On some large global solutions to the incompressible inhomogeneous nematic liquid crystal flows

ABSTRACT

In this paper, we investigate the Cauchy problem for the incompressible inhomogeneous nematic liquid crystal flows in ${\mathbb{R}}^n$ with initial data in some critical Besov spaces. Under the assumptions that initial density is close to the equilibrium state and the smallness on the initial direction field and nonlinear smallness on the linearized velocity, we obtain the global existence of solutions by fully using the Fourier localization technique and the advantage of weighted function generated by heat kernel. The meanings of this kind of initial data are that we can provide an example of large highly oscillating data satisfying that nonlinear smallness assumption, despite each component of the initial data could be arbitrarily large.

COMMUNICATED BY:

• George Jaiani

KEYWORDS:

• Global solutions
• nematic liquid crystal flows
• besov spaces

MATHEMATICS SUBJECT CLASSIFICATIONS (2010):

• 35Q35
• 35B65
• 76D03

Disclosure statement

This work was partially supported by NNSFC (Nos. 11601533, 11671407).

Additional information

Funding

This work was partially supported by NNSFC (Nos. 11601533, 11671407).