The Optimal Decay Rate of Strong Solution for the Compressible Nematic Liquid Crystal Equations with Large Initial Data

Abstract

This paper is devoted to establishing the optimal decay rate of the global large solution to compressible nematic liquid crystal equations when the initial perturbation is large and belongs to $L^1({\mathbb{R}}^3)\cap H^2({\mathbb{R}}^3)$. More precisely, we show that the first and second order spatial derivatives of large solution $(\rho -1,u,\nabla d)(t)$ converges to zero at the $L^2-$rate $(1+t{)}^{-\frac{5}{4}}$ and $L^2-$rate $(1+t{)}^{-\frac{7}{4}}$ respectively, which are optimal in the sense that they coincide with the decay rates of solution to the heat equation. Thus, we establish optimal decay rate for the second order derivative of global large solution studied in [12,18] since the compressible nematic liquid crystal flow becomes the compressible Navier-Stokes equations when the director is a constant vector. It is worth noticing that there is no decay loss for the highest-order spatial derivative of solution although the associated initial perturbation is large. Moreover, we also establish the lower bound of decay rates of $(\rho -1,u,\nabla d)(t)$ itself and its spatial derivative, which coincide with the upper one. Therefore, the decay rates of global large solution ${\nabla }^2(\rho -1,u,\nabla d)(t)$ $(k=0,1,2)$ are actually optimal.

References in corpus (4)

• Well-posedness for the heat flow of harmonic maps and the liquid crystals flow with rough initial data
• Blow up criterion for compressible nematic liquid crystal flows in dimension three
• On Multi-dimensional Compressible Flows of Nematic Liquid Crystals with Large Initial Energy in a Bounded Domain
• The Optimal Decay Rate of Strong Solution for the Compressible Navier-Stokes Equations with Large Initial Data

Cited by in corpus (1)

• Optimal decay of compressible Navier-Stokes equations with or without potential force