First-order hyperbolic pseudodifferential equations with generalized symbols

Hörmann, Günther

doi:10.1016/j.jmaa.2003.12.013

Cited by 22 publications

(42 citation statements)

References 12 publications

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“…They have proved to be a valuable tool for treating partial differential equations with singular data or coefficients [11,14,15,16,20]. Also, they have found a wealth of applications to differential geometry [13,18] and relativity theory (cf.…”

Section: Introductionmentioning

confidence: 99%

Hilbert $\widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems

Garetto

Vernaeve

2011

Trans. Amer. Math. Soc.

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Abstract. We develop a theory of Hilbert C-modules which forms the core of a new functional analytic approach to algebras of generalized functions. Particular attention is given to finitely generated submodules, projection operators, representation theorems for C-linear functionals and C-sesquilinear forms. We establish a generalized Lax-Milgram theorem and use it to prove existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form. IntroductionOver the past thirty years, nonlinear theories of generalized functions have been developed by many authors [1,6,12,19], mainly inspired by the work of J. F. Colombeau [3,4]. They have proved to be a valuable tool for treating partial differential equations with singular data or coefficients [11,14,15,16,20]. Also, they have found a wealth of applications to differential geometry [13,18] and relativity theory (cf.[12] and the references therein). As a consequence of intense research in the field, increasing importance is attached to an understanding of algebraic [2,21] and topological structures [5,7] in spaces of generalized functions.This paper is part of a wider project which aims to introduce functional analytic methods into Colombeau algebras of generalized functions. Our intent is to deal with the general problem of existence and qualitative properties of solutions of partial differential equations in the Colombeau setting by means of functional analytic tools adapted and generalized to the range of topological modules over the ring C of generalized numbers. Starting from the topological background given in [7,8], in this paper we develop a theory of Hilbert C-modules. This will be the framework used to investigate variational equalities and inequalities generated by highly singular problems in partial differential equations.A first example of a Hilbert C-module is the Colombeau space G H of generalized functions based on a Hilbert space (H, (·|·)) [7, Definition 3.1], where the scalar product is obtained by letting (·|·) act componentwise at the representatives level. A number of theorems, such as projection theorem, Riesz-representation and the Lax-Milgram theorem, can be obtained in a direct way when we work on G H by applying the corresponding classical results at the level of representatives at first and then by checking the necessary moderateness conditions. This is a sort of

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Section: Introductionmentioning

confidence: 99%

Hilbert $\widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems

Garetto

Vernaeve

2011

Trans. Amer. Math. Soc.

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“…In the recent research on the subject a variety of algebras of generalized functions [3,12,13,17,19,22] have been introduced in addition to the original construction by Colombeau [6,7] and investigated in its algebraic and structural aspects as well as in analytic and applicative aspects. These investigations have produced a theory of point values in the Colombeau algebra G( ) and results of invertibility and positivity in the ring of constant generalized functions C [19,38,39] but also microlocal analysis in Colombeau algebras and regularity theory for generalized solutions to partial and (pseudo-)differential equations [11,13,15,16,[20][21][22][23][24][25][26][27]. Apart from some early and inspiring work by Biagioni, Pilipović, Scarpalézos [1,32,[46][47][48], topological questions have played a marginal role in the existing Colombeau literature.…”

Section: Introductionmentioning

confidence: 99%

Topological Structures in Colombeau Algebras: Topological $\widetilde{\mathbb{C}}$ -modules and Duality Theory

Garetto

2005

Acta Appl Math

177

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We study modules over the ring C of complex generalized numbers from a topological point of view, introducing the notions of C-linear topology and locally convex C-linear topology. In this context particular attention is given to completeness, continuity of C-linear maps and elements of duality theory for topological C-modules. As main examples we consider various Colombeau algebras of generalized functions. Mathematics Subject Classifications (2000): 46F30, 13J99, 46A20.Key words: modules over the ring of complex generalized numbers, algebras of generalized functions, topology, duality theory. IntroductionColombeau algebras of generalized functions have proved to be an analytically powerful tool in dealing with linear and nonlinear PDEs with highly singular coefficients [1-3, 8-10, 18, 21, 23, 24, 29, 30, 32-34, 36, 37, 41]. In the recent research on the subject a variety of algebras of generalized functions [3,12,13,17,19,22] have been introduced in addition to the original construction by Colombeau [6,7] and investigated in its algebraic and structural aspects as well as in analytic and applicative aspects. These investigations have produced a theory of point values in the Colombeau algebra G( ) and results of invertibility and positivity in the ring of constant generalized functions C [19, 38, 39] but also microlocal analysis in Colombeau algebras and regularity theory for generalized solutions to partial and (pseudo-)differential equations [11,13,15,16,[20][21][22][23][24][25][26][27]. Apart from some early and inspiring work by Biagioni, Pilipović, Scarpalézos [1,32,[46][47][48], topological questions have played a marginal role in the existing Colombeau literature. However, the recent papers on pseudodifferential operators acting on algebras of generalized functions [13,15,16] and a preliminary kernel theory introduced in

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“…The final section investigates regularity as well as compatibility of Colombeau-type solutions with classical and distributional solutions in case the coefficient matrices are sufficiently regular. Proposition 4.1 is an analog of the compatibility proposition in [23, p. 99] and [16,Corollary 5], whereas Proposition 4.2 is a G ∞ -variant of the regularity result [16,Proposition 6]. In Proposition 4.4 we establish convergence of the generalized solution to a weak solution for arbitrary Lipschitz continuous coefficients, thereby accompanying the case study with discontinuous coefficients in the acoustic transmission problem carried out in [26,Theorem 2.4 and Corollary 2.5].…”

Section: Introductionmentioning

confidence: 82%

“…In particular, certain log-type conditions on the coefficient matrices are essential in order to use a Gronwall-type argument in the proof (cf. [17,16,23,25,26]). The first theorem allows for the most general initial data and right-hand side, but requires the principal coefficients A j ε (t, x) to be bounded uniformly in ε and (t, x) for large |x|.…”

Section: Generalized Solutions To the Cauchy Problemmentioning

confidence: 99%

Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

Hörmann

Spreitzer

2012

Journal of Mathematical Analysis and Applications

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We study existence, uniqueness, and distributional aspects of generalized solutions to the Cauchy problem for first-order symmetric (or Hermitian) hyperbolic systems of partial differential equations with Colombeau generalized functions as coefficients and data. The proofs of solvability are based on refined energy estimates on lens-shaped regions with spacelike boundaries. We obtain several variants and also partial extensions of previous results in Oberguggenberger (1989), Lafon and Oberguggenberger (1991), and Hörmann (2004) [26,23,16] and provide aspects accompanying related recent work in Oberguggenberger (2009), Garetto and Oberguggenberger (2011) [28,10,9].

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First-order hyperbolic pseudodifferential equations with generalized symbols

Cited by 22 publications

References 12 publications

Hilbert $\widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems

Hilbert $\widetilde{\mathbb{C}}$-modules: Structural properties and applications to variational problems

Topological Structures in Colombeau Algebras: Topological $\widetilde{\mathbb{C}}$ -modules and Duality Theory

Symmetric hyperbolic systems in algebras of generalized functions and distributional limits

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