It is well-known that the Cauchy problem of the compressible liquid crystals flow admits a unique global-in-time solution, belonging to $C^0(R_0^{+},H^l\left(R^3\right))$ with $l⩾3$; moreover the lower-order or higher-order derivative of solution enjoys the same decay-in-time rate as well as the linear solution (i.e., the solution of the corresponding linear problem). In this paper we further prove that highest-order derivative of the unique solution also enjoys the same decay-in-time rate as well as the linear solution by developing new analytical skills. In other words, the optimal decay rate for the highest-order derivative of solution can be also obtained.

中文翻译：

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向列液晶可压缩流方程的衰减

众所周知，可压缩液晶流的柯西问题允许一个唯一的全局时间解决方案，该解决方案属于 $C^0（{\mathrm{[R}}_0^{+}，H^{升}（{\mathrm{[R}}^3））$ 和 $升⩾3$; 此外，解的低阶或高阶导数与线性解（即相应线性问题的解）具有相同的时间衰减率。在本文中，我们通过开发新的分析技巧进一步证明了唯一解的高阶导数也具有相同的时间衰减率和线性解。换句话说，还可以获得溶液的最高阶导数的最佳衰减率。