The Dirichlet problem for the Stokes system on Lipschitz domains

Layer potential theory for the anisotropic Stokes system with variable L∞ symmetrically elliptic tensor coefficient

“…The methods of layer potential theory play also a significant role in the study of elliptic boundary value problems with variable coefficients. Mitrea and Taylor 15 have obtained well-posedness results for the Dirichlet problem for the smooth coefficient Stokes system in L p spaces on arbitrary Lipschitz domains in a compact Riemannian manifold and extended the well-posedness results by Fabes et al 7 from the Euclidean setting to the compact Riemannian setting. Dindos and Mitrea 3 have used the mapping properties of Stokes layer potentials in Sobolev and Besov spaces to show well-posedness results for Poisson problems for the smooth coefficient Stokes and Navier-Stokes systems with Dirichlet boundary condition on C 1 and Lipschitz domains in compact Riemannian manifolds.…”

Integral representation of a solution to the Stokes–Darcy problem

“…The limit in the last equality is well defined for almost all x 2 @G, and K S G is a bounded linear operator on OEL 2 . @G/ 3 [29,33,35], which can be extended to a bounded linear operator on OEH 1=2 . @G/ 3 [36].…”

“…The potential theory for the hydrodynamics was first developed to study classical solutions of the Dirichlet and Neumann problems for the Stokes system [23][24][25][26][27]. Later, solutions of the Dirichlet problem, the Neumann problem, and the transmission problem for the Stokes system have been looked for in the form of hydrodynamical boundary layers also for p-integrable boundary conditions and for solutions from Sobolev and Besov spaces [28][29][30][31][32][33][34]. We have used this theory to study a solution .v, p/ 2 OEH 1 .…”

Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains

“…Concerning the 2D incompressible Navier-Stokes flows (f (t, u t ) = 0 in (1)) driven by non-homogeneous boundary conditions in regular domains, Miranville and Wang [27], [28] introduced a background flow to deal with the boundary conditions, and obtained the existence of the finite dimensional global attractor under the assumptions that ∂Ω ∈ C 3 and |∇ϕ| ∈ L ∞ (∂Ω). For the Lipschitz-like case, motivated by [27,28] and based on estimates on the Stokes problem in [13] and [32], Brown, Perry and Shen [3] introduced the background flow in Lipschitz-like domains and proved the existence of the finite (fractal) dimensional universal attractor. Using the theory of pullback attractors, Yang, Qin, Lu and Ma [39] deduced the existence and regularity of pullback attractors, based on the background flow in Lipschitz-like domains.…”