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arXiv · 2610.34780

Liouville theorems for the stationary Navier-Stokes equations under partial component conditions

Cuiyang Liu·Wei Luo
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Abstract

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 $L^{4,\infty}$, then we show that $u$ must vanish. The second result only requires that one component $u_1$ belongs to $L^{3/2,\infty}$. 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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Cuiyang Liu, Wei Luo. 2026-10-26. Liouville theorems for the stationary Navier-Stokes equations under partial component conditions. https://arxiv.org/abs/2610.34780

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