The Spin-Coating Process: Analysis of the Free Boundary Value Problem

In this paper, an accurate model of the spin-coating process is presented and investigated from the analytical point of view. More precisely, the spin-coating process is being described as a one-phase free boundary value problem for Newtonian fluids subject to surface tension and rotational effects. It is proved that for T > 0 there exists a unique, strong solution to this problem in (0, T) belonging to a certain regularity class provided the data and the speed of rotation are small enough in suitable norms. The strategy of the proof is based on a transformation of the free boundary value problem to a quasilinear evolution equation on a fixed domain. The keypoint for solving the latter equation is the so-called maximal regularity approach. In order to pursue in this direction one needs to determine the precise regularity classes for the associated inhomogeneous linearized equations. This is being achieved by applying the Newton polygon method to the boundary symbol.     

The Ornstein-Uhlenbeck semigroup in exterior domains

Let Ω be an exterior domain in It is shown that Ornstein-Uhlenbeck operators L generate C₀-semigroups on L^p(Ω) for p ∈ (1, ∞) provided ∂Ω is smooth. The method presented also allows to determine the domain D(L) of L and to prove L^p − L^q smoothing properties of e^tL. If ∂Ω is only Lipschitz, results of this type are shown to be true for p close to 2. Received: 16 December 2004; revised: 4 February 2005     

Invariant measures and maximal L2 regularity for nonautonomous Ornstein-Uhlenbeck equations

We characterize the domain of the realizations of a linear parabolicoperator defined in L² spaces with respect to a suitable measurethat is invariant for the associated evolution semigroup. Asa byproduct, we obtain optimal L² regularity results for evolutionequations with time-dependant Ornstein–Uhlenbeck operators.     

Maximal L p regularity for parabolic difference equations

In this paper, we consider a family of finite difference operators {Ah }_{h >0} on discrete L _q ‐spaces L _q (?^N _h ). We show that the solution u _h to u ′_h (t) – A _h u _h(t) = f _h (t), t > 0, u _h (0) = 0 satisfies the estimate ‖A _h u _h ‖equation/tex2gif-inf-15.gif ≤ C ‖f _h ‖equation/tex2gif-inf-21.gif, where C is independent of h and f _h . In this case, the family {A _h }_{h >0} is said to have discrete maximal L _p regularity on the discrete L _q ‐space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

L p -regularity for parabolic operators with unbounded time-dependent coefficients

We establish the maximal regularity for nonautonomous Ornstein–Uhlenbeck operators in L ^p -spaces with respect to a family of invariant measures, where ${p \in (1, +\infty)}We establish the maximal regularity for nonautonomous Ornstein–Uhlenbeck operators in L ^p -spaces with respect to a family of invariant measures, where p ? (1, +￥){p \in (1, +\infty)} . This result follows from the maximal L ^p -regularity for a class of elliptic operators with unbounded, time-dependent drift coefficients and potentials acting on L^p(\mathbbR^N ){L^{p}(\mathbb{R}^{N} )} with Lebesgue measure.     

Rate of weak convergence of the finite element method for the stochastic heat equation with additive noise

The stochastic heat equation driven by additive noise is discretized in the spatial variables by a standard finite element method. The weak convergence of the approximate solution is investigated and the rate of weak convergence is found to be twice that of strong convergence. M. Kovács and S. Larsson supported by the Swedish Research Council (VR). Part of this work was done at Institut Mittag-Leffler. S. Larsson supported by the Swedish Foundation for Strategic Research (SSF) through GMMC, the Gothenburg Mathematical Modelling Centre.     

Maximal L p – L q -Estimates for the Stokes Equation: a Short Proof of Solonnikov’s Theorem

Introducing a new localization method involving Bogovskiĭ's operator we give a short and new proof for maximal L^p – L^q-estimates for the solution of the Stokes equation. Moreover, it is shown that, up to constants, the Stokes operator is an R{\mathcal{R}}-sectorial operator in L^p_s(W)L^{p}_{\sigma}(\Omega), 1 < p < ￥1 < p < \infty, of R{\mathcal{R}}-angle 0, for bounded or exterior domains of Ω.     

Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

A General Approach to Time Periodic Incompressible Viscous Fluid Flow Problems

This article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations. It yields, on the one hand, a unified approach to various classical problems in incompressible fluid flow and, on the other hand, gives new results for periodic solutions to the Navier–Stokes–Oseen flow, the Navier–Stokes flow past rotating obstacles, and, in the geophysical setting, for Ornstein–Uhlenbeck and various diffusion equations with rough coefficients. The method is based on a combination of interpolation and topological arguments, as well as on the smoothing properties of the linearized equation.