论文标题
惯性的不平等和惯性Qian-sheng模型的夜间液晶模型
Entropy inequality and energy dissipation of inertial Qian-Sheng model for nematic liquid crystals
论文作者
论文摘要
对于$ q $ tensor框架中列液晶的惯性Qian-sheng模型，我们说明了熵不平等和能量耗散在我们采用能量方法时的平滑溶液中良好型的作用。我们首先从熵不平等中得出系数要求，并指出熵不平等不足以保证能量耗散。然后，我们引入了一种新的条件（H），以确保能量耗散。我们证明，当遵守熵不等式和条件（H）时，局部时间平滑解决方案存在用于大型初始数据。否则，我们只能获取小数据本地解决方案。此外，为了及时扩展全球溶液并获得溶液的衰减，我们至少需要两个条件之一：熵不等式或$ \tildeμ_2=μ_2$，这大大扩大了先前工作中系数的范围。
For the inertial Qian-Sheng model of nematic liquid crystals in the $Q$-tensor framework, we illustrate the roles played by the entropy inequality and energy dissipation in the well-posedness of smooth solutions when we employ energy method. We first derive the coefficients requirements from the entropy inequality, and point out the entropy inequality is insufficient to guarantee energy dissipation. We then introduce a novel Condition (H) which ensures the energy dissipation. We prove that when both the entropy inequality and Condition (H) are obeyed, the local in time smooth solutions exist for large initial data. Otherwise, we can only obtain small data local solutions. Furthermore, to extend the solutions globally in time and obtain the decay of solutions, we require at least one of the two conditions: entropy inequality, or $\tildeμ_2= μ_2$, which significantly enlarge the range of the coefficients in previous works.
