Weak solutions of the two-dimensional incompressible inhomogeneous Navier-Stokes equations in the presence of variable odd viscosity

We consider the two-dimensional incompressible inhomogeneous Navier-Stokes equations with odd viscosity, where the shear and the odd viscosity coefficients depend continuously on the unknown density function. We establish the existence of weak solutions in both the evolutionary and stationary cases. Furthermore, we investigate the limit of the weak solutions as the odd viscosity coefficient converges to a constant. Lastly, we consider examples of stationary solutions for parallel, concentric and radial flows.