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One dimensional Cherenkov processes in ferromagnetic isolators are studied with perturbation theory under the constraint condition of conservation of energy and momentum. It is shown that the magnon-phonon interaction channels are limited and wave number dependent, which result in respectively $1/k^{2}$ and $1/k^{4}$ dependence of the lifetime and the relaxation time of long wavelength magnons. The reciprocal of relaxation time between magnons and phonons, $1/\tau_{mp}$, is found to be a linearly increasing function of the temperature as $T>$ 70 K. Based on the Sanders-Walton model, we further show that when a thermal (phonon) gradient is applied along the system, the temperature difference between the phonon bath and the magnons with wave-vector $k$ becomes more pronounced as $k$ decreasing.

One dimensional Cherenkov processes in ferromagnetic insulator