We show that the endpoint Strichartz estimate for the kinetic transport
equation is false in all dimensions. We also present a new approach to proving
the non-endpoint cases using multilinear analysis.Comment: 5 page

Bennett, Jonathan

Bez, Neal

Gutierrez, Susana

Lee, Sanghyuk

Communications in Partial Differential Equations

English

arXiv

We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present an alternative approach to proving the non-endpoint cases using multilinear analysis

University of Birmingham Research Portal

  On the Strichartz estimates for the kinetic transportequationBennett, Jonathan; Bez, Richard; Gutierrez, Susana; Lee, SanghyukDOI:10.1080/03605302.2013.850880License:Creative Commons: Attribution (CC BY)Document VersionPublisher's PDF, also known as Version of recordCitation for published version (Harvard):Bennett, J, Bez, N, Gutierrez, S & Lee, S 2014, 'On the Strichartz estimates for the kinetic transport equation',Communications in Partial Differential Equations, vol. 39, no. 10, pp. 1821-1826.https://doi.org/10.1080/03605302.2013.850880Link to publication on Research at Birmingham portalPublisher Rights Statement:Eligibility for repository : checked 12/09/2014General rightsUnless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or thecopyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposespermitted by law.•	Users may freely distribute the URL that is used to identify this publication.•	Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of privatestudy or non-commercial research.•	User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?)•	Users may not further distribute the material nor use it for the purposes of commercial gain.Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.When citing, please reference the published version.Take down policyWhile the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has beenuploaded in error or has been deemed to be commercially or otherwise sensitive.If you believe that this is the case for this document, please contact UBIRA@lists.bham.ac.uk providing details and we will remove access tothe work immediately and investigate.Download date: 01. Feb. 2019This article was downloaded by: [University of Birmingham]On: 12 September 2014, At: 06:41Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UKCommunications in Partial Differential EquationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lpde20On the Strichartz Estimates for the Kinetic TransportEquationJonathan Bennett a , Neal Bez b , Susana Gutiérrez a & Sanghyuk Lee ca School of Mathematics , University of Birmingham , Birmingham , Englandb Department of Mathematics , Saitama University , Saitama , Japanc Department of Mathematical Sciences , Seoul National University , Seoul , KoreaPublished online: 19 Aug 2014.To cite this article: Jonathan Bennett , Neal Bez , Susana Gutiérrez & Sanghyuk Lee (2014) On the StrichartzEstimates for the Kinetic Transport Equation, Communications in Partial Differential Equations, 39:10, 1821-1826, DOI:10.1080/03605302.2013.850880To link to this article:  http://dx.doi.org/10.1080/03605302.2013.850880PLEASE SCROLL DOWN FOR ARTICLETaylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained inthe publications on our platform. Taylor & Francis, our agents, and our licensors make no representations orwarranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Versionsof published Taylor & Francis and Routledge Open articles and Taylor & Francis and Routledge Open Selectarticles posted to institutional or subject repositories or any other third-party website are without warrantyfrom Taylor & Francis of any kind, either expressed or implied, including, but not limited to, warranties ofmerchantability, fitness for a particular purpose, or non-infringement. Any opinions and views expressed in thisarticle are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. Theaccuracy of the Content should not be relied upon and should be independently verified with primary sourcesof information. Taylor & Francis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly inconnection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Terms & Conditions of access anduse can be found at http://www.tandfonline.com/page/terms-and-conditions It is essential that you check the license status of any given Open and Open Select article to confirmconditions of access and use.Communications in Partial Differential Equations, 39: 1821–1826, 2014Published with license by Taylor & FrancisISSN 0360-5302 print/1532-4133 onlineDOI: 10.1080/03605302.2013.850880On the Strichartz Estimates for the KineticTransport EquationJONATHAN BENNETT1, NEAL BEZ2, SUSANAGUTIÉRREZ1, AND SANGHYUK LEE31School of Mathematics, University of Birmingham, Birmingham, England2Department of Mathematics, Saitama University, Saitama, Japan3Department of Mathematical Sciences, Seoul National University,Seoul, KoreaWe show that the endpoint Strichartz estimate for the kinetic transport equation isfalse in all dimensions. We also present an alternative approach to proving the non-endpoint cases using multilinear analysis.Keywords Endpoint estimates; Kinetic transport equations; Multilinear analysis;Strichartz estimates.Mathematics Subject Classiﬁcation 42B37.1. IntroductionThe solution of the kinetic transport equationtft x v+ v · xft x v = 0 f0 x v = f 0x vfor t x v ∈ ×d ×d, satisﬁes the Strichartz estimatesfLqt LpxLrv  f 0Laxv  (1.1)where2q= d(1r− 1p)1a= 12(1r+ 1p) q > a p ≥ a (1.2)(We write X  Y and Y  X if X ≤ CY for some ﬁnite constant C dependingat most on the parameters q p r a d, and X ∼ Y if X  Y and X  Y .) With© Jonathan Bennett, Neal Bez, Susana Gutiérrez and Sanghyuk LeeReceived August 28, 2013; Accepted September 23, 2013Address correspondence to Neal Bez, Department of Mathematics, Graduate Schoolof Science and Engineering, Saitama University, Saitama 338-8570, Japan; E-mail:nealbez@mail.saitama-u.ac.jp1821Downloaded by [University of Birmingham] at 06:41 12 September 2014 1822 Bennett et al.q p r a satisfying (1.2), but with the further condition q > 2 ≥ a, this was provedby Castella and Perthame [1], and it was observed by Keel and Tao [5] that thislatter condition can be relaxed to q > a and hence (1.2) sufﬁces. In [5] it is tentativelyconjectured that the Strichartz estimate (1.1) holds at the endpoint q = a, at leastfor d > 1. Using the invariance under the transformationsf 0 ↔ f 0 f ↔ f q p r a ↔(qpra)(1.3)this conjectured endpoint can be (and usually is) stated for initial data in L2xv asfL2t L2dd−1x L2dd+1v f 0L2xv  (1.4)The main purpose of this paper is to disprove this conjecture.Theorem 1. The endpoint Strichartz estimate (1.4) for the kinetic transport equationfails for all d ≥ 1.The case d = 1 of Theorem 1 was proved by Guo and Peng [4] and later byOvcharov [7] using different arguments (where the norm in x on the left-hand sideis Lx ). The argument we present uses key ideas from recent work of Frank et al. [3].We remark that for the free Schrödinger propagator, the endpoint Strichartzestimate fails in the case d = 2 (see [6]) but is true for all d > 2, as shown in thelandmark paper of Keel and Tao [5]. Thus, Theorem 1 highlights a fundamentaldifference in the Strichartz estimates for these related equations.In the next section, we prove Theorem 1. In the ﬁnal section, we provide analternative proof of the Strichartz estimates (1.1) in all non-endpoint cases usingmultilinear analysis.2. Proof of Theorem 1Using the invariance under the transformation (1.3) with  = 2dd+1 , estimate (1.4) isequivalent tofLd+1dt Ld+1d−1x L1v f 0Ld+1dxv (2.1)Since ft x v = f 0x − tv v, it is clear that (2.1) impliesf 0Ld+1dt Ld+1d−1x f 0Ld+1dxv (2.2)where f 0 is the macroscopic density deﬁned by the linear mappingf 0t x =∫df 0x − tv v dvHence, by duality, (2.2) implies∗gLd+1xv  gLd+1t L d+12x  (2.3)Downloaded by [University of Birmingham] at 06:41 12 September 2014 On the Strichartz Estimates 1823where the adjoint ∗ is given by∗gx v =∫gt x + tv dtFrom here, the argument strongly uses ideas from the paper of Frank et al.