On the regularity of averages over spheres for kinetic transport equations in hyperbolic Sobolev spaces

Bournaveas, Nikolaos; Gutiérrez, Susana

doi:10.4171/rmi/503

Cited by 6 publications

(13 citation statements)

References 21 publications

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“…We shall see later that there is a natural way to unify the smoothing estimates we seek for these velocity domains, so for the sake of simplicity of the exposition, we focus this introductory discussion on the unit sphere. As proved in the work of In very recent work [1], the purely L 2 -based results in [8] and [7] were significantly extended to estimates of the form…”

Section: Introductionmentioning

confidence: 79%

Smoothing Estimates for the Kinetic Transport Equation at the Critical Regularity

Bez¹,

Cunanan²,

Lee³

2018

SIAM J. Math. Anal.

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We prove smoothing estimates for velocity averages of the kinetic transport equation in hyperbolic Sobolev spaces at the critical regularity, leading to a complete characterisation of the allowable regularity exponents. Such estimates will be deduced from some mixed-norm estimates for the cone multiplier operator at a certain critical index. Our argument is not particular to the geometry of the cone and we illustrate this by establishing analogous estimates for the paraboloid.

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Section: Introductionmentioning

confidence: 79%

Smoothing Estimates for the Kinetic Transport Equation at the Critical Regularity

Bez¹,

Cunanan²,

Lee³

2018

SIAM J. Math. Anal.

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“…These averages were studied in [6][7][8]. When the right hand side doesn't contain velocity derivatives it was shown that…”

Section: N Bournaveas and H Wang Nodeamentioning

confidence: 99%

“…We refer the reader to [6][7][8] for a detailed study. In this paper we study velocity averages of the form…”

Section: N Bournaveas and H Wang Nodeamentioning

confidence: 99%

“…In this paper we show how the methods of [6][7][8] can be used to prove velocity averaging lemmas in hyperbolic Sobolev spaces H s,δ for the kinetic transport equation…”

Section: Introductionmentioning

confidence: 99%

“…

We show how the methods of [6][7][8] can be used to prove velocity averaging lemmas in hyperbolic Sobolev spaces for the kinetic transportHere v is allowed to vary in the whole space R d and the velocity field a(v) = v |v| lies on the unit sphere. We work in dimensions d ≥ 3 and, in contrast with [6,8], we allow right-hand sides with velocity derivatives in any direction and not necessarily tangential to the sphere.

Mathematics Subject Classification (2000)

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confidence: 99%

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Velocity Averaging Lemmas in Hyperbolic Sobolev Spaces for the Kinetic Transport Equation with Velocity Field on the Sphere

Bournaveas

Wang

2009

Nonlinear differ. equ. appl.

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Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

Bennett

Bez

Gutiérrez

et al. 2018

Journal de Mathématiques Pures et Appliquées

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We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator ρ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents q and r, we find a characterisation of the exponents β + and β − , except possibly for an endpoint case, for which DHere, D + and D − are the classical and hyperbolic derivative operators, respectively. In fact, we shall provide an argument which unifies these velocity domains and the velocity averaging estimates in either case are shown to be equivalent to mixed-norm bounds on the cone multiplier operator acting on L 2 . We develop our ideas further in several ways, including estimates for initial data lying in certain Besov spaces, for which a key tool in the proof is the sharp ℓ p decoupling theorem recently established by Bourgain and Demeter. We also show that the level of permissible smoothness increases significantly if we restrict attention to initial data which are radially symmetric in the spatial variable.

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On the regularity of averages over spheres for kinetic transport equations in hyperbolic Sobolev spaces

Abstract: Abstract. We study the smoothing effect of averaging over spheres for solutions of kinetic transport equations in hyperbolic Sobolev spaces.

Cited by 6 publications

References 21 publications

Smoothing Estimates for the Kinetic Transport Equation at the Critical Regularity

Smoothing Estimates for the Kinetic Transport Equation at the Critical Regularity

Velocity Averaging Lemmas in Hyperbolic Sobolev Spaces for the Kinetic Transport Equation with Velocity Field on the Sphere

Estimates for the kinetic transport equation in hyperbolic Sobolev spaces

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