Searcharxiv⌕ Search

arXiv subjects

Rainer Picard

Publications and source records attributed to Rainer Picard.

35 records · Page 2Linked to original sources
arXiv2015

A Note on a Two-Temperature Model in Linear Thermoelasticity

We discuss the so-called two-temperature model in linear thermoelasticity and provide a Hilbert space framework for proving well-posedness of the equations under consideration. With the abstract perspective of evolutionary equations, the two-temperature model turns out to be a coupled system of the elastic equations and an abstract ode. Following this line of reasoning, we propose another model being entirely an abstract ode. We highlight also an alternative way for a two-temperature model, which might be of independent interest.

math.AP↗
arXiv2015

On Abstract $\mathrm{grad}-\mathrm{div}$ Systems

For a large class of dynamical problems from mathematical physics the skew-selfadjointness of a spatial operator of the form $A=\left(\begin{array}{cc} 0 & -C^{*}\\ C & 0 \end{array}\right)$, where $C:D\left(C\right)\subseteq H_{0}\to H_{1}$ is a closed densely defined linear operator, is a typical property. Guided by the standard example, where $C=\mathrm{grad}=\left(\begin{array}{c} \partial_{1}\\ \vdots\\ \partial_{n} \end{array}\right)$ (and $-C^{*}=\mathrm{div}$, subject to suitable boundary constraints), an abstract class of operators $C=\left(\begin{array}{c} C_{1}\\ \vdots\\ C_{n} \end{array}\right)$ is introduced (hence the title). As a particular application we consider a non-standard coupling mechanism and the incorporation of diffusive boundary conditions both modeled by setting associated with a skew-selfadjoint spatial operator $A$.

math.AP↗
arXiv2014

On Some Models in Linear Thermo-Elasticity with Rational Material Laws

We shall consider some common models in linear thermo-elasticity within a common structural framework. Due to the flexibility of the structural perspective we will obtain well-posedness results for a large class of generalized models allowing for more general material properties such as anisotropies, inhomogeneities, etc.

math-ph↗
arXiv2014

Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.

math.AP↗
arXiv2014

On a Connection between the Maxwell System, the Extended Maxwell System, the Dirac Operator and Gravito-Electromagnetism

Maxwell's equation, Dirac's equation and the equation of gravito-electromagnetism are shown to be particular instances of the extended Maxwell system. The equations are discussed in the framework of the theory of evolutionary equations. Their formal relationship are systematically analyzed. Applications to coupled systems such as the Maxwell-Dirac system are also discussed.

math-ph↗
arXiv2014

Mother Operators and their Descendants

A mechanism deriving new well-posed evolutionary equations from given ones is inspected. It turns out that there is one particular spatial operator from which many of the standard evolutionary problems of mathematical physics can be generated by this abstract mechanism using suitable projections. The complexity of the dynamics of the phenomena considered can be described in terms of suitable material laws. The idea is illustrated with a number of concrete examples.

math.AP↗
arXiv2013

On Some Models for Elastic Solids with Micro-Structure

We review the concept of well-posedness in the context of evolutionary problems from mathematical physics for a particular subclass of problems from elasticity theory. The complexity of physical phenomena appears as encoded in so called material laws. The usefulness of the structural perspective developed is illustrated by showing that many initial boundary value problems in the theory of elastic solids share the same type of solution theory. Moreover, interconnections of the respective models are discussed via a previously introduced mother/descendant mechanism.

math.AP↗
arXiv2013

A Note on Fractional Evolutionary Equations

A class of linear evolutionary equations with material laws involving fractional time-derivatives is considered. The main result is well-posedness and causality for this problem class. The approach is illustrated with two examples: a fractional Fokker-Planck type equation and a class of visco-elastic materials described via fractional derivatives. In conclusion the possibility of imposing initial conditions is discussed.

