On the Reynolds number expansion for the Navier-Stokes equations

In a previous paper of ours [Morosi and Pizzocchero, Nonlinear Analysis 2012] we have considered the incompressible Navier-Stokes (NS) equations on a d-dimensional torus T^d, in the functional setting of the Sobolev spaces H^n(T^d) of divergence free, zero mean vector fields (n > d/2+1). In the cited work we have presented a general setting for the a posteriori analysis of approximate solutions of the NS Cauchy problem; given any approximate solution u_a, this allows to infer a lower bound T_c on the time of existence of the exact solution u and to construct a function R_n such that || u(t) - u_a(t) ||_n <= R_n(t) for all t in [0,T_c). In certain cases it is T_c = + infinity, so global existence is granted for u. In the present paper the framework of [Morosi and Pizzocchero, Nonlinear Analysis 2012] is applied using as an approximate solution an expansion u^N(t) = Sum_{j=0}^N R^j u_j(t), where R is the Reynolds number. This allows, amongst else, to derive the global existence of u when R is below some critical value R_{*} (increasing with N in the examples that we analyze). After a general discussion about the Reynolds expansion and its a posteriori analysis, we consider the expansions of orders N=1,2,5 in dimension d=3, with the initial datum of Behr, Necas and Wu [ESAIM:M2AN, 2001]. Computations of order N=5 yield a quantitative improvement of the results previously obtained for this initial datum in [Morosi and Pizzocchero, Nonlinear Analysis 2012], where a Galerkin approximate solution was employed in place of the Reynolds expansion.