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Liouville-type theorems for steady solutions to the Navier–Stokes system in a slab

Published online by Cambridge University Press:  26 February 2025

J. Bang ,
C. Gui Open the ORCID record for C. Gui [Opens in a new window] ,
Y. Wang  and
C. Xie
J. Bang
Affiliation:
Institute of Theoretical Sciences, Westlake University, Hangzhou 310030, PR China
C. Gui*
Affiliation:
Department of Mathematics, Faculty of Science and Technology, University of Macau, Taipa, Macau 999078, PR China
Y. Wang
Affiliation:
School of Mathematical Sciences, Center for dynamical systems and differential equations, Soochow University, Suzhou 215031, PR China
C. Xie
Affiliation:
School of Mathematical Sciences, Institute of Natural Sciences, Ministry of Education Key Laboratory of Scientific and Engineering Computing, and CMA-Shanghai, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, PR China
*
†Email address for correspondence: Changfenggui@um.edu.mo
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Abstract

Liouville-type theorems for the steady incompressible Navier–Stokes system are investigated for solutions in a three-dimensional (3-D) slab with either no-slip boundary conditions or periodic boundary conditions. When the no-slip boundary conditions are prescribed, we prove that any bounded solution is trivial if it is axisymmetric or $ru^r$ is bounded, and that general 3-D solutions must be Poiseuille flows when the velocity is not big in $L^\infty$ space. When the periodic boundary conditions are imposed on the slab boundaries, we prove that the bounded solutions must be constant vectors if either the swirl or radial velocity is independent of the angular variable, or $ru^r$ decays to zero as $r$ tends to infinity. The proofs are based on the fundamental structure of the equations and energy estimates. The key technique is to establish a Saint-Venant type estimate that characterizes the growth of the Dirichlet integral of non-trivial solutions.

JFM classification

Instability: Shear-flow instability

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Type
JFM Papers
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Journal of Fluid Mechanics , Volume 1005 , 25 February 2025 , A6
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© The Author(s), 2025. Published by Cambridge University Press

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