Optimal Decay Rates For Higher-order Derivatives Of Solutions To Two Classes Of Compressible Fluid Systems

In this paper,we primarily focus on the 3D compressible micropolar fluid equation and Navier-Stokes-Poisson(NSP)equation with the potential external force.More precisely,we prove the optimal decay rates for higher-order spatial derivatives of the solutions to the Cauchy problem of these two systems in the framework of small perturbations.For the 3D compressible micropolar fluid equation,we prove the optimal decay rates for the highest order spatial derivatives of the solutions by making full use of the properties of low-frequency and high-frequency decomposition of functions and the energy estimate.The main novelty of this work is three-fold:First,for any integer N≥3,we show that the highest-order(i.e N-order)spatial derivatives of both the density and the velocity tend time-asymptotically to the corresponding equilibrium state with the L2-rate(1+t)-3/4-N/2,which is the same as that of the heat equation.Second,we prove that the N-1-order and N-order spatial derivatives of the micro-rotational velocity converge to the equilibrium state with the L2-rate(1+t)-5/4-(N-1)/2 and L2-rate(1+t)-5/4-N/2 respectively,which are faster than ones of the density and the velocity.Third,by a product,it turns out that the high-frequency part of the N-order spatial derivatives decay estimate of the density and the velocity in L2 norm is(1+t))-5/4-N/2,which are faster than ones of themselves.Particularly,these decay rates are totally new and optimal as compared to the previous results.For the 3D compressible NSP equation with the potential external force,we investigate the optimal decay rates for higher-order spatial derivatives of the solutions when the initial value is sufficiently close to the steady-state solution(ρ,u,▽φ).The main novelty of this work is that we prove the first and second order spatial derivatives of the solutions converge to the steady-state solutions with the L2-rate(1+t)-5/4,which makes full use of the properties of low-frequency and high-frequency decomposition of functions and the energy estimate.Compared with the previous results,we obtain the optimal decay rate of the first order spatial derivative of the solutions,and improve the decay rate of the second order spatial derivative of the solutions.