On the Reynolds number expansion for the Navier–Stokes equations

In a previous paper of ours (Morosi and Pizzocchero (2012) [1]) we have considered the incompressible Navier–Stokes (NS) equations on a dd-dimensional torus Td, in the functional setting of the Sobolev spaces HΣ0n(Td) of divergence free, zero mean vector fields (n>d/2+1n>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 ua, this allows to infer a lower bound Tc on the time of existence of the exact solution uu and to construct a function RnRn such that ‖u(t)−ua(t)‖n⩽Rn(t) for all t∈[0,Tc). In certain cases it is Tc=+∞, so global existence is granted for uu. In the present paper the framework of Morosi and Pizzocchero (2012) [1] is applied using as an approximate solution an expansion uN(t)=∑j=0NRjuj(t), where RR is the Reynolds number. This allows, amongst else, to derive the global existence of uu when RR is below some critical value R∗R∗ (increasing with NN 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=1,2,5N=1,2,5 in dimension d=3d=3, with the initial datum of Behr, Nečas and Wu (2001) [11]. Computations of order N=5N=5 yield a quantitative improvement of the results previously obtained for this initial datum in Morosi and Pizzocchero (2012) [1], where a Galerkin approximate solution was employed in place of the Reynolds expansion.