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

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