On the existence and regularity of the solutions to the incompressible Navier-Stokes equations in presence of mass diffusion
This paper is devoted to the study of the incompressible Navier-Stokes equations with mass diffusion in a bounded domain in $R^3$ with $C^3$ boundary. We prove the existence of weak solutions, in the large, and the behavior of the solutions as the diffusion parameter $\lambda \rightarrow 0.$ Moreover, the existence of $L^2$-strong solution, in the small, and in the large for small data, is proved. Asymptotic regularity (the regularity after a finite period) of a weak solution is studied. Finally, using the Dore-Venni theory, the problem of the $L^q$-maximal regularity is investigated.