  The superluminal neutrinos detected by OPERA indicates Lorentz invariance
violation (LIV) of the neutrino sector at the order of $10^{-5}$. We study the
implications of the result in this work. We find that such a large LIV implied
by OPERA data will make the neutrino production process $\pi \to \mu + \nu_\mu$
kinematically forbidden for neutrino energy greater than about 5 GeV. The OPERA
detection of neutrinos at 40 GeV can constrain the LIV parameter to be smaller
than $3\times 10^{-7}$. Furthermore the neutrino decay in the LIV framework
will modify the neutrino spectrum greatly. The atmospheric neutrino spectrum
measured by IceCube can constrain the LIV parameter to the level of $10^{-12}$.
The future detection of astrophysical neutrinos of Galactic sources is expected
to be able to give even stronger constraint on the LIV parameter of neutrinos.

High Energy Physics - Phenomenology;High Energy Astrophysical Phenomena;High Energy Physics - Experiment