Matthias Hieber

Bio: Matthias Hieber is an academic researcher from Technische Universität Darmstadt. The author has contributed to research in topics: Bounded function & Primitive equations. The author has an hindex of 28, co-authored 113 publications receiving 4599 citations. Previous affiliations of Matthias Hieber include Fudan University & University of Pittsburgh.

Vector-Valued Laplace Transforms and Cauchy Problems

Abstract: Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central tool is an “integrated version” of Widder’s theorem (characterizing Laplace transforms of bounded functions). It holds in any Banach space (whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the Radon-Nikodým property). The Hille-Yosida theorem and other generation theorems are immediate consequences. The method presented here can be applied to operators whose domains are not dense.

R-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type

Abstract: Introduction Notations and conventions $\mathcal R$-Boundedness and Sectorial Operators: Sectorial operators The classes ${\mathcal{BIP}}(X)$ and $\mathcal H^\infty(X)$ $\mathcal R$-bounded families of operators $\mathcal R$-sectorial operators and maximal $L_p$-regularity Elliptic and Parabolic Boundary Value Problems: Elliptic differential operators in $L_p(\mathbb{R}^n E)$ Elliptic problems in a half space: General Banach spaces Elliptic problems in a half space: Banach spaces of class $\mathcal{HT}$ Elliptic and parabolic problems in domains Notes References.

Optimal L p - L q -estimates for parabolic boundary value problems with inhomogeneous data

Abstract: In this paper we investigate vector-valued parabolic initial boundary value problems \({(\mathcal A(t,x,D)}\) , \({\mathcal B_j(t,x,D))}\) subject to general boundary conditions in domains G in \({\mathbb R^n}\) with compact C 2m -boundary. The top-order coefficients of \({\mathcal A}\) are assumed to be continuous. We characterize optimal L p -L q -regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on \({\mathcal A}\) and the Lopatinskii–Shapiro condition on \({(\mathcal A, \mathcal B_1,\dots, \mathcal B_m)}\) are necessary for these L p -L q -estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.

$ L^p $-Theory of the Stokes equation in a half space

Abstract: In this paper, we investigate \(L^p\)-estimates for the solution of the Stokes equation in a half space H where \( 1\leq p \leq \infty \). It is shown that the solution of the Stokes equation is governed by an analytic semigroup on \( BUC_\sigma(H), C_{0,\sigma}(H) \) or \( L^\infty_\sigma(H) \). From the operatortheoretical point of view it is a surprising fact that the corresponding result for \( L^1_\sigma(H) \) does not hold true. In fact, there exists an \( L^1 \)-function f satisfying \( {\it div} f = 0 \) such that the solution of the corresponding resolvent equation with right hand side f does not belong to \(L^1\). Taking into account however a recent result of Kozono on the nonlinear Navier-Stokes equation, the \( L^1 \)-result is not surprising and even natural. We also show that the Stokes operator admits a R-bounded \( H^\infty \)-calculus on \( L^p \) for 1 < p <\( \infty \) and obtain as a consequence maximal \( L^p-L^q \)-regularity for the solution of the Stokes equation.

On the Equation div u = g and Bogovskii’s Operator in Sobolev Spaces of Negative Order

Abstract: Consider the divergence problem with homogeneous Dirichlet data on a Lipschitz domain. Two approaches for its solutions in the scale of Sobolev spaces are presented. The first one is based on Calderon-Zygmund theory, whereas the second one relies on the Stokes equation with inhomogeneous data.

On the Navier-Stokes equations

Abstract: A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

Geophysical fluid dynamics.

Abstract: The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Observation and Control for Operator Semigroups

Abstract: The evolution of the state of many systems modeled by linear partial difierentialequations (PDEs) or linear delay-difierential equations can be described by operatorsemigroups. The state of such a system is an element in an inflnite-dimensionalnormed space, whence the name \inflnite-dimensional linear system".The study of operator semigroups is a mature area of functional analysis, which isstill very active. The study of observation and control operators for such semigroupsis relatively more recent. These operators are needed to model the interactionof a system with the surrounding world via outputs or inputs. The main topicsof interest about observation and control operators are admissibility, observability,controllability, stabilizability and detectability. Observation and control operatorsare an essential ingredient of well-posed linear systems (or more generally systemnodes). Inthisbookwedealonlywithadmissibility, observabilityandcontrollability.We deal only with operator semigroups acting on Hilbert spaces.This book is meant to be an elementary introduction into the topics mentionedabove. By \elementary" we mean that we assume no prior knowledge of flnite-dimensional control theory, and no prior knowledge of operator semigroups or ofunbounded operators. We introduce everything needed from these areas. We doassume that the reader has a basic understanding of bounded operators on Hilbertspaces, difierential equations, Fourier and Laplace transforms, distributions andSobolev spaces on