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A Real Space Method for Averaging Lemmas

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47 pages
A Real Space Method for Averaging Lemmas Pierre-Emmanuel Jabin? email: , Ecole Normale Superieure Departement de Mathematiques et Applications, CNRS UMR 8553 45 rue d'Ulm, 75230 Paris Cedex 05, France Luis Vega email: , Universidad del Paıs Vasco Departamento de Matematicas Bilbao 48080 Spain Abstract. We introduce a new method to prove averaging lemmas, i.e. prove a regularizing effect on the average in velocity of a solution to a kinetic equation. The method does not require the use of Fourier transform and the whole procedure is performed in the 'real space'. We are consequently able to improve the known result when the integrability of the solution (or the right hand side of the equation) is different in space and in velocity. We also present a few counterexamples to test the optimality of the new results. Resume. Nous presentons une nouvelle methode pour obtenir des lemmes de moyenne, c'est-a-dire un effet regularisant sur les moyennes en vitesse d'une equation cinetique. Cette methode ne fait pas appel a la transformee de Fourier et toute la preuve se fait dans l'espace reel. Par consequent, nous sommes capables d'ameliorer les resultats connus quand l'integrabilite de la solution (ou du second membre de l'equation) est differente en espace et en vitesse.

  • optimal regularity

  • phase space

  • depends only

  • following result

  • estimations de disper- sion

  • equation


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A
Real
Space
Method for Averaging
Pierre-Emmanuel Jabinemail: jabin@dma.ens.fr, ´ EcoleNormaleSupe´rieure
Lemmas
D´epartementdeMathe´matiquesetApplications,CNRSUMR8553
45 rue d’Ulm, 75230 Paris Cedex 05, France Luis Vega
email: mtpvegol@lg.ehu.es,
UniversidaddelPaı´sVasco
DepartamentodeMatem´aticas
Bilbao 48080 Spain
Abstract.introduce a new method to prove averaging lemmas,We i.e. prove a regularizing effect on the average in velocity of a solution to a kinetic equation. The method does not require the use of Fourier transform and the whole procedure is performed in the ’real space’. We are consequently able to improve the known result when the integrability of the solution (or the right hand side of the equation) is different in space and in velocity. We also present a few counterexamples to test the optimality of the new results. Re´sum´e.semmebtroouepsldeirenotnenusnpsuose´r´eemodthouenllveN demoyenne,cest-`a-direuneetre´gularisantsurlesmoyennesenvitesse dune´equationcin´etique.Cettem´ethodenefaitpasappela`latransforme´e deFourierettoutelapreuvesefaitdanslespacere´el.Parconse´quent,nous sommescapablesdame´liorerlesresultatsconnusquandlinte´grabilite´dela ´ solution(oudusecondmembredel´equation)estdi´erenteenespaceeten vitesse.Nousdonnonse´galementquelquescontre-exemplespourv´erierle caracte`reoptimaldesnouveauxre´sultats.
projects PICS 1014 and HPRN-CT-2002-00282 and apartially supported by the MCyT grant
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Key words.Regularizing effects, averaging lemmas, dispersion estimates, X-ray transform. Mots-cle´s.-erdensspdiemmesdemrisant,ltsmitaoiyoneene,aluge´rsteE sion,transforme´eauxrayonsX. Mathematics Subject Classification35B65, 82C40, 47G10.
1 Introduction
1.1 Main results
We study the following stationary kinetic equation v∙ rxf(x, v) = Δα/2g(x, v), xRd, vRd,0α <1.
(1.1)
As a transport equation, (1.1) has typically no regularizing effects (although in some cases it does, see at the end of the paper). However in many applica-tions, the important physical quantity is notfitself but some of its moments so that we are interested in the optimal regularity of a quantity like Z
ρ(x) =f(x, v)φ(v)dv, φCc(Rd) given. Rd
(1.2)
It is also possible to consider an average on the sphere, with the same gain in regularity, Z
