A Geometric Characterization of Steady Laminar Flow
- 1Department of Mathematics Stony Brook University Stony Brook New York USA
- 2Instituto de Ciencias Matemáticas Consejo Superior de Investigaciones Científicas Madrid Spain
Abstract
We study the steady states of the Euler equations on the periodic channel or annulus. We show that if these flows are laminar (layered by closed non‐contractible streamlines which foliate the domain), then they must be either parallel or circular flows. We also show that a large subset of these shear flows are isolated from non‐shear stationary states. For Poiseuille flow, , our result shows that all stationary solutions in a sufficiently small neighbourhood are shear flows. We then show that if with , then in any neighbourhood, there exist smooth non‐shear steady states, traveling waves, and quasiperiodic solutions of any number of non‐commensurate frequencies. This proves the rigidity near Poiseuille is sharp. Finally, we prove that on general compact doubly connected domains, laminar steady Euler flows with constant velocity on the boundary must also be either parallel or circular, and the domain a periodic channel or an annulus. This shows that laminar free boundary Euler solutions must have Euclidean symmetry.
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BibTeX
@article{drivas2026geometric,
title = {{A Geometric Characterization of Steady Laminar Flow}},
author = {Theodore D. Drivas and Marc Nualart},
journal = {Communications on Pure and Applied Mathematics},
year = {2026},
month = jul,
doi = {10.1002/cpa.70055},
url = {https://doi.org/10.1002/cpa.70055},
note = {CC BY},
}
RIS
TY - JOUR
TI - A Geometric Characterization of Steady Laminar Flow
AU - Theodore D. Drivas
AU - Marc Nualart
PY - 2026
DA - 2026/07/07
JO - Communications on Pure and Applied Mathematics
DO - 10.1002/cpa.70055
UR - https://doi.org/10.1002/cpa.70055
AB - We study the steady states of the Euler equations on the periodic channel or annulus. We show that if these flows are laminar (layered by closed non‐contractible streamlines which foliate the domain), then they must be either parallel or circular flows. We also show that a large subset of these shear flows are isolated from non‐shear stationary states. For Poiseuille flow, , our result shows that all stationary solutions in a sufficiently small neighbourhood are shear flows. We then show that if with , then in any neighbourhood, there exist smooth non‐shear steady states, traveling waves, and quasiperiodic solutions of any number of non‐commensurate frequencies. This proves the rigidity near Poiseuille is sharp. Finally, we prove that on general compact doubly connected domains, laminar steady Euler flows with constant velocity on the boundary must also be either parallel or circular, and the domain a periodic channel or an annulus. This shows that laminar free boundary Euler solutions must have Euclidean symmetry.
SN - 0010-3640
SN - 1097-0312
ER -