In these notes we review the theory of weak solutions of the Navier-Stokes equations for compressible flows in near equilibrium flow regime. The study of solutions of this type was initiated by D. Hoff [Arch. Rational Mech. Anal. 114 (1991)] with the motivation of studying the dynamics of an interface of discontinuity of solutions. We demonstrate three results. The first is the existence of weak solutions with uniformly bounded density for the typical initial-boundary value problems. The second result improves the regularity of such weak solutions in the case when the density is piecewise Hölder continuous with a jump discontinuity across a $C^{1+\alpha }$ hypersurface. We show that the flow generated by the velocity $u$ transports the discontinuity surface and preserve its regularity. In the last result, we consider the fluid flows in which the density, at time $t=0,$ is piecewise smooth and the interface of discontinuity is a surface with a corner-type singularity. We show that solutions in Hoff's regularity are well suited for the analysis of this problem and describe the phenomenon of an instantaneous change of the geometry of the interface at the point of the singularity.