Long time behavior and attractors for energetically insulated fluid systems (Q977958)

The author continues previous work on the fluid problems and specifically the Navier-Stokes-Fourier system [\textit{E. Feireisl} and \textit{A. Novotný}, Singular limits in thermodynamics of viscous fluids. Advances in Mathematical Fluid Mechanics. Basel: Birkhäuser (2009; Zbl 1176.35126); \textit{E. Feireisl} and \textit{H. Petzeltová}, J. Dyn. Differ. Equations 19, No. 3, 685--707 (2007; Zbl 1168.35032); \textit{E. Feireisl}, Comput. Math. Appl. 53, No. 3--4, 461--490 (2007; Zbl 1122.76075)]. This system consists of the conservation law of mass, the balance of momentum, an entropy production equation and Fourier's law. It is consistent with the second law of thermodynamics and the author emposes insulating boundary conditions for the energy flux. As a first result, the static states are shown to consist of a two-parameter family and to maximize total entropy among all states with the same mass and total energy. The main result is that (for the weak formulation of the Navier-Stokes-Fourier system) the distance of trajectories to the set of static solutions tends to zero \textit{uniformly}. The author concludes with a number of remarks on modifications and extensions, for instance that non-insulating boundary conditions can have a dramatic effect on the long term dynamics.