Genuine Hydrodynamic Analysis to the 1-D QHD System: Existence, Dispersion and Stability

Abstract

In this paper we consider a genuinely hydrodynamic approach for the one dimensional quantum hydrodynamics system. In the recent years, the global existence of weak solutions with large data has been obtained in Antonelli and Marcati (Commun Math Phys 287(2):657–686, 2009; Arch Ration Mech Anal 203:499–527, 2012), in several space dimensions, by using the connection between the hydrodynamic variables and the Schrödinger wave function. One of the main purposes of the present paper is to overturn this point of view, avoiding the need to postulate the a priori existence of a wave function that generates the hydrodynamic data. In a first result, we are able to demonstrate the existence of finite energy weak solutions with large data. Next we introduce a functional based on a new notion of generalized chemical potential, that allows us to prove the existence of more regular weak solutions under the assumption that the initial data have a finite generalized chemical potential and that the energy density is continuous on the boundaries of the vacuum set. These solutions enjoy an appropriate entropy inequality. We then obtain dispersive estimates for which the mass densities vanish and the speed formally tends to a rarefaction wave with a mechanism reminiscent of the Landau Damping. We are also able to show, by using genuinely hydrodynamic arguments, that for finite energy and finite mass weak solutions Morawetz-type estimates hold and therefore by means of these estimates, in the case of solutions that have bounded generalized chemical potential and satisfies the entropy inequality, we get additional regularity. By virtue of these properties it is possible to obtain a result of strong stability in the energy norm, in the class of solutions with finite mass, finite energy and bounded generalized chemical potential. Moreover we also analyze the possibility of extending the generalized chemical potential and a Radon measure that, in the atomic parts, takes into account the potential differences on the phase boundaries.