Analysis and Mathematical Physics
, Volume 11 (2) – Feb 17, 2021

/lp/springer-journals/periodic-solutions-to-navier-stokes-equations-on-non-compact-einstein-K4J4ZN7Kig

- Publisher
- Springer Journals
- Copyright
- Copyright © The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature 2021
- ISSN
- 1664-2368
- eISSN
- 1664-235X
- DOI
- 10.1007/s13324-021-00488-2
- Publisher site
- See Article on Publisher Site

Consider the Navier-Stokes Equations (NSE) for viscous incompressible fluid flows on a non-compact, smooth, simply-connected and complete Einstein manifold (M,g)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\mathbf {M},g)$$\end{document} with negative Ricci curvature tensor. We prove the existence and uniqueness of a time-periodic solution to NSE for vector fields on (M,g)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\mathbf {M},g)$$\end{document}. 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 (M,g)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\mathbf {M},g)$$\end{document}. We also prove the stability of the periodic solution.

Analysis and Mathematical Physics – Springer Journals

**Published: ** Feb 17, 2021

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