Existence and Uniqueness of Time-Periodic Solutions with Finite Kinetic Energy for the Navier-Stokes Equations in R^3

Nonlinearity, 25(1) (2012), 37-55
http://dx.doi.org/10.1088/0951-7715/25/1/37

Our aim is to prove existence and uniqueness of time-periodic strong solutions with finite kinetic energy for the Navier–Stokes equations in {\mathbb R}^3 . For this, appropriate conditions are imposed on the external force, together with a smallness condition involving the viscosity of the fluid and the period of motion. We extend the method we have recently used to construct steady states with finite kinetic energy to the time-periodic case. First, existence and uniqueness of strong solutions with finite kinetic energy are established for a linearized version of the problem, using the Galerkin method and the Fourier transform in the space variables. Then, a strong solution with finite kinetic energy for the nonlinear problem is obtained by means of the contraction mapping principle. We also show that such a solution satisfies the energy equality and is unique within a class of weak solutions.