Wave Equations and Symmetric First-order Systems in Case of Low Regularity

Hanel, Clemens; Hörmann, Günther; Spreitzer, Christian; Steinbauer, Roland

doi:10.1007/978-3-0348-0585-8_15

Cited by 3 publications

(2 citation statements)

References 22 publications

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“…Remark 3.4. The required conditions for the existence of a distributional shadow (and weak solution) are weaker than those that would be required in a similar result based on transforming the equation (3.1) into a first-order system as in [24], since the lower-order coefficients of this system would contain derivatives of the principal coefficients of equation (3.1) and therefore…”

Section: We Consider the Corresponding Generalised Cauchy Problem Obt...mentioning

confidence: 94%

Green operators in low regularity spacetimes and quantum field theory

Hoermann

Sanchez²,

Spreitzer

et al. 2020

Class. Quantum Grav.

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In this paper we develop the mathematics required in order to provide a description of the observables for quantum fields on low-regularity spacetimes. In particular we consider the case of a massless scalar field φ on a globally hyperbolic spacetime M with C 1,1 metric g. This first entails showing that the (classical) Cauchy problem for the wave equation is well-posed for initial data and sources in Sobolev spaces and then constructing low-regularity advanced and retarded Green operators as maps between suitable function spaces. In specifying the relevant function spaces we need to control the norms of both φ and g φ in order to ensure that g • G ± and G ± • g are the identity maps on those spaces. The causal propagator G = G + − G − is then used to define a symplectic form ω on a normed space V (M ) which is shown to be isomorphic to ker g . This enables one to provide a locally covariant description of the quantum fields in terms of the elements of quasi-local C * -algebras.

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Section: We Consider the Corresponding Generalised Cauchy Problem Obt...mentioning

confidence: 94%

Green operators in low regularity spacetimes and quantum field theory

Hoermann

Sanchez²,

Spreitzer

et al. 2020

Class. Quantum Grav.

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show abstract

“…However, we employ an algorithm that also guarantees the symmetry of the system, cf. [18]. Indeed, by setting the state vector…”

Section: Transformation Into Conservation Formmentioning

confidence: 99%

Neural Tissue Response to Impact - Numerical Study of Wave Propagation at Level of Neural Cells

Hozman

Bradáč²,

Kovanda

2014

NNW

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This tutorial is based on modification of the professor nomination lecture presented two years ago in front of the Scientific Council of the Czech Technical University in Prague [16].It is devoted to the techniques for the models developing suitable for processes forecasting in complex systems. Because of the high sensitivity of the processes to the initial conditions and, consequently, due to our limited possibilities to forecast the processes for the long-term horizon, the attention is focused on the techniques leading to practical applications of the short term prediction models. The aim of this tutorial paper is to bring attention to possible difficulties which designers of the predicting models and their users meet and which have to be solved during the prediction model developing, validation, testing, and applications. The presented overview is not complete, it only reflects the author's experience with developing of the prediction models for practical tasks solving in banking, meteorology, air pollution and energy sector. The paper is completed by an example of the global solar radiation prediction which forms an important input for the electrical energy production forecast from renewable sources. The global solar radiation forecasting is based on numerical weather prediction models. The time-lagged ensemble technique for uncertainty quantification is demonstrated on a simple example.

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Propagation of singularities for generalized solutions to wave equations with discontinuous coefficients

Deguchi

Oberguggenberger²

2015

Preprint

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Wave Equations and Symmetric First-order Systems in Case of Low Regularity

Cited by 3 publications

References 22 publications

Green operators in low regularity spacetimes and quantum field theory

Green operators in low regularity spacetimes and quantum field theory

Neural Tissue Response to Impact - Numerical Study of Wave Propagation at Level of Neural Cells

Propagation of singularities for generalized solutions to wave equations with discontinuous coefficients

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