Steady Solutions with Finite Kinetic Energy for a Perturbed Navier-Stokes System in R-3

Journal of Differential Equations, 247(7) (2009), 2124-2139
http://dx.doi.org/10.1016/j.jde.2009.07.003

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.