Global solution to the three-dimensional compressible flow of liquid crystals (Q2870585)

Liquid crystals are substances that exhibit properties between those of a conventional liquid and those of a solid crystal. The three-dimensional flow of nematic liquid crystal can be governed by the following system of partial differential equations: NEWLINE\[NEWLINE{\partial\rho\over\partial t}+ \text{div}(\rho u)= 0,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\partial(\rho u)\over\partial t}+ \text{div}(\rho u\otimes u)-\mu\Delta u- (\mu+\lambda)\nabla\text{\,div\,}u+\nabla P(\rho)\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE=-\xi\text{\,div}(\nabla d\odot\nabla d-{1\over 2}|\nabla d|^2),NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\partial d\over\partial t}+ u\cdot\nabla d= \theta(\Delta d+ |\nabla d|^2d),\tag{3}NEWLINE\]NEWLINE where \(\rho\in \mathbb R\) is the density function of the fluid, \(u\in \mathbb R^3\) is the velocity, and \(d\in S^2\) represents the director field for the averaged macroscopic molecular orientations. The scalar function \(P\in \mathbb R\) is the pressure, which is an increasing and convex functon in \(\rho\). All these functions depend on the spatial variable \(x= (x_1, x_2, x_3)\in \mathbb R^3\) and the time variable \(t>0\). The constants \(\mu\) and \(\nu\) are shear viscosity and the bulk viscosity coefficiens of the field, respectively, that satisfy the physical assumption \(\mu> 0\), \(2\mu+ 3\lambda\geq 0\). The contants \(\xi\), \(\theta\) stand for the competition between kinetic energy and potential energy. The symbol \(\otimes\) denotes the Kronecker tensor product, and \(\odot\) denotes a matrix whose \(ij\)th entry is \(\partial_{x_i}\partial_{x_j}\). \(I\) is a \(3\times 3\) identity matrix. The initial data are given NEWLINE\[NEWLINE\rho|_{t=0}= \rho_0(x),\;u|_{t=0}= u_0,\;d|_{t=0}= d_0,\tag{4}NEWLINE\]NEWLINE with \(d_0\in S^2\).NEWLINENEWLINEThe main result of this paper is as follows:NEWLINENEWLINE Theorem 0.1 Let \(d\in S^2\). If there exist two positive constants \(\eta\) and \(\Gamma\) such that if \(\mu_0-1\in B^{{1\over 2},{3\over 2}}\), \(u_0\in B^{{1\over 2}}\), \(u_0\in d_0-\widehat d\in B^{{1\over 2},{3\over 2}}\) satisfying NEWLINE\[NEWLINE\|\rho_0- 1\|_{B^{{1\over 2},{3\over 2}}}\leq\eta,NEWLINE\]NEWLINE then system (1)--(4) has a unique global strong solution \((\rho,u,d)\) with \((\rho-1, u,d-\widehat d)\in B^{{1\over 2}}\) satisfying NEWLINE\[NEWLINE\|(\rho- 1, u,d-\widehat d)\|_{B^{{1\over 2}}}\leq \Gamma(\|\rho_0- 1\|_{B^{{1\over 2},{3\over 2}}}+\| u_0\|_{B^{{1\over 2}}}+\| d_0-\widehat d\|_{B^{{1\over 2},{3\over 2}}}).NEWLINE\]