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Stability of spatially inhomogeneous solutions to the Vlasov equation isinvestigated for the Hamiltonian mean-field model to provide the spectralstability criterion and the formal stability criterion in the form of necessaryand sufficient conditions. These criteria determine stability of spatiallyinhomogeneous solutions whose stability has not been decided correctly by usinga less refined formal stability criterion. It is shown that some of suchsolutions can be found in a family of stationary solutions to the Vlasovequation, which is parametrized with macroscopic quantities and has a two-phasecoexistence region in the parameter space.

Spectral and formal stability criteria of spatially inhomogeneous stationary solutions to the Vlasov equation for the Hamiltonian mean-field model