A note on a global strong solution to the 2D Cauchy problem of density-dependent nematic liquid crystal flows with vacuum (Q1747406)

In this note, the author considers the following simplified version of the Ericksen-Leslie model used to describe the evolution of density-dependent nematic liquid crystals in the whole plane \(\mathbb{R}^2\), \[ \rho_t+\operatorname{div}(\rho\mathbf{u})=0, \tag{1} \] \[ (\rho\mathbf{u})_t+\operatorname{div}(\rho\mathbf{u}\otimes\mathbf{u})-{\triangle}\mathbf{u}+\nabla P=-\operatorname{div}(\nabla\mathbf{d}\odot\nabla\mathbf{d}), \tag{2} \] \[ \mathbf{d}_t+(\mathbf{u}\cdot{\nabla})\mathbf{d}={\triangle}\mathbf{d}+|{\nabla}\mathbf{d}|^2\mathbf{d}, \tag{3} \] \[ \operatorname{div}\mathbf{u}=0,\qquad |\mathbf{d}|=1. \tag{4} \] System (1)--(4) is supplemented with the following initial conditions in \(\mathbb{R}^2\), \[ \rho=\rho_0,\qquad \rho\mathbf{u}=\rho_0\mathbf{u}_0,\qquad \mathbf{d}=\mathbf{d}_0,\qquad |\mathbf{d}_0|=1. \tag{5} \] The notation used here is the following: \(\rho\) is density, \(\mathbf{u}=(u_1,u_2)\) is the velocity field, \(P\) is the pressure, and \(\mathbf{d}=(d_1,d_2,d_3)\) is the (averaged) macroscopic/continuum molecule orientation of the nematic liquid crystal flow. In Theorem~1.2, the author shows that the smallness condition on the initial data needed to prove the global-in-time existence of a unique strong solution to the Cauchy problem (1)--(5), established in [\textit{L. Li} et al., Nonlinearity 30, No. 11, 4062--4088 (2017; Zbl 1386.76016)], can be sharpened in the following sense, \[ \|{\nabla}\mathbf{d}_0\|_2^2\exp\left[2\Lambda^2\left(\|\sqrt{\rho_0}\mathbf{u}_0\|_2^2+\|{\nabla}\mathbf{d}_0\|_2^2\right)\right]\leq\frac{1}{8\Lambda^2}, \] where \(\Lambda\) is the best possible constant in the 2D Ladyzhenskaya inequality for \({\nabla}\mathbf{d}_0\).