Periodic solutions to Navier-Stokes equations on non-compact Einstein manifolds with negative curvature

Abstract

Consider the Navier-Stokes Equations (NSE) for viscous incompressible fluid flows on a non-compact, smooth, simply-connected and complete Einstein manifold $$(\mathbf {M},g)$$ with negative Ricci curvature tensor. We prove the existence and uniqueness of a time-periodic solution to NSE for vector fields on $$(\mathbf {M},g)$$ . Our method is based on the dispersive and smoothing properties of the semigroup generated by the linearized Stokes equations to construct a bounded (in time) solution of the nonhomogeneous Stokes equation and on the ergodic method to obtain the periodic solution to Stokes equation. Then, using the fixed point arguments, we can pass from the Stokes equations to Navier-Stokes equations to obtain periodic solutions to NSE on the Einstein manifold $$(\mathbf {M},g)$$ . We also prove the stability of the periodic solution.