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June, 2022 On Periodic Solutions of the Incompressible Navier–Stokes Equations on Non-compact Riemannian Manifolds
Thieu Huy Nguyen, Truong Xuan Pham, Thi Van Nguyen, Thi Ngoc Ha Vu
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Taiwanese J. Math. 26(3): 607-633 (June, 2022). DOI: 10.11650/tjm/211205

Abstract

In this paper, we study the existence, uniqueness and stability of the time periodic mild solutions to the incompressible Navier–Stokes equations on the non-compact manifolds with negative Ricci curvature tensor. In our strategy, we combine the dispersive and smoothing estimates for Stokes semigroups and Massera-type theorem to establish the existence and uniqueness of the time periodic mild solution to Stokes equation on Riemannian manifolds. Then using fixed point arguments, we can pass to semilinear equations to obtain the existence and uniqueness of the periodic solution to the imcompressible Navier–Stokes equations under the action of a periodic external force. The stability of the solution is also proved by using the cone inequality.

Funding Statement

The works of the first two authors and the last author were partially supported by Vietnam Institute for Advance Study in Mathematics (VIASM). The work of the last author is partly supported by by the Project of Vietnam Ministry of Education and Training under Project B2022-BKA-06. This work is financially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02-2021.04.

Citation

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Thieu Huy Nguyen. Truong Xuan Pham. Thi Van Nguyen. Thi Ngoc Ha Vu. "On Periodic Solutions of the Incompressible Navier–Stokes Equations on Non-compact Riemannian Manifolds." Taiwanese J. Math. 26 (3) 607 - 633, June, 2022. https://doi.org/10.11650/tjm/211205

Information

Received: 11 March 2021; Revised: 17 November 2021; Accepted: 15 December 2021; Published: June, 2022
First available in Project Euclid: 23 December 2021

Digital Object Identifier: 10.11650/tjm/211205

Subjects:
Primary: 35B10 , 35Q30
Secondary: 76D07

Keywords: Navier–Stokes equations , negative Ricci curvature tensors , non-compact Riemannian manifolds , periodic solutions , stability

Rights: Copyright © 2022 The Mathematical Society of the Republic of China

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Vol.26 • No. 3 • June, 2022
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