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

Abstract

Consider a Navier-Stokes liquid filling the three-dimensional space exterior to a moving rigid body and subject to an external force. Using a coordinates system attached to the body, the equations of the fluid can be written in a time-independent domain, which results in a perturbed Navier-Stokes system where the extra terms depend on the velocity of the rigid body. In this paper, we consider the related whole space problem and construct a strong solution with finite kinetic energy for the corresponding steady-state equations. For this, appropriate conditions on the external force have to be imposed (for instance, that it is a function with compact support and null average) together with a smallness condition involving the viscosity of the fluid. First, a linearized version of the problem is analysed by means of the Fourier transform, and then a strong solution to the full nonlinear problem is obtained by a fixed point procedure. We also show that such a solution satisfies the energy equation and is unique within a certain class.

Existence and uniqueness of time-periodic solutions with finite kinetic energy for the Navier–Stokes equations in {\mathbb R}^3

“…Hence, following our previous work [15] (see also the work of Bjorland and Schonbek [1]), in order to solve (1.2) and fulfil the condition v ∈ C([0, T ]; L 2 (R 3 ) 3 ), we will assume the following:…”

“…As in [1,15], our aim is to solve problem (1.2) with an appropriate external force f , in the L 2 -framework and show, in particular, that v(t) ∈ L 2 (R 3 ) 3 for all t ∈ [0, T ]. Furthermore, in the spirit of [1,15], all the summability properties of v and p should be established without resorting to potential theoretic methods.…”

“…As in [1,15], our aim is to solve problem (1.2) with an appropriate external force f , in the L 2 -framework and show, in particular, that v(t) ∈ L 2 (R 3 ) 3 for all t ∈ [0, T ]. Furthermore, in the spirit of [1,15], all the summability properties of v and p should be established without resorting to potential theoretic methods. Under the hypotheses we have specified for f , the method we will use to construct a strong solution with finite kinetic energy of (1.2) consists essentially of solving a linearized version of problem (1.2) via the Galerkin method, as suggested by Yudovich [13], and then in applying the Fourier transform directly to that linear problem to deduce that v ∈ L 2 (0, T ; L 2 (R 3 ) 3 ).…”