Vanishing Angular Viscosity Limit For Micropolar Fluid Model In $\mathbb{R}_+^2$: Boundary Layer And Optimal Convergence Rate

Abstract: We consider the initial-boundary value problem for the incompressible two-dimensional micropolar fluid model with angular viscosity in the upper half-plane. This model describes the motion of viscous fluids with microstructure. The global well-posedness of strong solutions for this problem with positive angular viscosity can be established via the standard energy method, as presented in the classical monograph [\L kaszewicz, {\it Micropolar fluids: Theory and applications.} Birkh\"auser, 1999]. Corresponding results for the zero angular viscosity case were established recently in [Liu, Wang, {\it Commun. Math. Sci.} 16 (2018), no. 8, 2147-2165]. However, the link between the positive angular viscosity model (the full diffusive system) and the zero angular viscosity model (the partially diffusive system) via the vanishing diffusion limit remains unknown. In this work, we first construct Prandtl-type boundary layer profiles. We then provide a rigorous justification for the vanishing angular viscosity limit of global strong solutions, without imposing smallness assumptions on the initial data. Our analysis reveals the emergence of a strong boundary layer in the angular velocity field (micro-rotation velocity of the fluid particles) during this vanishing viscosity process. Moreover, we also obtain the optimal $L^\infty$ convergence rate as the angular viscosity tends to zero. Our approach combines anisotropic Sobolev spaces with careful energy estimates to address the nonlinear interaction between the velocity and angular velocity fields.