) is a corrector term that we compute explicitely.