This paper is devoted to analyzing the solution framework of the gas-dynamic equations for a three-dimensional unbounded homentropic sheared flow using the Lie group approach. An extensive symmetry analysis of the system of governing PDEs is performed to decrease the number of independent variables. The classification of inequivalent subalgebras into an optimal set called the optimal set of subalgebras, is essential. We have constructed the one-dimensional, two-dimensional, and three-dimensional optimal set of subalgebras for the model PDEs. The three-dimensional optimal set of subalgebras is very useful as it directly transforms the system of governing PDEs into a system of ODEs. Consequently, we obtain closed-form exact solutions of the governing model. Alternatively, the two-dimensional optimal subalgebras yield some solution ansatz, which describes various physical modes such as Kelvin mode and certain other modes and their typical combinations. The three-dimensional normal mode approach is justified using a combined ansatz in the limiting case. Moreover, we acquire the conserved quantities corresponding to the governing model by performing the conservation laws multiplier technique.

中文翻译：

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三维气体动力学方程的李对称性和采用某些模态方法的最优分类

本文致力于使用李群方法分析三维无界垂线剪切流气体动力学方程的解框架。对偏微分方程的控制系统进行了广泛的对称性分析，以减少自变量的数量。将不等价子代数分类为称为最优子代数集的最优集合是至关重要的。我们为模型偏微分方程构建了一维、二维和三维最优子代数集。三维最优子代数集非常有用，因为它将控制偏微分方程组直接转换为常微分方程组。因此，我们获得了控制模型的闭式精确解。或者，二维最优子代数产生一些解 ansatz，它描述各种物理模式，例如开尔文模式和某些其他模式及其典型组合。在极限情况下使用组合模拟来证明三维简正模方法是合理的。此外，我们通过执行守恒定律乘法技术获得了与控制模型相对应的守恒量。