Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

Aragona, J.; García, Ana E. Delgado; Juriaans, S. O.

doi:10.1016/j.na.2009.04.070

Cited by 15 publications

(17 citation statements)

References 21 publications

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“…This definition appears in [3] for example. Now, in the case of Ω, the topological sheaf of Ω ⊂ R n we have the Colombeau's full generalized functions on Ω,…”

Section: Definitions Results and Notationsmentioning

confidence: 99%

A Discontinuous Differential Calculus in the Framework Colombeau's Full Algebra

Cortes¹,

García²,

Silva³

2017

Preprint

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Starting from the Colombeau's full generalized functions, the sharp topologies and the notion of generalized points, we introduce a new kind differential calculus (for functions between totally disconnected spaces). We study generalized pointvalues, Colombeau's differential algebra, holomorphic and analytic functions. We show that the Embedding Theorem and the Open Mapping Theorem hold in this framework. Moreover, we study some applications in differential equations.

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“…This definition appears in [3] for example. Now, in the case of Ω, the topological sheaf of Ω ⊂ R n we have the Colombeau's full generalized functions on Ω,…”

Section: Definitions Results and Notationsmentioning

confidence: 99%

A Discontinuous Differential Calculus in the Framework Colombeau's Full Algebra

Cortes¹,

García²,

Silva³

2017

Preprint

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“…P r o o f. It follows from the completude of algebras G Ω and G(Ω) (see [3]) and from the fact that K f is the ring of the constants of such algebras. Proposition 1.10 K f , τ sf is not:…”

Section: Corollary 18 For Givenmentioning

confidence: 99%

Colombeau's Algebra of full Generalized Numbers

Aragona,

Garcia,

Juriaans

2009

Preprint

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Let K f denote the commutative unital ring of Colombeau's full generalized numbers. This ring can be endowed with an ultra-metric in such a way that it becomes a topological ring. There are many interesting question about K f in the framework of Commutative Algebra and General Topology as well as of the superposition of these two subjects. The purpose of this paper aims to give an initial step toward the study of this ring.

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“…An interesting point is the fact that M. Grosser [28] has proved that statement (7) is equivalent to the particular case n = 0. We define the algebra of nonlinear generalized functions as the quotient…”

Section: We Setmentioning

confidence: 99%

“…It was called "sharp topology" by D. Scarpalezos [34,41,42] who gave the impulse for its use. The sharp (or Scarpalezos) topology was at the origin of numerous works [10,41,42,8,3,5,6,22,24,25,26,9,2,7,36]. Besides its good properties it is not a usual Hausdorff locally convex algebra topology on the field of real or complex numbers as this would have been welcome for a development of functional analysis in the context of nonlinear generalized functions in order to benefit of compactness and of the well developed mathematical theory of locally convex spaces.…”

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confidence: 99%

Locally convex topological algebras of generalized functions: Compactness and nuclearity in a nonlinear context

Aragona

Juriaans

Colombeau

2015

Trans. Amer. Math. Soc.

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In this paper we introduce Hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions. These topologies are strong duals of Fréchet-Schwartz space topologies and even strong duals of nuclear Fréchet space topologies. In particular any bounded set is relatively compact and one benefits from all deep properties of nuclearity. These algebras of generalized functions contain most of the classical irregular functions and distributions. They are obtained by replacing the mathematical tool of C ∞ functions in the original version of nonlinear generalized functions by the far more evolved tool of holomorphic functions. This paper continues the nonlinear theory of generalized functions in which such locally convex topological properties were strongly lacking up to now.Key words: Functional analysis, nonlinear generalized functions, sharp topology, differential calculus of nonlinear generalized functions, nonlinear operations on distributions, locally convex topological algebras, compact maps, Schwartz locally convex spaces, nuclear maps, nuclear spaces, .AMS classification: 46F30, 46F10. (+) the work of this author was done under financial support of FAPESP, 1 processo 2011/12532-1, and thanks to the hospitality of the IME-USP. (+) corresponding author jfcolombeau@ime.usp.br 1. Introduction. We denote by G(Ω) the special (or simplified) algebra of nonlinear generalized functions on Ω. We construct subalgebras of G(Ω) that enjoy Hausdorff locally convex topologies suitable for the development of a functional analysis: they have very good topological properties in particular concerning compactness (strong duals of Fréchet-Schwartz spaces) at the same time as they are compatible both with partial derivatives and nonlinearity. These properties even extend to nuclearity. Nuclear spaces [38,40,45] are an extension of finite dimensional spaces in which finite sums are replaced by convergent series in a natural topology: a difference with Hilbert spaces lies in that in the nuclear (DFN) algebras the bounded sets are relatively compact (even much more: they resemble to some extent the finite dimensional bounded sets) at the price of the use of nonmetrizable topologies which are topological inductive limits of a sequence of separable Hilbert spaces linked by nuclear inclusion maps.

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Generalized solutions of a nonlinear parabolic equation with generalized functions as initial data

Cited by 15 publications

References 21 publications

A Discontinuous Differential Calculus in the Framework Colombeau's Full Algebra

A Discontinuous Differential Calculus in the Framework Colombeau's Full Algebra

Colombeau's Algebra of full Generalized Numbers

Locally convex topological algebras of generalized functions: Compactness and nuclearity in a nonlinear context

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