Optimal time decay of the compressible micropolar fluids

This paper primarily studies the large-time behavior of solutions to the Cauchy problem on the compressible micropolar fluid system which is a generalization of the classical Navier–Stokes system. The asymptotic stability of the steady state with the strictly positive constant density, the vanishing velocity, and micro-rotational velocity is established under small perturbation in regular Sobolev space. Moreover, it turns out that both the density and the velocity tend time-asymptotically to the corresponding equilibrium state with rate (1+t)−3/4(1+t)−3/4 in L2L2 and the micro-rotational velocity also tends to the equilibrium state with the faster rate (1+t)−5/4(1+t)−5/4 in L2L2 norm. The proof is based on the spectrum analysis and time-weighted energy estimate.