Analysis of the adiabatic piston problem via methods of continuum mechanics

• Published in: Annales de l'Institut Henri Poincaré. Analyse non linéaire Vol. 35; no. 5; pp. 1377 - 1408
• Main Authors: Feireisl, Eduard, Mácha, Václav, Nečasová, Šárka, Tucsnak, Marius
• Format: Journal Article
• Language: English
• Published: Elsevier Masson SAS 01.08.2018; EMS
• Subjects: Analysis of PDEs; Free boundary problem; Mathematics; Navier–Stokes–Fourier system; Optimization and Control; Piston problem; Piston problem; Navier–Stokes–Fourier system; Free boundary problem
• ISSN: 0294-1449; 1873-1430
• DOI: 10.1016/j.anihpc.2017.11.008

We consider a system modelling the motion of a piston in a cylinder filled by a viscous heat conducting gas. The piston is moving longitudinally without friction under the influence of the forces exerted by the gas. In addition, the piston is supposed to be thermally insulating (adiabatic piston). This fact raises several challenges which received a considerable attention, essentially in the statistical physics literature. We study the problem via the methods of continuum mechanics, specifically, the motion of the gas is described by means of the Navier–Stokes–Fourier system in one space dimension, coupled with Newton's second law governing the motion of the piston. We establish global in time existence of strong solutions and show that the system stabilizes to an equilibrium state for t→∞.