Micropolar fluids with initial angular velocities in non-homogeneous Sobolev spaces of order $-1\mathrm{/}2$

In this paper, we investigate fractional energy methods for Micropolar fluids, starting with an initial angular velocity of negative Sobolev regularity. For the initial angular velocity assumption, we consider a non-homogeneous Sobolev norm of negative order. The regularity -1/2 studied here corresponds to the critical scaling of a simplified associated system, and the general framework can also be applied to the Boussinesq system with viscosity. Since our approach differs from those based on mild solutions and does not rely on a projected system, this work provides new tools for studying the Caffarelli-Kohn-Nirenberg theory of singularities in coupled variables within the Navier-Stokes equations.