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In this paper we prove sufficient conditions for the Fredholm property of a non-smooth pseudodifferential operator $P$ which symbol is in a Holder space with respect to the spatial variable. As a main ingredient for the proof we use a suitable symbol-smoothing.

We show the short-time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume-preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non-local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface.

For a large class of initial data, we prove the existence of classical solutions locally in time to a modified Hele-Shaw problem that takes elastic effects into account. The system arises as sharp interface model of a Cahn-Hilliard system coupled with linearized elasticity. By using the Hanzawa transformation, we can reduce the system to a single evolution equation for the height function. Then short time existence is proven by inverting the linearized operator and applying the contraction mapping principle.

The asymptotic behavior of the solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. We discuss the long time existence and convergence to solutions of the time-dependent von Karman and linear plate equation under appropriate scalings of the applied force and of the initial values in terms of h (© 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We consider the Navier–Stokes system with variable density and variable viscosity coupled to a transport equation for an order-parameter c. Moreover, an extra stress depending on c and ∇c, which describes surface tension like effects, is included in the Navier–Stokes system. Such a system arises, e.g. for certain models of granular flows and as a diffuse interface model for a two-phase flow of viscous incompressible fluids. The so-called density-dependent Navier–Stokes system is also a special case of our system. We prove short-time existence of strong solution in Lq-Sobolev spaces with q>d. We consider the case of a bounded domain and an asymptotically flat layer with a combination of a Dirichlet boundary condition and a free surface boundary condition. The result is based on a maximal regularity result for the linearized system. Copyright © 2010 John Wiley & Sons, Ltd.

In this paper we prove unique solvability of the generalized Stokes resolvent equations in an infinite layer Ω0 = ℝn –1 × (–1, 1), n ≥ 2, in Lq -Sobolev spaces, 1 < q < ∞, with slip boundary condition of on the “upper boundary” ∂Ω+0 = ℝn –1 × {1} and non-slip boundary condition on the “lower boundary” ∂Ω–0 = ℝn –1 × {–1}. The solution operator to the Stokes system will be expressed with the aid of the solution operators of the Laplace resolvent equation and a Mikhlin multiplier operator acting on the boundary. The present result is the first step to establish an Lq -theory for the free boundary value problem studied by Beale [9] and Sylvester [22] in L2-spaces. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)