Averaging lemmas with a force term in the transport equation F Berthelin and S Junca

Description

Niveau: Supérieur, Doctorat, Bac+8
Averaging lemmas with a force term in the transport equation F. Berthelin and S. Junca November 24, 2008 Abstract We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. We compare the obtained regularities according to the space and velocity variables. Our results are mainly in L2, and for constant force, in Lp for 1 < p ≤ 2. Key-words: averaging lemma – force term – kinetic equation – stationary phase – Fourier series – Hardy space Mathematics Subject Classification: 1

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Sujets

• Navier?Stokes equations

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 15
• Langue: English

Extrait

Averaging

lemmas with a force term transport equation

F. Berthelin and S. Junca

LaboratoireJ.A.Dieudonn´e,CNRSUMR6621, Universite´deNiceSophia-Antipolis, Parc Valrose, 06108, Nice, France, [e-mail], [e-mail]

Abstract

in

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We obtain several averaging lemmas for transport operator with a force term. These lemmas improve the regularity yet known by not considering the force term as part of an arbitrary right-hand side. Two methods are used: local variable changes or stationary phase. These new results are subjected to two non degeneracy assumptions. We character- ize the optimal conditions of these assumptions to compare the obtained regularities according to the space and velocity variables. Our results are mainly inL 2 , and for constant force, inL p for 1< p≤2.

R´ ´ esume

Nousobtenonsplusieurslemmesdemoyennepourdese´quationsde transportavecuntermedeforce.Cesre´sultatsam´eliorentlare´gularite´ connueenneconsid´erantpasletermedeforcecommeuntermesource arbitraire.Deuxtechniquessontutilise´es:deschangementsdevari- ableslocauxoudesphasesstationnaires.Cesr´esultatssontquantifie´es pardeuxhypothe`sesdenond´eg´en´erescence.Nouscaract´erisonsles conditionsoptimalesdeceshypothe`sespourcomparerlesregularit´es ´ obtenues,parrapportauxvariablesd’espaceetdevitesse.Lesr´esultats sont principalement dansL 2 , et pour le cas constant, dansL p pour 1< p≤2.

Key-words lemma – force term – kinetic equation – stationary: averaging phase – non degeneracy conditions – Fourier series – Hardy space

Mathematics Subject Classification: 35B65, 42B20, 82C40.

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Contents

1 Introduction

2 First Theorem in theL 2 framework

3

Case of a constant force field

2

6

10

4 About non degeneracy conditions 16 4.1M= 1, one dimensional velocity . 17. . . . . . . . . . . . . . . . . 4.2M=N. . . . . . . . . . . . . . . 20. . . . . . . . . . . . . . . . 5 Theorem in theL p framework 20

1 Introduction

Averaging lemma is a major tool to get compactness from a kinetic equation. ([7], ...). Such results have been used in a lot of papers during these last years. Among this literature, an important result using an averaging lemma as a key argument is the proof of the hydrodynamic limits of the Boltzmann or BGK equations to the incompressible Euler or Navier-Stokes equations ([16]). Another major application consists in obtaining the compactness for nonlin- earscalarconservationlaws(in[25] s ) ([ w 6 h ] i ) challows,forinstance,tostudythe propagation of high frequency wave . Basically, averaging lemma is a result which says that the macroscopic quanti- ties Z f(t, x, v)ψ(v)dvhave a better regularity with respect to (t, x) than the F m o i r cr e o x s a co m p p i l c e, qua [ nt ] it a y n f( d t [ , 2] x , , v) wherefis solution of a kinetic equation. in 9 the following result is established. Theorem[DiPerna,Lions,Meyer–B´ezard] Letf,g k ∈L p ( R t × R xN × R M v )with1< p≤2such that ∂ t f+ div x [a(v)f] = X ∂ v k g k ,(1.1) |k|≤m witha∈W m,∞ ( R M , R N )form∈ N . Letψ∈W m,∞ ( R M )with compact support. LetA >0such that the support ofψis included in[−A, A] M  assume the. We following non-degeneracy fora(.): there exists0< α≤1andC >0such that for any(u, σ)∈S N andε >0, meas  {v∈[−A, A] M ;u−ε < a(v)∙σ < u+ε}  ≤Cε α .

Then ρ ψ (t, x) = Z R M f(t, x, v)ψ(v)dv is inW s,p ( R t × R xN )wheres= (m+ α 1)p 0 ,p 0 being the conjugated exponent forp. Regarding equation (1.1), the obtained regularity is proved to be optimal, see [23] and [24]. In [11], the gain of a half-derivative inL 2 context was proved as

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