Large-time behavior of compressible polytropic fluids and nonlinear Schr{ö}dinger equation

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 Schrödinger equation, allowing the presence of a semi-classical parameter as well as long range nonlinearities. Contents 1. Introduction 1.1. Main results 1.2. Nonlinear Schrödinger equation 1.3. Outline of the paper 2. Proof of Theorem 1.2 3. Nonlinear Schrödinger equation 3.1. A priori estimates 3.2. Interpretation 3.3. Proof of Proposition 1.4 4. Proof of Theorem 1.5 4.1. Euler 4.2. Euler-Korteweg 4.3. Navier-Stokes Appendix A. Computations of formal energy estimates Appendix B. Proof of Lemma 1.1 References