Smoothness of Malliavin derivatives and dissipativity of solutions to two-dimensional micropolar fluid system

Micropolar fluids have microstructure and belong to a class of fluids with nonsymmetric stress tensor. We study the incompressible two-dimensional micropolar fluid system with periodic boundary condition forced by random noise that is white-in-time. In particular, we obtain a sufficient condition on the size of the angular viscosity coefficients in comparison to the vortex and kinematic viscosity coefficients so that the solution to this system is smooth in the Malliavin sense. In addition, we prove a result concerning an orthogonal projection onto a finite number of Fourier modes, taking advantage of the dissipative nature of the system.