Global and exponential attractors for a Ginzburg-Landau model of superfluidity

TL;DR

This paper studies the long-term behavior of a non-isothermal Ginzburg-Landau model for superfluidity in liquid helium-4, establishing the existence of global and exponential attractors that describe the system's asymptotic states.

Contribution

It proves the existence of global and exponential attractors for a thermodynamically consistent superfluidity model, advancing understanding of its long-term dynamics.

Findings

Existence of a global attractor comprising stationary solutions.

Existence of a finite-dimensional exponential attractor.

The attractors describe the asymptotic behavior of the superfluidity model.

Abstract

The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techniques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.