Iterative method for mass diffusion model with density dependent viscosity

Abstract

The aim of this work is to study the existence of strong solutions for $3D$ fluids models with mass diffusion (also called Kazhikhov-Smagulov type system) assuming density dependent viscosity. The considered system represents a pollutant model. <br>&nbsp; &nbsp We use an iterative method to approach regular solutions. Moreover, some convergence rates are obtained, depending on weak, strong and more regular norms. This work extend to [1], where this technique has been used for the model with constant viscosity. <br>&nbsp; &nbsp The model has a diffusive operator $-\lambda$div$(\rho (\nabla v +\nabla v^t))$ with $v$ the velocity field, which not allows us to use direct Stokes regularity (as has been done in [1]. Thus, it becomes more difficult to obtain the $H^2\times H^1$ and $H^3\times H^2$ regularity for the velocity-pressure pair $(v,p)$. The key is to use a new regularity result for a Stokes type problem with $\rho\Delta v$ as diffusion term.