Intermolecular forces of infinite range and the Boltzmann equation

Arkeryd, Leif

doi:10.1007/bf00280403

Cited by 108 publications

(108 citation statements)

References 3 publications

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“…The existence theorem for very soft solutions to the linear Boltzmann equation is based on a compactness lemma, which is analogous to that given by Arkeryd in [14], (cf, also Lemma 4.1 in [5]). The proof is omitted here; see Lemma 2.5 in [13].…”

Section: Proposition 3 A) Let For the Coulomb Case (Kmentioning

confidence: 98%

On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions

Pettersson¹

2011

AIP Conference Proceedings

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Abstract. This paper considers the time-and space-dependent linear Boltzmann equation with general boundary conditions in the case of inelastic (granular) collisions. First, in the (angular) cut-off case, mild L 1 -solutions are constructed as limits of the iterate functions and boundedness of higher velocity moments are discussed in the case of inverse power collisions forces. Then the problem of the weak solutions, as weak limit of a sequence of mild solutions, is studied for a bounded body, in the case of very soft interactions (including the Coulomb case). Furthermore, strong convergence of weak solutions to the equilibrium, when time goes to infinity, is discussed, using a generalized H-theorem, together with a translation continuity property.

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Section: Proposition 3 A) Let For the Coulomb Case (Kmentioning

confidence: 98%

On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions

Pettersson¹

2011

AIP Conference Proceedings

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“…Even more advanced examples such as the proof of the invariant subspace conjecture are explained in your book [45]. For an application to the Bolzmann equation, see Arkeryd [4,5].…”

Section: Question 811 Given What Relativity Has Taught Us About Timmentioning

confidence: 99%

“…The earlier reconstruction was implemented by the great triumvirate 3 of Cantor, Dedekind, and Weierstrass. The ontological target of the triumvirate reconstruction was the abstract entity called the infinitesimal , a basic building block of a continuum, 4 according to a line of investigators harking back to the Greek antiquity. 5 To place these historical developments in context, it is instructive to examine Felix Klein's remarks dating from 1908.…”

Section: Introductionmentioning

confidence: 99%

“…C. Boyer refers to Cantor, Dedekind, and Weierstrass as "the great triumvirate", see [25, p. 298]. 4 For an entertaining history of infinitesimals see P. Davis and R. Hersh [46, p. 237-254]. For an analysis of a variety of competing theories of the continuum, see P. Ehrlich [51] as well as R. Taylor [157].…”

Section: Introductionmentioning

confidence: 99%

“…A discussion of infinitesimal optics is in K. Stroyan [149], J. Keisler [93], D. Tall [154], and L. Magnani and R. Dossena [109,48], and Bair & Henry [7]. Applications of the B-continuum range from aid in teaching calculus [54,84,85,155,156] (see illustration in Figure 1) to the Bolzmann equation (see L. Arkeryd [4,5]); modeling of timed systems in computer science (see H. Rust [135]); mathematical economics (see Anderson [3]); mathematical physics (see Albeverio et al [1]); etc.…”

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confidence: 99%

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A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

Katz

2011

Found Sci

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Abstract. We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of an ontologically limitative disposition on historiography, teaching, and research.

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On the Boltzmann equation for long‐range interactions

Reynaud

Villani

2001

Comm Pure Appl Math

197

238

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We study the Boltzmann equation without Grad's angular cutoff assumption. We introduce a suitable renormalized formulation that allows the cross section to be singular in both the angular and the relative velocity variables. Angular singularities occur as soon as one is interested in long-range interactions, while singularities in the relative velocity variable occur in the study of soft potentials, in particular, Coulomb interaction. Together with several new estimates, this new formulation enables us to prove existence of weak solutions and to give a proof of a conjecture by Lions (appearance of strong compactness) under general, fully realistic assumptions.

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Intermolecular forces of infinite range and the Boltzmann equation

Cited by 108 publications

References 3 publications

On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions

On Weak Solutions to the Linear Boltzmann Equation with Inelastic Coulomb Collisions

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

On the Boltzmann equation for long‐range interactions

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