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.

中文翻译：

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从平衡律系统到各向同性抛物线系统的收敛速度，一个周期情况

证明了当初始数据是平滑的并且足够接近恒定平衡状态时，部分耗散的双曲型系统在较慢的时间尺度上全局地收敛到抛物线型系统。基于此结果，我们建立了部分耗散双曲系统的光滑解与各向同性抛物线极限系统的光滑解之间的全局时间误差估计，该估计位于三维环面中，而不是在一维整个空间中（Appl。解剖学杂志100（5）（2021）1079-1095）。这避免了为通量和源项之间的强联系而提出的条件，并使获得的结果更加概括。在证明中，我们提供了一种类似的流函数技术，该技术对三维周期情况有效。对于一维周期性情况，提供了类似的方法。