Averaging lemmas with a force term in the transport equation

Description

Niveau: Supérieur, Doctorat, Bac+8
Averaging lemmas with a force term in the transport equation F. Berthelin and S. Junca Laboratoire J.A. Dieudonne, CNRS UMR 6621, Universite de Nice Sophia-Antipolis, Parc Valrose, 06108, Nice, France, , 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. 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 in L2, and for constant force, in Lp for 1 < p ≤ 2. Resume Nous obtenons plusieurs lemmes de moyenne pour des equations de transport avec un terme de force. Ces resultats ameliorent la regularite connue en ne considerant pas le terme de force comme un terme source arbitraire. Deux techniques sont utilisees : des changements de vari- ables locaux ou des phases stationnaires. Ces resultats sont quantifiees par deux hypotheses de non degenerescence. Nous caracterisons les conditions optimales de ces hypotheses pour comparer les regularites obtenues, par rapport aux variables d'espace et de vitesse. Les resultats sont principalement dans L2, et pour le cas constant, dans Lp pour 1 < p ≤ 2.

• changements de vari- ables locaux


• equation ∂tf


• sobolev exponent when


• transport equation


• valid when


• force term


• when considering


• navier stokes equations


• rt ?

Sujets

• Generic scalar transport equation
• Navier?Stokes equations

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 16
• 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

the

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

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6

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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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