On temporal decay of solution to the three‐dimensional compressible flow of nematic liquid crystal in Besov space

Abstract

In this paper, we investigate large time behavior of global‐in‐time strong solution to the three‐dimensional compressible flow of nematic liquid crystal with low regularity assumptions on initial datum. More precisely, we show that the negative Besov space ‐norms (s ≥ 0) of solution are preserved along time evolution; by using this fact together with the conventional energy estimates in Besov space framework and the interpolation inequalities, we establish that, for the initial perturbation just small in homogeneous Besov space , the global‐in‐time strong solution to the Cauchy problem of the compressible flow of nematic liquid crystal has the following optimal temporal decay rate: urn:x-wiley:mma:media:mma5176:mma5176-math-0004 provided that we further assume that still belongs to . Here, is a constant unite vector. To illustrate our methods clearly, we also revisit the optimal temporal decay of solutions to the heat equation in the framework of Besov space.