Existence and stability of steady-state solutions with finite energy for the Navier–Stokes equation in the whole space

Abstract

Abstract. We consider the steady-state Navier-Stokes equation in the whole space R 3 driven by a forcing function f . The class of source functions f under consideration yield the existence of at least one solution with finite Dirichlet integral ( ∇U 2 < ∞). Under the additional assumptions that f is absent of low modes and the ratio of f to viscosity is sufficiently small in a natural norm we construct solutions which have finite energy (finite L 2 norm). These solutions are unique among all solutions with finite energy and finite Dirichlet integral. The constructed solutions are also shown to be stable in the following sense: If U is such a solution then any viscous, incompressible flow in the whole space, driven by f and starting with finite energy, will return to U .

Steady solutions with finite kinetic energy for a perturbed Navier–Stokes system in R3

“…Like in the recent articles [1,2] of Bjorland and Schonbek, we will assume that 3 and that there exist α > 1/2, > 0 and M > 0 such that f (ξ ) M|ξ | α , for a.a. ξ with |ξ | < , (1.4) wheref denotes the Fourier transform of f . As we will see in Section 3, this class of functions…”

“…In [1], a new method is proposed for constructing a weak solution v with finite kinetic energy for system (1.3) with ζ = 0 and ω = 0. Specifically, for f small enough in a natural norm and with low frequencies sufficiently controlled, the approach of Bjorland and Schonbek consists in first showing that a linearized version of the problem (1.2) with ζ = 0, ω = 0, F = 0 and u 0 = f possesses a unique solutionũ such that ũ(t) 2 C (1 + t) −β (β > 1), and then in taking the steady stateṽ(x) := ∞ 0ũ (x, t) dt as an "approximate solution" for (1.3).…”

“…In the spirit of the approaches of [20] and [1], our objective is to solve the more general steady problem (1.3) in the L 2 -framework, avoiding potential theoretic methods. Under the hypotheses we have specified above for the external force f , we propose, as an alternative to the potential theoretic methods, a simple approach for obtaining a strong solution to (1.3), that consists essentially in applying the Fourier transform directly to an appropriate linearized version of the problem.…”

“…Under the hypotheses we have specified above for the external force f , we propose, as an alternative to the potential theoretic methods, a simple approach for obtaining a strong solution to (1.3), that consists essentially in applying the Fourier transform directly to an appropriate linearized version of the problem. This procedure is more direct than the one of Bjorland and Schonbek [1] because it avoids the study of the related unsteady problem (1.2). On the other hand, it also allows a straightforward analysis of the terms depending on ζ and ω which are not present in the problem studied in [1].…”

“…This procedure is more direct than the one of Bjorland and Schonbek [1] because it avoids the study of the related unsteady problem (1.2). On the other hand, it also allows a straightforward analysis of the terms depending on ζ and ω which are not present in the problem studied in [1]. Basically, it will be restriction (1.4) onf that will ensure v ∈ L 2 (R 3 ) 3 in a steady flow.…”