Optimal decay rates for higher-order derivatives of solutions to the 3D compressible micropolar fluids system

Abstract

We investigate optimal decay rates for higher-order spatial derivatives of solutions to the 3D Cauchy problem of the compressible micropolar fluids system, and the main novelty of this work is three-fold: First, for any integer N≥3, we show that N-order spatial derivatives of the density and the velocity converge to zero at the L2-rate (1+t)−(34+N2), which is the same as that of the heat equation. Second, we prove that N−1-order and N-order spatial derivatives of the micro-rotational velocity converge to zero at the L2-rate (1+t)−(54+N−12) and L2-rate (1+t)−(54+N2) respectively, which are faster than ones of the density and the velocity. Third, by a product, we also show that the high-frequency part of the N-order spatial derivatives of the density and the velocity converge to zero at the L2-rate (1+t)−(54+N2), which are faster than ones of themselves. Particularly, these decay rates are totally new as compared to Liu-Zhang (2016) [15] and Tong-Pan-Tan (2021) [32].