Large-bulk-viscosity limit for two-dimensional compressible micropolar fluids with vacuum

Abstract

We consider the Cauchy problem for the two-dimensional compressible micropolar fluid system on the torus $\mathbb T^2$. The initial density is bounded and nonnegative and may contain vacuum, while the initial velocity and microrotation belong to $H^1(\mathbb T^2)$. For sufficiently large bulk viscosity $ν$, we prove the global-in-time existence of solutions without a smallness assumption on the $H^1$ size of the data, provided that the initial compressible component is well prepared in the sense that $\sqrtν\|\operatorname{div}u_0\|_{L^2}$ is uniformly bounded. The proof combines the effective viscous flux with a coupled energy scheme that exploits the vorticity--microrotation dissipation at both the basic and material-derivative levels. For the linear pressure law, we also prove uniqueness in the stated solution class by means of a coupled backward parabolic problem, logarithmic interpolation, and Osgood's lemma. For fixed initial data with $\operatorname{div}u_0=0$, we also justify the limit $ν\to\infty$: every sequence of the corresponding solutions admits a subsequence converging to a solution of the two-dimensional inhomogeneous incompressible micropolar system.