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

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.