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

TL;DR

This paper determines the optimal decay rates for large solutions to the compressible nematic liquid crystal equations, showing they match heat equation decay rates and are optimal without decay loss for highest derivatives.

Contribution

It establishes the optimal decay rates for large solutions of the equations, including second order derivatives, and proves these rates are sharp and match lower bounds.

Findings

Decay rates of $( ho-1, u, abla d)$ are $(1+t)^{-5/4}$ and $(1+t)^{-7/4}$ for first and second derivatives.

No decay loss occurs for the highest-order derivatives despite large initial perturbations.

Decay rates of solutions and derivatives are proven to be optimal, matching lower bounds.

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 $(1+t)^{-\frac54}$ and $L^2-$rate $(1+t)^{-\frac74}$ 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.