On the temporal decay of solutions to the two-dimensional nematic liquid crystal flows

TL;DR

This paper investigates how the energy of solutions to 2D nematic liquid crystal flows diminishes over time, establishing conditions for decay and determining the exact decay rates.

Contribution

It provides new results on the decay behavior and rates of weak solutions to 2D nematic liquid crystal flows under certain initial conditions.

Findings

Energy norm of solutions tends to zero as time approaches infinity.

Exact decay rates of the energy norm are established.

Decay behavior depends on initial data conditions.

Abstract

We consider the temporal decay estimates for weak solutions to the two-dimensional nematic liquid crystal flows, and we show that the energy norm of a global weak solution has non-uniform decay \begin{align*} \|u(t)\|_{L^{2}}+\|\nabla d(t)\|_{L^{2}}\rightarrow 0\quad \text{ as } t\rightarrow \infty, \end{align*} under suitable conditions on the initial data. We also show the exact rate of the decay (uniform decay) of the energy norm of the global weak solution.