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Potentials and transmission problems in weighted Sobolev spaces for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor

, , and . Complex Variables and Elliptic Equations, 65 (1): 109-140 (2020)
DOI: 10.1080/17476933.2019.1631293

Abstract

We obtain well-posedness results in Lp-based weighted Sobolev spaces for a transmission problem for anisotropic Stokes and Navier–Stokes systems with L∞ strongly elliptic coefficient tensor, in complementary Lipschitz domains of Rn, n≥3. The strong ellipticity allows to explore the associated pseudostress setting. First, we use a variational approach that reduces the anisotropic Stokes system in the whole Rn to an equivalent mixed variational formulation with data in Lp-based weighted Sobolev spaces. We show that such mixed variational formulation is well-posed in the space Hp1(Rn)n×Lp(Rn), n≥ 3, for any p in an open interval containing 2. Then similar well-posedness results are obtained for two linear transmission problems. These results are used to define the Newtonian and layer potential operators for the considered anisotropic Stokes system and to obtain mapping properties of these operators. 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 a nonlinear transmission problem for the anisotropic Stokes and Navier–Stokes systems in Lp-based weighted Sobolev spaces, whenever the given data are small enough.

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