On the Reynolds number expansion for the Navier-Stokes equations

Abstract

In a previous paper of ours [Nonlinear Anal. 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 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 [Nonlinear Anal., 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 [M2AN, 2001]. Computations of order N=5 yield a quantitative improvement of the results previously obtained for this initial datum in [Nonlinear Anal. 2012], where a Galerkin approximate solution was employed in place of the Reynolds expansion.

Author's note. To appear in Nonlinear Analysis TMA. In the preliminaries and in the final paragraph there is some overlap with previous works of ours (arXiv:1203.6865, arXiv:0709.1670, arXiv:0909.3707, arXiv:1009.2051, arXiv:1104.3832, arXiv:1007.4412); these overlaps aim to make the paper self-contained, and do not involve the main results about the Reynolds expansion

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Cited by in corpus (2)

• Smooth solutions of the Euler and Navier-Stokes equations from the a posteriori analysis of approximate solutions
• Large order Reynolds expansions for the Navier-Stokes equations