arXiv subjects
Publications and source records attributed to Rainer Picard.
We consider the Dirac equation in a real setting to show its close relationship to Maxwell's equation. In addition to analysing the classical Dirac mass term, we also inspect the alternative Majorana mass term and discuss their quaternionic structure and its consequences.
We provide certain compatibility conditions for m-accretive operators such that the adjoint of the sum is given by the closure of the sum of the respective adjoint. We revisit the proof of well-posedness of the abstract class of partial differential-algebraic equations known as evolutionary equations. We show that the general mechanism provided here can be applied to establish well-posedness for non-autonomous evolutionary equations with $L_{\infty}$-coefficients thus not only generalising known results but opening up new directions other methods such as evolution families have a hard time to come by.
In this short note we show that Hilbert complexes are strongly related to what we shall call annihilating sets of skew-selfadjoint operators. This provides for a new perspective on the classical topic of Hilbert complexes viewed as families of commuting normal operators.
We consider two operators $A_{0},B_{0}$ between two Hilbert spaces satisfying $A_{0}\subseteq-B_{0}^{\ast}$ and $B_{0}\subseteq-A_{0}^{\ast}$ and inspect extensions $A^{\#}$ and $B^{\#}$ of $A_{0}$ and $B_{0}$, respectively, whose domain consists of those elements satisfying an abstract periodic boundary condition. The motivating example is the derivative on some interval, where the so-defined realisation gives the classical derivative with periodic boundary conditions. We derive necessary and sufficient conditions for the operator equality $A^{\#}=-\left(B^{\#}\right)^{\ast}$ and illustrate our findings by applications to the classical vector analytic operators $\mathrm{grad},\,\mathrm{div}$ and $\mathrm{curl}$. In particular, the realisation $\mathrm{curl}^{\#}$ naturally arises in the study of so-called Beltrami fields.
We consider skew-symmetric operators $A_{0}$ on a Hilbert space $H$ and characterise all (nonlinear) m-accretive restrictions of $A:=-A_{0}^{\ast}$ in terms of the "deficiency spaces" $\ker(1\pm A)$. The results are illustrated by several examples and applied to a partial differential equation with an impedance type boundary condition.
We study port-Hamiltonian systems on a familiy of intervals and characterise all boundary conditions leading to $m$-accretive realisations of the port-Hamiltonian operator and thus to generators of contractive semigroups. The proofs are based on a structural observation that the port-Hamiltonian operator can be transformed to the derivative on a familiy of reference intervals by suitable congruence relations allowing for studying the simpler case of a transport equation. Moreover, we provide well-posedness results for associated control problems without assuming any additional regularity of the operators involved.
An abstract first order differential equation of hyperbolic type with drift term on a Riemannian manifold is considered. For proving its well-posedness, transmutator and commutator relations are needed, which are studied in a general functional analytic setting.
We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus using the Fourier transform. Main tools are extrapolation- and interpolation spaces. Main results are the existence and uniqueness of solutions and the causality of solution operators for non-linear fractional differential equations.
We present an abstract framework for parabolic type equations which possibly degenerate on certain spatial regions. The degeneracies are such that the equations under investigation may admit a type change ranging from parabolic to elliptic type problems. The approach is an adaptation of the concept of so-called evolutionary equations in Hilbert spaces and is eventually applied to a degenerate eddy current type model. The functional analytic setting requires quite minimal assumptions on the boundary and interface regularity. The degenerate eddy current model is justified as a limit model of non-degenerate hyperbolic models of Maxwell's equations.
We consider general difference equations $u_{n+1} = F(u)_n$ for $n \in \mathbb{Z}$ on exponentially weighted $\ell_2$ spaces of two-sided Hilbert space valued sequences $u$ and discuss initial value problems. As an application of the Hilbert space approach, we characterize exponential stability of linear equations and prove a stable manifold theorem for causal nonlinear difference equations.
We further elaborate on the solvability of stochastic partial differential equations (SPDEs). We shall discuss non-autonomous partial differential equations with an abstract realization of the stochastic integral on the right-hand side. Our approach allows the treatment of equations with mixed type, where classical solution strategies fail to work. The approach extends prior observations in [S\"u\ss, A. \& Waurick, M. A Solution Theory for a General Class of SPDEs. \emph{Stochastics and Partial Differential Equations: Analysis and Computations}, 2017, 5, 278-318], where the respective results were obtained for linear autonomous equations and (multiplicative) white noise.
We consider a coupled system describing the interaction between acoustic and elastic regions, where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes. Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions. The results are obtained in a real Hilbert space setting incurring no regularity constraints on the boundary and almost none on the interface of the underlying regions.
We consider a coupled system describing the interaction between acoustic and elastic regions, where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes. Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions. The results are obtained in a real Hilbert space setting incurring no regularity constraints on the boundary and almost none on the interface of the underlying regions.
We consider a coupled system of Maxwell's equations and the equations of elasticity, which is commonly used to model piezo-electric material behavior. The boundary influence is encoded as a separate dynamics on the boundary data spaces coupled to the partial differential equations. Evolutionary well-posedness, i.e. Hadamard well-posedness and causal dependence on the data, is shown for the resulting model system.
The issue of justifying the eddy current approximation of Maxwell's equations is re-considered in the time-dependent setting. Convergence of the solution operators is shown in the sense of strong operator limits.
The issue of so-called maximal regularity is discussed within a Hilbert space framework for a class of evolutionary equations. Viewing evolutionary equations as a sums of two unbounded operators, showing maximal regularity amounts to establishing that the operator sum considered with its natural domain is already closed. For this we use structural constraints of the coefficients rather than semi-group strategies or sesqui-linear form methods, which would be difficult to come by for our general problem class. Our approach, although limited to the Hilbert space case, complements known strategies for approaching maximal regularity and extends them in a different direction. The abstract findings are illustrated by re-considering some known maximal regularity results within the framework presented.
We consider the model equations for the Timoshenko beam as a first order system in the framework of evolutionary equations. The focus is on boundary damping, which is implemented as a dynamic boundary condition. A change of material laws allows to include a large class of cases of boundary damping. By choosing a particular material law, it is shown that the first order approach to Sturm-Liouville problems with boundary damping is also covered.
There is an increasing reliance on mathematical modelling to assist in the design of piezoelectric ultrasonic transducers since this provides a cost-effective and quick way to arrive at a first prototype. Given a desired operating envelope for the sensor the inverse problem of obtaining the associated design parameters within the model can be considered. It is therefore of practical interest to examine the well-posedness of such models. There is a need to extend the use of such sensors into high temperature environments and so this paper shows, for a broad class of models, the well-posedness of the magneto-electro-thermo-elastic problem. Due to its widespread use in the literature, we also show the well-posedness of the quasi-electrostatic case