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Our concern in this paper lies with trace spaces for weighted Sobolev spaces, when the weight is a power of the distance to a point at the boundary. For a large range of powers we give a full description of the trace space.

Abstract  We consider a generalization of the Stokes resolvent equation, where the constant viscosity is replaced by a general given positive function. Such a system arises in many situations as linearized system, when the viscosity of an incompressible, viscous fluid depends on some other quantities. We prove that an associated Stokes-like operator generates an analytic semi-group and admits a bounded H ∞ -calculus, which implies the maximal L q -regularity of the corresponding parabolic evolution equation. The analysis is done for a large class of unbounded domains with $${W^{2-\frac1r}_r}$$ -boundary for some r > d with r ≥ q, q′. In particular, the existence of an L q -Helmholtz projection is assumed. [ABSTRACT FROM AUTHOR]

Abstract The present contribution deals with the Stokes operator Aq on Lqσ(Ω), 1

Abstract. We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In this second part, we use pseudodifferential operator techniques to construct a parametrix to the reduced Stokes equations, which solves the system in Lq-Sobolev spaces, 1 < q < 8, modulo terms which get arbitrary small for large resolvent parameters ?. This parametrix can be analyzed to prove the existence of a bounded H8-calculus of the (reduced) Stokes operator. [ABSTRACT FROM AUTHOR]

Abstract. We study the generalized Stokes equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer $\Omega _0 = \mathbb{R}^{n - 1} \times ( - 1,1).$ Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. In the first part we prove the unique solvability in Lq-Sobolev spaces, 1 < q < 8, by extending the known results in the case of an infinite layer O0 via a perturbation argument to asymptotically flat layers which are sufficiently close to O0. Combining this result with standard cut-off techniques and the parametrix constructed in the second part, we prove the unique solvability for an arbitrary asymptotically flat layer. Moreover, we show equivalence of unique solvability of the generalized and the reduced Stokes resolvent equations, which is essential for the second part of this contribution. [ABSTRACT FROM AUTHOR]

We consider weak solutions for a diffuse interface model of two non-Newtonian viscous, incompressible fluids of power-law type in the case of different densities in a bounded, sufficiently smooth domain. This leads to a coupled system of a nonhomogenouos generalized Navier–Stokes system and a Cahn–Hilliard equation. For the Cahn–Hilliard part a smooth free energy density and a constant, positive mobility is assumed. Using the -truncation method we prove existence of weak solutions for a power-law exponent , d  =  2, 3. [ABSTRACT FROM AUTHOR]