LARGE-TIME BEHAVIOR OF COMPRESSIBLE POLYTROPIC FLUIDS AND NONLINEAR SCHRÖDINGER EQUATION

In this paper we analyze the large-time behavior of weak solutions to polytropic fluid models possibly including quantum and viscous effects. Formal a priori estimates show that the density of solutions to these systems should disperse with time. Scaling appropriately the system, we prove that, under a reasonable assumption on the decay of energy, the density of weak solutions converges in large times to an unknown profile. In contrast with the isothermal case, we also show that there exists a large variety of asymptotic profiles. We complement the study by providing existence of global-in-time weak solutions satisfying the required decay of energy.