Global Dynamics of 3D Compressible Viscous and Heat-Conducting Micropolar Fluids with Vacuum at Infinity

Abstract

In this paper, we are concerned with the Cauchy problem of 3D viscous and heat-conducting micropolar fluids with far field vacuum. Compared with the case of non-vacuum at infinity (Huang and Li in Arch Ration Mech Anal 227:995–1059, 2018; Huang et al. in J Math Fluid Mech 23(1):50, 2021), due to \((\rho (t, x), \theta (t, x))\rightarrow (0, 0)\) as \(|x|\rightarrow \infty \), we don’t have useful energy equality (or inequality), which is very important to establish a priori estimates in Huang and Li (Arch Ration Mech Anal 227:995–1059, 2018) and Huang et al. (J Math Fluid Mech 23(1):50, 2021). Thus, a new assumption of a priori estimates and more complicated calculations will be needed. On the other hand, we need to deal with some strong nonlinear terms which come from the interactions of velocity and micro-rotation velocity. Finally, we show that the global existence and uniqueness of strong solutions provided that the initial energy is suitably small. In particular, large-time behavior and a exponential decay rate of the strong solution are obtained, which generalizes the incompressible case (Ye in Dicret Contin Dyn Syst Ser B 24:6725–6743, 2019) to the full compressible case.