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

Abstract

Key words Nematic liquid crystal flow, temporal decay, Fourier splitting method MSC (2010) 76A15, 35B65, 35Q35We 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 decayunder 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.

On decay estimates of the 3D nematic liquid crystal flows in critical Besov spaces

“…For the issue of the large time behavior of solutions, Du and Wang and Liu proved that the small global strong solution obtained by is arbitrary spacetime regularity and is algebraically decay as time goes to infinity. Recently, base on the Fourier splitting method introduced by Schonbek for the incompressible Navier–Stokes equations, Liu showed that the L 2 ‐norm decay of weak solutions to the Cauchy problem of the two‐dimensional incompressible version of – with initial date with 1≤ p < 2 is Liu and Xu established that for initial data with m ≥3 and satisfies for sufficiently small enough η > 0, then system – has a unique global‐in‐time smooth solution ( u , d ) and satisfies that …”

“…Here, M is a positive constant. In this paper, motivated by the papers , we intend to consider decay rates for the strong solutions of system – to the constant stationary solution as t → ∞ when the initial perturbation is sufficiently small in critical Besov spaces with 1 < p < ∞ and our main result is as follows:Theorem Let 1 < p < ∞ , with div u 0 =0, and , such that for some sufficiently small η > 0. Then there exists a unique global‐in‐time strong solution ( u , d ) to system – satisfying …”

“…For the issue of the large time behavior of solutions, Du and Wang [21] and Liu [22] proved that the small global strong solution obtained by [7] is arbitrary spacetime regularity and is algebraically decay as time goes to infinity. Recently, base on the Fourier splitting method introduced by Schonbek [23,24] for the incompressible Navier-Stokes equations, Liu [25] showed that the L 2 -norm decay of weak solutions to the Cauchy problem of the two-dimensional incompressible version of (1.1)-(1.4) with initial date .u 0 , rd 0 / 2…”

“…Here, M is a positive constant. In this paper, motivated by the papers [23][24][25][26], we intend to consider decay rates for the strong solutions of system (1.1)-(1.4) to the constant stationary solution…”

Space-time decay estimates of solutions to liquid crystal system in <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^3$\end{document}</tex-math></inline-formula>

“…In the papers of Lin, Lin and Wang [22], Hong and Xin [13], the authors studied the existence of global Leray-Hopf type weak solutions for the above system in 2D case. Liu [25], Liu and Xu [24] considered the decay estimates for the above system. For the other results, we suggest the reader to Wang [32], Lin and Liu [23], Fan [8], Qian [27] and the references therein.…”