˜(x) =f(x, v)φ(v)(v), φCc(Sd1) given. ρ |v|=1
(1.3)
It turns out that the averageρis more regular thanf(as long asα <1 of course) as it was first noticed in [16] in anL2framework. Since that paper numerous works have been devoted to proving the optimal regularity for the average. The study is motivated by a large class of kinetic equations where the non linear term may be controlled by some average of the solution and by kinetic formulations where the average is the only important quantity.
The gain in regularity depends on the smoothness offandgthemselves. In comparison with previous works, we will use different spaces in velocity and space (see a more detailed discussion after the presentation of the results).
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Consequently the functionsfandg, defined in the phase space, are assumed to be in the following spaces fWp,βv1(Rd, Lpx2(Rd)), β0, gWγq,v1(Rd, Lq2(Rd)),−∞< γ <1.(1.4) ˙ We denote byBr,st,uthe space which is obtained by real interpolation of two Besov spacesBurs,much like the classical Besov spaces can be obtained by real interpolation of Sobolev spaces. The first result which we prove is the following theorem
Theorem 1.1Letfandgsatisfy(1.1)and(1.4)with1< p2, q2<, 1p1min(p2, p2)and1q1min(q2, q2)where for a generalp,pis the dual exponent ofp, and assume moreover thatγ1/q1<0. Then, , kρkB˙sr,,Ckfk1Wθ,p1(Lpx2)× kgkWθvγ,q1(Lxq2)
with
1 1θ = +s,θq= (1α)θ, r p2 2 θ1 +β11+/β1/p1 = p1./ γ+ 1q1
(1.5)
Remarks. 1. This theorem contains most of the previous results (in particular the ones in [11] and [19]). It extends naturally the result given in [19] forβ <1/2. 2. We do not know whether in this case the average belongs to the true Sobolev spaceWs,r. This optimal space was obtained in [3] for the usual case (p1=p2,q1=q2andβ This= 0). is certainly true ifp1=p2and q1=q2but some difficulties could arise when the exponents are different. In any case, the simple but rough method of interpolation which we choose s, r here cannot do better thanB,. 3. We do not have any trouble with exponentsp1orq1equal to 1, only with p2orq2. 4. The gain of regularity depends only on the regularity and integrability in velocity. This corresponds to [31] where the average is obtained in a space weaker but with the same homogeneity as ours by Sobolev embedding.
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However contrary to [31], we have a limitation on the exponent in velocity (see the section about optimality).
Since we work with different spaces in space and velocity, the order in which the norms are taken is very important. In (1.4) we take first the norm inx and then the norm inv. Asp1p2orq1q2, this is a stronger assumption than the contrary (the norm invfirst). So a natural question is whether it is possible to invert the order of the spaces. We are able to give a full answer only in dimension two.
Proposition 1.1Ifd= 2, Letfandgsatisfy(1.1)but assumegis like in (1.4)butfinLxp2(Wvβ,p1)(respectivelygLxq2(Wvβ,q1)andflike in(1.4)) provided we still havep1p2and moreoverp22(resp.q1q2and q22) then kρkW˙s,rCkfk1Lxp2θ(Wvβ,p1)× kgkθLxq2(Wvγ,q1), with
r1=1p2θ+qθ2, s= (1α)θ <(1α)θ0, 1 θ0=1+β+1/pβ11γp/1+ 1/q1.
(1.6)
Remark. This result is optimal in the sense that the conclusion is false ifp2>2 or q2>equivalent in higher dimensions, but we can cannot prove an 2. We show that the limit onp2orq2is in generaldwith 1/d= 11/d, see the discussion at the end of the proof of the proposition.
Theorem 1.1 exhibits a sort of saturation: The regularity of the average does not improve whenp1grows beyondp2this point, it is very interesting to . At invert the norms because that means we work in the strongest space. So let us assume now thatfandgsatisfy fLpx2(Rd, Wp,vβ1(Rd) 0 gLq2(Rd, Wvq,γ1(Rd))),,β<γ<,1.(1.7) With this new framework, we can prove (but for the moment only in dimen-sion two) the
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