Local and global solutions for the compressible navier-stokes equations near equilibria via the energy method

In this chapter we present the classical energy approach for existence of regular solutions to the equations of compressible, heat-conducting fluids in a bounded three-dimensional domain. Firstly, we provide a state of the art and recall representative results in this field. Next, we give a proof of one of them, concerning Dirichlet boundary conditions for velocity and temperature. The result and thus the proof is divided into two main parts. A local-in-time existence result in high-regularity norms, via a method of successive approximations, occupies the former one. In the latter part, a differential inequality is derived, which allows us to extend the local-in-time solution to the global-in-time solution, provided a certain smallness condition is satisfied. This smallness condition is in fact an equilibrium proximity condition, since it involves differences between data and constants, whereas the data for temperature and density may be large themselves. All our considerations are performed within the L²-approach. The proved result is close to that of Valli and Zajaczkowski (Commun Math Phys 103:259-296, 1986), but the techniques used here: the method of successive approximations (instead of a Leray-Schauder fixed-point argument there) as well as a clear continuation argument renders our exposition more traceable. Moreover, one may easily derive now an explicit smallness condition via our approach. Besides, the thermodynamic restriction on viscosities is relaxed, certain technicalities are improved and a possibly useful approach to deal with certain difficulties at the boundary in similar problems is provided.