The Navier-Stokes flow with linearly growing initial velocity in the whole space

Abstract

In this paper, the uniqueness of the solutions to the Navier-Stokesequations in the whole space is constructed, provided that the velocity grows linearly at infinity. The velocity can be chosen as Mx + u(x) for some constant matrix M and some function u. The perturbation u is taken in some homogeneous Besov spaces, which contain some nondecaying functions at space infinity, typically, somealmost periodic functions. It is also proved that a locally-in-time solution exists, when M is essentially skew-symmetric which demonstrates the rotating fluid in 2- dor 3-dimension.

Keywords

• Navier-Stokes equations
• linearly growing data
• mild solution
• homogeneous Besov spaces
• almost periodic function.