Global smooth solution to the n-dimensional liquid crystal equations with fractional dissipation

Abstract

Abstract In this article, we focus on the global regularity of n-dimensional liquid crystal equations with fractional dissipation terms ( − Δ ) α u {\left(-\Delta )}^{\alpha }u and ( − Δ ) β d {\left(-\Delta )}^{\beta }d . We show that the equations have a unique global smooth solution if α ≥ 1 2 + n 4 \alpha \ge \frac{1}{2}+\frac{n}{4} and β ≥ 1 2 + n 4 \beta \ge \frac{1}{2}+\frac{n}{4} .