The Navier-Stokes equations on manifolds with boundary
We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold M with boundary. The motion on M is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on ∂M. We establish existence and...
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| Published in | Journal of Differential Equations Vol. 416; pp. 1602 - 1659 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Elsevier Inc
25.01.2025
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| ISSN | 0022-0396 |
| DOI | 10.1016/j.jde.2024.10.030 |
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| Abstract | We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold M with boundary. The motion on M is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on ∂M. We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on M that satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case M is two-dimensional we show that solutions with divergence free initial condition in L2(M;TM) exist globally and converge to an equilibrium exponentially fast. |
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| AbstractList | We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold M with boundary. The motion on M is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on ∂M. We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on M that satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case M is two-dimensional we show that solutions with divergence free initial condition in L2(M;TM) exist globally and converge to an equilibrium exponentially fast. |
| Author | Simonett, Gieri Wilke, Mathias Shao, Yuanzhen |
| Author_xml | – sequence: 1 givenname: Yuanzhen surname: Shao fullname: Shao, Yuanzhen email: yshao8@ua.edu organization: The University of Alabama, Tuscaloosa, AL, USA – sequence: 2 givenname: Gieri orcidid: 0000-0002-2211-9032 surname: Simonett fullname: Simonett, Gieri email: gieri.simonett@vanderbilt.edu organization: Vanderbilt University, Nashville, TN, USA – sequence: 3 givenname: Mathias orcidid: 0000-0002-9845-8823 surname: Wilke fullname: Wilke, Mathias email: mathias.wilke@mathematik.uni-halle.de organization: Martin-Luther-Universität Halle-Wittenberg, Halle (Saale), Germany |
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| Cites_doi | 10.1007/s000210050018 10.1016/j.matpur.2014.03.005 10.4171/ifb/405 10.1007/s00028-014-0226-6 10.1137/18M1166183 10.1007/BF02567605 10.1007/s00028-017-0382-6 10.1619/fesi.47.361 10.24033/asens.1178 10.1016/j.jde.2017.10.010 10.1016/j.geomphys.2019.103543 10.1002/fld.3835 10.1016/j.geomphys.2017.07.015 10.4064/sm-44-1-47-60 10.1080/03605300601088740 10.1007/978-3-319-27698-4 10.2307/1970024 10.2996/kmj/1138845076 10.1090/S0002-9947-08-04827-7 10.1007/s00021-017-0342-5 10.1063/1.5005142 10.1007/s00028-020-00648-0 10.1007/BF00276188 10.1137/080716074 10.1007/s00208-003-0493-y 10.1007/s00205-008-0202-9 10.1007/s00021-022-00742-y 10.1002/mana.201100157 10.1007/BF00276875 |
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| Keywords | secondary Navier boundary conditions Stability Killing fields H∞-calculus and critical spaces Ricci curvature primary Well-posedness |
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| SubjectTerms | [formula omitted]-calculus and critical spaces Killing fields Navier boundary conditions Ricci curvature Stability Well-posedness |
| Title | The Navier-Stokes equations on manifolds with boundary |
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