Optimal Decay Rate for the Compressible Flow of Liquid Crystals in $$L^p$$ L p Type Critical Spaces

Abstract

The purpose of this paper is to establish the optimal time decay rate of solutions for the compressible flow of nematic liquid crystals in $$L^p$$ type critical framework and any dimension $$N\ge 2$$ . Concretely, if the low frequencies of the initial data are in e.g. $$L^{p/2}(\mathbb {R}^N)$$ , we could obtain that the $$L^p$$ norm of the critical global solution $$(\rho -1,\mathbf{u})$$ decays like $$t^{-\frac{N}{p}+ \frac{N}{4}}$$ , while the $$L^p$$ norm of $$\mathbf{d}-\hat{\mathbf{d}}$$ decays like $$t^{-\frac{N}{2p}}$$ for $$t\rightarrow \infty $$ , exactly as shown in the work by Gao et al. (J Differ Equ 261:2334–2383, 2016) when $$p=2$$ and $$N=3$$ , for solutions with high Sobolev regularity. As a by-product, we derive an accurate description of the time decay rates, not only for Lebesgue spaces, but also for a family of Besov norms with negative or nonnegative regularity exponents, which improves the decay results in high Sobolev regularity. The main tools used here are the Littlewood–Paley theory and refined time weighted inequalities in Fourier space.