Existence of solutions to incompressible biaxial nematic liquid crystals flows

Abstract

The recent paper considers the incompressible hydrodynamic flow of biaxial nematic liquid crystal in dimensions N with N = 2 or 3. The unique local solutions are firstly derived for ${\rho }_0\ge 0$ in dimensions 2 and 3. Then the unique global solution is obtained for $0<M_1\le {\rho }_0$ and small initial energy: ${\int }_{\mathrm{\Omega }}({\rho }_0|v_0{|}^2+|\mathrm{\nabla }{n}_0{|}^2+|\mathrm{\nabla }{m}_0{|}^2)\mathrm{d}x$.

Keywords:

• Liquid crystal
• biaxial nematic
• incompressible flow
• global solution

2010 Mathematics Subject Classifications:

• 35A01
• 35A09
• 35Q35
• 76D03

Acknowledgments

The second author is partially supported by NSF of China (No. 11571117) and by the Science and Technology Program of Guangzhou (No. 201707010040).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author is partially supported by the NSF of China [grant number 11571117] and by the Science and Technology Program of Guangzhou [grant number 201707010040].