Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

Hörmann, Günther; Spreitzer, Christian

doi:10.1016/j.jmaa.2011.11.014

Cited by 7 publications

(9 citation statements)

References 24 publications

(80 reference statements)

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“…In the last part of the proof we show that the distributional shadow u of the generalised solution is the unique weak solution to the Cauchy problem. The proof follows the line of arguments in the proof of [32,Corollary 4.6]. First note that both u and ∂ t u are continuous and thus, by construction of u, the initial conditions are satisfied.…”

Section: The Smooth Settingmentioning

confidence: 84%

“…In Section 3 we establish the results we need to prove existence and uniqueness of solutions to the forward (and backward) initial value problem for the wave equation on R n+1 for C 1,1 metrics. Rather than rework the entire theory of the wave equation for metrics of low-regularity we use a method of regularising the coefficients [32], using the smooth theory to obtain the corresponding solutions of the Cauchy problem and then using a compactness argument to show that this converges to a weak H 2 loc (R n+1 ) solution of the original equation. This proceeds via the theory of Colombeau generalised functions [13] and is related to the work of [20] on very weak solutions.…”

Section: Introductionmentioning

confidence: 99%

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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: The Smooth Settingmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Green operators in low regularity spacetimes and quantum field theory

Hoermann

Sanchez²,

Spreitzer

et al. 2020

Class. Quantum Grav.

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“…In the following theorem we will give conditions on the coefficients of a wave equation (1) that guarantee the existence of a unique generalized solution to the corresponding first-order problem and, hence, existence and uniqueness of a generalized solution to the wave equation (1). To this end, we are going to invoke the existence theory for symmetric hyperbolic systems developed in [Obe88, Obe89, CoOb90, LaOb91, Hor04a] and, in particular, the existence results of [HoSp12], which we will now briefly summarize. We start by recalling the essential asymptotic conditions.…”

Section: Existence and Uniqueness For The Cauchy Problemsmentioning

confidence: 99%

“…Uniqueness of generalized solutions amounts to stability of the family of smooth solutions under negligible perturbations of the data. For convenience of the reader, we combine results from [HoSp12], adjusted to the situation at hand, in the following theorem (cf. [HoSp12, Theorems 3.1, 3.2, and 3.4]).…”

Section: Existence and Uniqueness For The Cauchy Problemsmentioning

confidence: 99%

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

Hanel

Hörmann

Spreitzer

et al. 2013

Pseudo-Differential Operators, Generalized Functions and Asymptotics

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We analyse an algorithm of transition between Cauchy problems for secondorder wave equations and first-order symmetric hyperbolic systems in case the coefficients as well as the data are non-smooth, even allowing for regularity below the standard conditions guaranteeing well-posedness. The typical operations involved in rewriting equations into systems are then neither defined classically nor consistently extendible to the distribution theoretic setting. However, employing the nonlinear theory of generalized functions in the sense of Colombeau we arrive at clear statements about the transfer of questions concerning solvability and uniqueness from wave equations to symmetric hyperbolic systems and vice versa. Finally, we illustrate how this transfer method allows to draw new conclusions on unique solvability of the Cauchy problem for wave equations with non-smooth coefficients.

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“…The Colombeau algebra G(R m+1 ) is a differential algebra containing the space of distributions D (R m+1 ) as a subspace, and thus it provides a framework in which all operations arising in (1.1) are meaningful. Indeed, existence and uniqueness of solutions to (1.1) in the Colombeau algebra (as well as in its dual) have been proved under various conditions: when A j and B are real-valued n × n matrices with entries in G(R m+1 ), for A j symmetric [35,40]; for symmetric hyperbolic systems of pseudodifferential operators with Colombeau symbols [32,46]; for strictly hyperbolic systems of pseudodifferential operators [24].…”

Section: Introductionmentioning

confidence: 99%

Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities

Garetto

Oberguggenberger²

2013

Proceedings of the Edinburgh Mathematical Society

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This article addresses linear hyperbolic partial differential equations and pseudodifferential equations with strongly singular coefficients and data, modelled as members of algebras of generalised functions. We employ the recently developed theory of generalised Fourier integral operators to construct parametrices for the solutions and to describe propagation of singularities in this setting. As required tools, the construction of generalised solutions to eikonal and transport equations is given and results on the microlocal regularity of the kernels of generalised Fourier integral operators are obtained.This section gives some background on Colombeau techniques used in the sequel of this paper. As main sources we refer to [8,11,12,15].Nets of complex numbers. A net (u ε ) ε in C (0,1] is said to be strictly nonzero if there exist r > 0 and η ∈ (0, 1] such that |u ε | ≥ ε r for all ε ∈ (0, η]. For several regularity issues we will make use of the concept of slow scale net (s.s.n). A slow scale net is a net (r ε ) ε ∈ C (0,1] such thatA net (u ε ) ε in C (0,1] is said to be slow scale-strictly nonzero if there exist a slow scale net (s ε ) ε and η ∈ (0, 1] such that |u ε | ≥ 1/s ε for all ε ∈ (0, η].C-modules of generalised functions based on a locally convex topological vector space. The most common algebras of generalised functions of Colombeau type as well as the spaces of generalised symbols we deal with are introduced by referring to the following general models.Let E be a locally convex topological vector space topologised through the family of seminorms {p i } i∈I .

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Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

Cited by 7 publications

References 24 publications

Green operators in low regularity spacetimes and quantum field theory

Green operators in low regularity spacetimes and quantum field theory

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

Generalized Fourier Integral Operator Methods for Hyperbolic Equations with Singularities

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