Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier-Stokes systems with $L_{\infty}$ strongly elliptic coefficient tensor

Abstract

We obtain well-posedness results in $L_p$-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier-Stokes systems with $L_{\infty}$ strongly elliptic coefficient tensor, in complementary Lipschitz domains of ${\mathbb R}^n$, $n\ge 3$. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces two linear transmission problems for the anisotropic Stokes system to equivalent mixed variational formulations with data in $L_p$-based weighted Sobolev and Besov spaces. We show that such a mixed variational formulation is well-posed in the space ${\mathcal H}^1_{p}({\mathbb R}^n)^n\times L_p({\mathbb R}^n)$, {$n\geq 3$}, for any $p$ in an open interval containing $2$. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system. Various mapping properties of these operators are also obtained. The potentials are employed to show the well-posedness of some linear transmission problems, which then is combined with a fixed point theorem in order to show the well-posedness of the nonlinear transmission problem for the anisotropic Stokes and Navier-Stokes systems in $L_p$-based weighted Sobolev spaces, whenever the given data are small enough.