In this paper, we review a recently offered notion of quasi-static evolution of the axial self-gravitating structures at large scales and the criterium to characterize the corresponding evolutionary aspects under the influence of strong curvature regimes. In doing so, we examine the axial source’s dynamic and quasi-static behavior within the parameters of various modified gravity theories. We address the formalism of these notions and their possible implications in studying the dissipative and anisotropic configuration. We initiate by considering higher-order curvature gravity. The Palatini formalism of $f(R)$ gravity is also taken into consideration to analyze the behavior of the kinematical as well as the dynamical variables of the proposed problem. The set of invariant velocities is defined to comprehend the concept of quasi-static approximation that enhances the stability of the system in contrast to the dynamic mode. It is identified that vorticity and distinct versions of the structure scalars $Y_{\mathrm{\text{I}}}$, $Y_{\mathrm{\text{II}}}$ and $Y_{\mathrm{\text{KL}}}$ play an important role in revealing the significant effects of a fluid’s anisotropy. As another example of evolution, we check the influence of Palatini-based factors on the shearing motion of the object. A comparison-based study of the physical nature of distinct curvature factors on the propagation of the axial source is exhibited. This provides the intriguing platform to grasp the notion of quasi-static evolution together with the distinct curvature factors at current time scenario. The importance of slowly evolving axially symmetric regimes will be addressed through the distinct modified gravitational context. Finally, we share a list of queries that, we believe, deserve to be addressed in the near future.

在本文中，我们回顾了最近提出的大尺度轴向自重力结构准静态演化的概念和标准，以表征强曲率状态影响下的相应演化方面。在此过程中，我们在各种修正重力理论的参数范围内检查了轴向源的动态和准静态行为。我们讨论这些概念的形式主义及其在研究耗散和各向异性配置时可能的含义。我们首先考虑高阶曲率引力。还考虑了 $f(R)$ 重力的帕拉蒂尼形式主义来分析所提出问题的运动学行为以及动力学变量。定义不变速度集是为了理解准静态近似的概念，与动态模式相比，准静态近似增强了系统的稳定性。研究发现，涡度和不同版本的结构标量 $Y_{\mathrm{\text{I}}}$ 、 $Y_{\mathrm{\text{II}}}$ 和 $Y_{\mathrm{\text{KL}}}$ 在揭示流体各向异性的显着影响方面发挥着重要作用。作为进化的另一个例子，我们检查基于帕拉蒂尼的因素对物体剪切运动的影响。展示了对轴向源传播的不同曲率因子的物理性质的基于比较的研究。这提供了一个有趣的平台来掌握准静态演化的概念以及当前场景下不同的曲率因子。缓慢演化的轴对称体系的重要性将通过独特的修改引力背景来解决。最后，我们分享了一系列问题，我们认为这些问题值得在不久的将来得到解决。