Weak Solutions of the Linear Boltzmann Equation with Very Soft Interactions

Chvála, František; Pettersson, Robert

doi:10.1006/jmaa.1995.1135

Cited by 3 publications

(9 citation statements)

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“…The purpose of this paper is to generalize our earlier results in [13] on existence of weak solutions for very soft elastic collisions (including the Coulomb forces), in the Cauchy problem case, to inelastic (granular) collisions in a bounded body, and (for simplicity) without external forces.…”

Section: Introductionmentioning

confidence: 84%

“…To get Proposition B, start with differentiation along the characteristics of iterate functions f n N in (16)- (18), multiply by (1 + v 2 ) σ /2 and integrate, with some change of variables, cf. [13]. Then, by the proposition (above) on velocities after a binary collision, using only the positive part (at the right hand side) of the estimate, one get the integrals of type…”

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

confidence: 96%

“…We start our calculations with a lemma, estimating the difference between the test functions in the collision term(16); cf. Lemma 2.3. in [13].…”

Section: Weak Solutions In the Coulomb Casementioning

confidence: 99%

“…Now, using a Gronwall type estimate, the (local) boundedness of velocity moments are reached for the iterates f n N , and finally the monotonicity property f n N ↑ f n , n → ∞, gives the result; cf. [13] and further references.…”

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

confidence: 99%

“…Proof. (Sketch) See mainly the paper [13]. Define first a sequence of functions by (18) and then use the weak compactness lemma together with boundedness of higher moments.…”

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

confidence: 99%

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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: Introductionmentioning

confidence: 84%

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

confidence: 96%

“…We start our calculations with a lemma, estimating the difference between the test functions in the collision term(16); cf. Lemma 2.3. in [13].…”

Section: Weak Solutions In the Coulomb Casementioning

confidence: 99%

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

confidence: 99%

“…Proof. (Sketch) See mainly the paper [13]. Define first a sequence of functions by (18) and then use the weak compactness lemma together with boundedness of higher moments.…”

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

confidence: 99%

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