Rigidity results in generalized isothermal fluids

TL;DR

This paper studies the long-time behavior of solutions to various isothermal fluid equations, revealing universal dispersion rates and profiles, and establishing compactness of weak solutions in the isothermal case.

Contribution

It introduces a novel scaling approach to analyze the asymptotic behavior and develops modified entropies for weak solutions, extending understanding of isothermal fluid dynamics.

Findings

Solutions exhibit universal dispersion rates.

Asymptotic profiles are characterized for generalized equations.

Compactness of weak solutions is proven in the isothermal case.

Abstract

We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier Stokes equations, as well as generalizations of these equations where the convex pressure law is asymptotically linear near vacuum. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. In the exactly isothermal case, we establish the compactness of bounded sets of such weak solutions, by introducing modified entropies adapted to the new unknown functions.