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The first aim of this paper is to develop a layer potential theory in$L_2$-based weighted Sobolev spaces on Lipschitz bounded and exterior domainsof ${\mathbb R}^n$, $n\geq 3$, for the anisotropic Stokes system with$L_{\infty }$ viscosity coefficient tensor satisfying an ellipticity conditionfor symmetric matrices. To do this, we explore equivalent mixed variationalformulations and prove the well-posedness of some transmission problems for theanisotropic Stokes system in Lipschitz domains of ${\mathbb R}^n$, with thegiven data in $L_2$-based weighted Sobolev spaces. These results are used todefine the Newtonian and layer potentials and to obtain their properties. Thenwe analyze well-posedness of the exterior Dirichlet, Neumann and mixed problemsfor the Stokes system with $L_{\infty }$ symmetrically elliptic coefficienttensor. Solutions of some of these problems are also represented in terms ofthe anisotropic Stokes Newtonian and layer potentials. Finally, we prove theexistence of a weak solution for a transmission problem in complementaryLipschitz domains in ${\mathbb R}^3$ for the anisotropic Navier-Stokes systemwith general data in $L_2$-based weighted Sobolev spaces. The analysis relieson an existence result for a Dirichlet problem for the anisotropicNavier-Stokes system in a family of bounded domains, and on the Leray-Schauderfixed point theorem.

Variational approach for layer potentials of the Stokes system with  $L_{\infty }$ symmetrically elliptic coefficient tensor and applications to  Stokes and Navier-Stokes boundary problems