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Based on existence results for the Stokes operator and its solution properties in manifolds with cylindrical ends by Große et al. and Kohr et al., the Stokes flow in a three‐dimensional compact domain Ω+$$ {\Omega}^{+} $$ with circular openings Σj(j=1,2)$$ {\Sigma}_j\kern0.1em \left(j=1,2\right) $$ through which the fluid enters and leaves Ω+$$ {\Omega}^{+} $$ through unbounded cylindrical pipes the Stokes flow is modeled as a mixed boundary value problem Ω+$$ {\Omega}^{+} $$ whereas in the cylindrical ends, the velocities and pressures are constant on every straight line in the cylindrical directions with initial values from the openings Σj$$ {\Sigma}_j $$ of Ω+$$ {\Omega}^{+} $$. These values equal the velocities and pressures which are obtained from the mixed boundary values' solution in Ω+$$ {\Omega}^{+} $$ at the openings Σj$$ {\Sigma}_j $$. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a layer potential theory in L2‐based weighted Sobolev spaces on Lipschitz bounded and exterior domains of ℝn, n ≥ 3, for the anisotropic Stokes system with L∞ viscosity tensor coefficient satisfying an ellipticity condition for symmetric matrices with zero matrix trace. To do this, we explore equivalent mixed variational formulations and prove the well‐posedness of some transmission problems for the anisotropic Stokes system in Lipschitz domains of ℝn, with the given data in L2‐based weighted Sobolev spaces. These results are used to define the volume (Newtonian) and layer potentials and to obtain their properties. Then, we analyze the well‐posedness of the exterior Dirichlet and Neumann problems for the anisotropic Stokes system with L∞ symmetrically elliptic tensor coefficient by representing their solutions in terms of the obtained volume and layer potentials. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show existence of a solution of the Dirichlet problem for a nonlinear generalized Darcy- Forchheimer-Brinkman system in a bounded Lipschitz domain in Rn(n = 2, 3), with small boundary datum in L2-based Sobolev spaces. A useful intermediary result is thewell-posedness of the Poisson problem for a generalized Brinkman system in a bounded Lipschitz domain in Rn(n ⩾ 2), with Dirichlet boundary condition and data in L2-based Sobolev spaces. We obtain this well-posedness result by showing that thematrix type operator associated with the Poisson problem is an isomorphism. Then,we combine thewell-posedness result from the linear case with a fixed point theoremin order to show the existence of a solution of the Dirichlet problem for the nonlinear generalized Darcy-Forchheimer-Brinkman system. Some applications are also included. [ABSTRACT FROM AUTHOR]