广义向列型液晶方程的能量等式

Energy Equality of the Generalized Nematic Liquid Crystal Equations

摘要: 针对广义向列型液晶方程，研究其弱解满足能量等式的充分条件。采用磨光函数技术，通过对非线性项进行精细估计，在分数阶耗散参数介于1与5/2之间时，证明若速度场与分子指向向量的梯度满足适当时空可积性，则能量等式成立。结果表明，该条件可保证系统在三维全空间中能量守恒，并涵盖了经典Navier-Stokes方程中的已知结论，为广义液晶流动模型的能量守恒性提供了理论依据。

Abstract: For the generalized nematic liquid crystal equations, we study sufficient conditions under which weak solutions satisfy the energy equality. Using the mollifier technique and performing delicate estimates on the nonlinear terms, it is proved that when the fractional dissipation parameter lies between 1 and 5/2, the energy equality holds provided that the velocity field and the gradient of the molecular orientation vector satisfy appropriate space-time integrability conditions. The results show that these conditions guarantee energy conservation in the three-dimensional whole space, and they cover known conclusions for the classical Navier-Stokes equations, providing a theoretical basis for the energy identity of generalized liquid crystal flow models.