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Liouville theorems for the stationary Navier-Stokes equations under partial component conditions
Abstract
We prove Liouville theorems for smooth stationary incompressible Navier-Stokes flows in R3 with finite Dirichlet integral and velocity vanishing uniformly at infinity. If the partial components of velocity (u1,u2) belong to L4,∞, then we show that u must vanish. The second result only requires that one component u1 belongs to L3/2,∞. The proofs are based on the head-pressure sign and a localized energy identity, combined with different pressure estimates for the two cases.
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BibTeX
@misc{liu2026liouville,
title = {{Liouville theorems for the stationary Navier-Stokes equations under partial component conditions}},
author = {Cuiyang Liu and Wei Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34780},
eprint = {2610.34780},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34780},
}
RIS
TY - UNPB
TI - Liouville theorems for the stationary Navier-Stokes equations under partial component conditions
AU - Cuiyang Liu
AU - Wei Luo
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34780
UR - https://arxiv.org/abs/2610.34780
AB - We prove Liouville theorems for smooth stationary incompressible Navier-Stokes flows in R^3 with finite Dirichlet integral and velocity vanishing uniformly at infinity. If the partial components of velocity (u_1,u_2) belong to L^4,∞, then we show that u must vanish. The second result only requires that one component u_1 belongs to L^3/2,∞. The proofs are based on the head-pressure sign and a localized energy identity, combined with different pressure estimates for the two cases.
ER -