Matthias Geissert

L p-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle

Abstract Consider the equations of Navier-Stokes in the exterior of a rotating domain. It is shown that, after rewriting the problem on a fixed domain Ω, the solution of the corresponding Stokes equation is governed by a C 0-semigroup on L σ p (Ω), 1 < p < ∞, with generator . Moreover, for and initial data u 0 ∈ L σ p (Ω), we prove the existence of a unique local mild solution to the Navier-Stokes problem.

Invariant measures and maximal L2 regularity for nonautonomous OrnsteinUhlenbeck equations

We characterize the domain of the realizations of a linear parabolic operator defined in L2 spaces with respect to a suitable measure that is invariant for the associated evolution semigroup. As a byproduct, we obtain optimal L2 regularity results for evolution equations with time-dependant Ornstein�Uhlenbeck operators.

Weak Neumann implies Stokes

Abstract Consider a domain Ω ⊂ ℝn with possibly non compact but uniform C3-boundary and assume that the Helmholtz projection P exists on Lp(Ω) for some 1 < p < ∞. It is shown that the Stokes operator in Lp(Ω) generates an analytic semigroup on admitting maximal Lq-Lp-regularity. Moreover, for there exists a unique local mild solution to the Navier–Stokes equations on domains of this form provided p > n.

Applications of discrete maximal L p regularity for finite element operators

In this paper, we present applications of discrete maximal Lp regularity for finite element operators. More precisely, we show error estimates of order h2 for linear and certain semilinear problems in various Lp(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 Lp regularity formerly proved by the author.

Asymptotic behavior and hypercontractivity in non-autonomous Ornstein–Uhlenbeck equations

In this paper we investigate a class of non-autonomous linear parabolic problems with time-depending Ornstein�Uhlenbeck operators. We study the asymptotic behavior of the associated evolution operator and evolution semigroup in the periodic and non-periodic situation. Moreover, we show that the associated evolution operator is hypercontractive

Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise

Abstract We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean- p Holder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space–time (Sobolev–Holder) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family.

Lp-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids

Consider the system of equations describing the motion of a rigid body immersed in a viscous, incompressible fluid of Newtonian or generalized Newtonian type. The class of fluids considered includes in particular shearthinning or shear-thickening fluids of power-law type of exponent d ≥ 1. We develop a method to prove that this system admits a unique, local, strong solution in the Lp-setting. The approach presented in the case of generalized Newtonian fluids is based on the theory of quasi-linear evolution equations and requires that the exponent p satisfies the condition p > 5.

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.

Strong LP-solutions to the Navier-Stokes flow past moving obstacles: The case of several obstacles and time dependent velocity

Consider the Navier-Stokes flow past several moving or rotating obstacles with possible time-dependent velocity. It is shown that under suitable assumptions on the data, there exists a unique, local strong solution in the L P - L q -setting for suitable p,q ∈ (1,∞). Moreover, it is proved that this strong solution coincides with the known mild solution in the very weak sense.

Stability results for fluids of Oldroyd-B type on exterior domains

Consider the set of equations describing fluids of Oldroyd-B type on an exterior domain. It is shown that the solution of the linearized equation is governed by a bounded analytic semigroup on Lp(Ω) × W1, p(Ω), 1<p<∞, which is strongly stable. Moreover, it is shown that the trivial solution of the full system is asymptotically stable in the sense that any solution starting in small ball around (0, 0) converges towards 0 as t → ∞.