In this paper, we construct global weak solutions{`a} la Hoff (i.e. intermediate regularity) for the compressible Navier-Stokes system governing a barotropic fluid with a pressure law p($rho$) = a$rho$ $gamma$ where a > 0 and $gamma$ $ge$ d/(4 -- d)) and with an anisotropic fourth order symmetric viscous tensor with smooth coefficients under the assumption that the norms of the initial data ($rho$0 -- M, u0) $in$ L 2$gamma$ T d x (H 1 (T d)) d are sufficiently small, where M denotes the total mass of the fluid. We consider periodic boundary conditions for simplicity i.e. a periodic box $Omega$ = T d with d = 2, 3 with |$Omega$| = 1. The main technical contribution of our paper is the extension of the Hoff solutions framework by relaxing the integrability needed for the initial density which is usually assumed to be L $infty$ (T d). In this way, we are able to cover the case of viscous tensors that depend on the time and space variables. Moreover, when comparing to the results known for the global weak solutions{`a} la Leray (i.e. obtained assuming only the basic energy bounds), we obtain a relaxed condition on the range of admissible adiabatic coefficients $gamma$.
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