We present a quantitative study of the $ u N$ cross section in the neutrino energy range $10^4<E_{ u}<10^{14}$ GeV within two transversal QCD approaches: NLO DGLAP evolution using different sets of PDFs and BK small-$x$ evolution with running coupling and kinematical corrections. We show that the non-linear effects embodied in the BK equation yield a slower raise in the cross section for $E_{ u}gtrsim 10^{8}$ GeV than the usual DGLAP based calculation. Finally, we translate this theoretical uncertainty into upper bounds for the ultra-high-energy neutrino flux for different experiments.
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