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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.