We present a general concept that is suitable for studying the stability of equilibria for open systems in continuum thermodynamics. We apply such concept to a generalized Newtonian incompressible heat conducting fluid with prescribed nonuniform temperature on the boundary and with the no-slip boundary conditions for the velocity in three dimensional domain. For large class of constitutive relation for the Cauchy stress, we identify a class of proper solutions converging to the equilibria exponentially in a suitable metric and independently of the distance to equilibria at the initial time. Consequently, the equilibrium is nonlinearly stable and attracts all weak solutions from that class. The proper solutions exist and satisfy entropy (in)equality.

In the past two decades, starting with the pioneering work of Bourgain, Brezis, and Mironescu, there has been widespread interest in characterizing Sobolev and BV (bounded variation) functions by means of non-local functionals. In my recent work I have studied two such functionals: a BMO-type (bounded mean oscillation) functional, and a functional related to the fractional Sobolev seminorms. I will discuss some of my results concerning the limits of these functionals, the concept of Gamma-convergence, and also open problems.