On the existence and regularity of the solutions to the incompressible Navier-Stokes equations in presence of mass diffusion

Abstract. This paper is devoted to the study of the incompressible Navier-Stokes equations with mass diffusion in a bounded domain in R3 with C3- boundary. We prove the existence of weak solutions, in the large, and the behavior of the solutions as the diffusion parameter ! 0. Moreover, the existence of L2-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 Lq-maximal regularity is investigated.