It is proved that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish the global-in-time error estimates between the smooth solutions to the partially dissipative hyperbolic systems and those to the isotropic parabolic limiting systems in a three dimensional torus, rather than in the one dimensional whole space (Appl. Anal.100(5) (2021) 1079–1095). This avoids the condition raised for the strong connection between the flux and the source term and make the result obtained more generalized. In the proof, we provide a similar stream function technique which is valid for the three dimensional periodic case. Similar method is provided for the one-dimensional periodic case. As applications of the results, we give several examples arising from physical models at the end of the paper.
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