On compactness of solutions to the Navier-Stokes equations of compressible flow
The papers deals with the system describing the motion of a compressible flow in the following form: \[ \rho_t+\text{ div}(\rho u)=0,\quad (\rho u)_t+\text{ div}(\rho u\otimes u)+\nabla P=\text{ div} T(u)+\rho f \] for the velocity of the fluid \(u=(u_1,u_2,u_3)\) and for the mass density \(\rho\). Here, the pressure \(P\) and the density \(\rho\) are functionally dependent: \(P=p(\rho)\) for \(p\) nondecreasing, and locally Lipschitz continuous. This system is supplemented with periodic boundary conditions. The main goal of this paper is to show that any sequence of weak solutions satisfying the natural a priori estimates contains a subsequence converging (not necessarily strongly) to another weak solution of the same problem. Such a result is crucial for proving the existence of solutions for the Cauchy problem, the existence of time periodic solutions, or when the long-time behavior of solutions is examined. Assumptions imposed in this paper are weaker than those in Section 1 of the book of \textit{P. L. Lions} [Mathematical topics in fluid mechanics. Vol. 2: Compressible models, Oxford Lecture Series in Mathematics and its Applications. 10. Oxford: Clarendon Press. xiv, 348 p. (1998; Zbl 0908.76004).].
scientific article; zbMATH DE number 1454726