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In this paper, a compressible viscous-dispersive Euler system in one spacedimension in the context of quantum hydrodynamics is considered. The purpose ofthis study is twofold. First, it is shown that the system is locallywell-posed. For that purpose, the existence of classical solutions which areperturbation of constant states is established. Second, it is proved that inthe particular case of subsonic equilibrium states, sufficiently smallperturbations decay globally in time. In order to prove this stabilityproperty, the linearized system around the subsonic state is examined. Using anappropriately constructed compensating matrix symbol in the Fourier space, itis proved that solutions to the linear system decay globally in time,underlying a dissipative mechanism of regularity gain type. These linear decayestimates, together with the local existence result, imply the global existenceand the decay of perturbations to constant subsonic equilibrium states assolutions to the full nonlinear system.

Well-posedness and decay structure of a quantum hydrodynamics system with Bohm potential and linear viscosity