Large-time behavior of compressible polytropic fluids and nonlinear Schr{\"o}dinger equation

TL;DR

This paper investigates the long-term behavior of solutions to compressible polytropic fluids with quantum and capillary effects, showing convergence to profiles and analyzing related nonlinear Schr{"o}dinger equations.

Contribution

It establishes the convergence of weak solutions to asymptotic profiles and provides existence results, extending understanding of fluid and Schr{"o}dinger dynamics.

Findings

Density disperses over time under energy decay assumptions

Existence of global weak solutions with decay properties

Multiple asymptotic profiles in the non-isothermal case

Abstract

In this paper we analyze the large-time behavior of weak solutions to polytropic fluid models possibly including quantum and capillary 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. As a byproduct of our method, we also obtain results concerning the large-time behavior of solutions to nonlinear Schr{\"o}dinger equation, allowing the presence of a semi-classical parameter as well as long range nonlinearities.