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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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Liu et al. (2026) studied this question.
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