math.AP↗
arXiv2013

On a Comprehensive Class of Linear Control Problems

We discuss a class of linear control problems in a Hilbert space setting. This class encompasses such diverse systems as port-Hamiltonian systems, Maxwell's equations with boundary control or the acoustic equations with boundary control and boundary observation. The boundary control and observation acts on abstract boundary data spaces such that the only geometric constraint on the underlying domain stems from requiring a closed range constraint for the spatial operator part, a requirement which for the wave equation amounts to the validity of a Poincare-Wirtinger-type inequality. We also address the issue of conservativity of the control problems under consideration.

math.OC↗
arXiv2013

On Non-Autonomous Evolutionary Problems

The paper extends well-posedness results of a previously explored class of time-shift invariant evolutionary problems to the case of non-autonomous media. The Hilbert space setting developed for the time-shift invariant case can be utilized to obtain an elementary approach to non-autonomous equations. The results cover a large class of evolutionary equations, where well-known strategies like evolution families may be difficult to use or fail to work. We exemplify the approach with an application to a Kelvin-Voigt-type model for visco-elastic solids.

math.AP↗
arXiv2012

On a Class of Boundary Control Problems

We discuss a class of linear control problems in a Hilbert space setting, which covers diverse systems such as hyperbolic and parabolic equations with boundary control and boundary observation even including memory terms. We introduce abstract boundary data spaces in which the control and observation equations can be formulated without strong geometric constraints on the underlying domain. The results are applied to a boundary control problem for the equations of visco-elasticity.

math.OC↗
arXiv2012

A Functional Analytic Perspective to Delay Differential Equations

We generalize the solution theory for a class of delay type differential equations developed in a previous paper, dealing with the Hilbert space case, to a Banach space setting. The key idea is to consider differentiation as an operator with the whole real line as the underlying domain as a means to incorporate pre-history data. We focus our attention on the issue of causality of the differential equations as a characterizing feature of evolutionary problems and discuss various examples. The arguments mainly rely on a variant of the contraction mapping theorem and a few well-known facts from functional analysis.

math.FA↗
arXiv2012

A Hilbert Space Perspective on Ordinary Differential Equations with Memory Term

We discuss ordinary differential equations with delay and memory terms in Hilbert spaces. By introducing a time derivative as a normal operator in an appropriate Hilbert space, we develop a new approach to a solution theory covering integro-differential equations, neutral differential equations and general delay differential equations within a unified framework. We show that reasonable differential equations lead to causal solution operators.

math.CA↗
arXiv2012

A Class of Evolutionary Problems with an Application to Acoustic Waves with Impedance Type Boundary Conditions

A class of evolutionary operator equations is studied. As an application the equations of linear acoustics are considered with complex material laws. A dynamic boundary condition is imposed which in the time-harmonic case corresponds to an impedance or Robin boundary condition. Memory and delay effects in the interior and also on the boundary are built into the problem class.

math.AP↗
arXiv2012

The Elusive Drude-Born-Fedorov Model for Chiral Electromagnetic Media

Electro-magnetic wave propagation in more complex linear materials such as bi-anisotropic media have come to a considerable attention within the last fifteen to twenty years. The Drude-Born-Fedorov model has been extensively studied mostly in the time-harmonic case as a model for chiral media. In the physically relevant time-dependent case the record is much less convincing. In this paper we focus on this case and analyze the Drude-Born-Fedorov model in the light of recently developed Hilbert space approach to evolutionary problems. The solution theory will be developed in the framework of extrapolation spaces (Sobolev lattices).

math.AP↗
arXiv2012

Evolutionary Problems Involving Sturm-Liouville Operators

The purpose of this paper is to further exemplify an approach to evolutionary problems originally developed in earlier works for a special case and later extended to more general evolutionary problems. We are here concerned with the $(1+1)$ -dimensional evolutionary case, which in a particular case results in a hyperbolic partial differential equation with a Sturm-Liouville type spatial operator constrained by an impedance type boundary condition.

math.AP↗