三维可压缩向列型液晶系统解的衰减率的改进

An Improvement on Decay Rates for Solutions to the 3D System of Compressible Nematic Liquid Crystal

摘要: 本文的主要目的在于提高三维可压缩向列型液晶系统解的最高阶(S阶)空间导数的衰减率。如果初值的$H^S\left(S\ge 3\right)$ 和范数都是有界的，并且其$H^3$ 范数足够小，则应用纯能量法，我们给出了解的最高阶空间导数$L^2$ 范数的最优衰减率为${\left(1+t\right)}_{}^{-\left(\frac{S}{2}+\frac{\alpha }{2}\right)}$ ，而在魏，李和姚的研究中其衰减率仅为${\left(1+t\right)}_{}^{-\left(\frac{S-1}{2}+\frac{\alpha }{2}\right)}$ 。

Abstract: Abstract: The major objective of this thesis lies in improving the decay rates for the highest order (S-order) of spatial derivative of the solutions to the 3D system of compressible nematic liquid crystal. If the norms of both $H^S\left(S\ge 3\right)$ and for the initial value are bounded, as well as the norm of $H^3$ for that is small enough, with applying pure energy method, we give that the optimal decay rates for the highest order of spatial derivative of the solutions in norm of $L^2$ are ${\left(1+t\right)}_{}^{-\left(\frac{S}{2}+\frac{\alpha }{2}\right)}$ , while that is just ${\left(1+t\right)}_{}^{-\left(\frac{S-1}{2}+\frac{\alpha }{2}\right)}$ in Wei, Li and Yao’s study.