Global Existence and Large-Time Behavior for 3D Full Compressible Magneto-Micropolar System Without Heat Conductivity

The system of full compressible magneto-micropolar flows is discussed in 3D bounded domains with slip boundary conditions. Based on the energy method, after establishing some key a priori exponential decay-in-times rates of the strong solutions, we obtain both the global existence and exponential stability of strong solutions. In particular, it should be pointed out that the estimates of ${\parallel (\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w})\parallel }_{L^2}$ and ${\parallel (\mathrm{div}\mathbf{u},\mathrm{div}\mathbf{w})\parallel }_{L^2}$ are established separately, which implies that the growth rate of $(\mathrm{div}\mathbf{u},\mathrm{div}\mathbf{w})$ in $L^2$ are faster than that of $(\mathrm{curl}\mathbf{u},\mathrm{curl}\mathbf{w})$ under the condition that the diameter of the domain is suitably large. Compared with previous works, we no longer consider the pressure P as $\rho \theta$, but as variable in $(x,t)$, and directly deal with ${\parallel \nabla P\parallel }_{L^2}$. Based on slip boundary conditions, we established the $L^p$-norm for the gradient of effective viscous flux, and the term ${\parallel \nabla P\parallel }_{L^2}$ can be controlled by $\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}$. Through precise calculations, we found that $\parallel (\nabla {\mathbf{u}}_t,\nabla {\mathbf{w}}_t,\nabla {\mathbf{b}}_t){\parallel }_{L^2}$ is dependent on ${\parallel \nabla P\parallel }_{L^2}$. Therefore, the smallness condition we propose does not depend on the $L^r$-norm of the density gradient, which means that density can contain large oscillations.