[3] concerning reﬁned Strichartz estimates for the free Schrödinger propagatorassociated with orthonormal initial data.Suppose g ∈  ×d \ 	0 is nonnegative and such that ĝ ∈ Cc ×d isalso nonnegative. Here, we use ĝ to denote the space-time Fourier transform of ggiven byĝ  =∫×dgt xe−it+x· dtdxIn this proof, we shall also use c to denote a constant depending on at most d, whichmay change from line to line.Proceeding formally, using Fourier inversion we get∗gd+1Ld+1xv=∫ d+1∏j=1gtj x + tjv d	tdxdv= c∫ d+1∏j=1ĝj jd+1∏k=1eitkk+v·keix·∑d+1=1  d	d	d	tdxdv= c∫ d+1∏j=1ĝj jd+1∏k=1k + v · k ( d+1∑=1)d	d	dv= c∫ d+1∏j=1ĝ−v · j j ( d+1∑k=1k)d	dvand hence∗gd+1Ld+1xv= c∫ d∏j=1ĝ−v · j j ̂g(v ·d∑k=1k−d∑=1)d	dv (2.4)We remark that by appropriately truncating the integrals in the above identitiesand limiting arguments, (2.4) makes sense in 0 for the class of g underconsideration.Deﬁne K to be the d by d matrix whose consecutive rows are −1    −d.Using the change of variables w = Kv, so that wj = −j · v for each 1 ≤ j ≤ d, weobtain∗gd+1Ld+1xv= c∫ d∏j=1ĝwj j ̂g(−d∑k=1wk−d∑=1)1 detKd 	wd	Writing each j = rjj in polar coordinates, we have detK =( d∏j=1rj) det1 · · · dDownloaded by [University of Birmingham] at 06:41 12 September 2014 1824 Bennett et al.Since ĝ0 0 > 0 and ĝ is continuous, it follows that∗gd+1Ld+1xv∫r1∫d−1d( d∏j=1rd−2j)1 det1 · · · dd	rd	 (2.5)For d = 1 the radial integral is inﬁnite, and for d ≥ 2,∫d−1d1 det1 · · · dd	 = so the angular integral is inﬁnite. Hence, for all d ≥ 1 we have shown that (2.3), andconsequently (1.4), cannot hold.Remark. The above argument shows that the endpoint estimate (2.3) fails rathergenerically. For example, the space-velocity norm ∗gd+1 is inﬁnite whenever g ∈ ×d \ 	0 is nonnegative and such that ĝ ∈ Cc ×d is also nonnegative.3. A Multilinear Approach to the Non-Endpoint CasesFix  > 1. In this section, the notation  allows, in addition, the implicit constantto depend on .We shall prove∗gLd+1xv gLqt Ld+12x (3.1)for all g ∈ Lqt Ld+12x , where the exponent q satisﬁes1q+ dd + 1 = 1Our argument is a multilinear variant of that in [1] and [9] (see also [8]), which dealswith all non-endpoint cases directly. Similar multilinear arguments have been usedrecently in [3].Using the invariance under transformations in (1.3), to prove the full range ofnon-endpoint Strichartz estimates, it sufﬁces to consider q p a satisfyingq > a p ≥ a 2q= d(1− 1p)1a= 12(1+ 1p)(3.2)and show that (1.1) holds with r = 1, or equivalently, thatf 0Lqt Lpx  f 0Laxvholds for all f 0 ∈ Laxv. By duality, this is equivalent to∗gLa′xv  gLq′t Lp′x (3.3)Downloaded by [University of Birmingham] at 06:41 12 September 2014 On the Strichartz Estimates 1825for all g ∈ Lq′t Lp′x . Note that (3.2) implies that a′ = 2p′ and 1q′ + da′ = 1, in which case(3.3) reads∗gLa′xv  gLq′t La′2xand the condition q > a is equivalent to a′ > d + 1. Therefore, (3.1) with  = a′d+1 >1 implies the full range of non-endpoint Strichartz estimates (1.1).Proof of (3.1). Without loss of generality, suppose g is nonnegative. By multiplyingout and using Minkowski’s integral inequality, we get∗gd+1Ld+1xv=( ∫ ( ∫ d+1∏j=1gtj x + tjvd	t)dxdv)1/≤∫ ( ∫ d+1∏j=1gtj x + tjvdxdv)1/d	tNow ﬁx t1     td+1 and consider the multilinear form∫ d+1∏j=1gjtj x + tjvdxdvA straightforward estimate via the change of variables x v → x + tiv x + tjvgives∫ d+1∏j=1gjtj x + tjvdxdv 1ti − tjdgiti ·L1xgjtj ·L1x∏k =ijgktk ·Lxfor each 1 ≤ i < j ≤ d. A multilinear interpolation argument yields∫ d+1∏j=1gjtj x + tjvdxdv ∏1≤i<j≤dti − tj− 2d+1d+1∏k=1gktk ·Ld+12xApplying this with g for each gj we get∗gd+1Ld+1xv∫d+1∏1≤i<j≤dti − tj−2d+1d+1∏k=1gtk ·Ld+12xd	t gd+1Lqt Ld+12xThe last inequality is a consequence of the multilinear Hardy–Littlewood–Sobolevinequality due to Christ [2]. Remark. It seems natural to conjecture that ∗ satisﬁes the weak-type estimate∗gLd+1xv  gLd+1t L d+12x Downloaded by [University of Birmingham] at 06:41 12 September 2014 1826 Bennett et al.A similar comment was made in an operator-theoretic context in [3].Certain replacements for the endpoint are already known. For example, withq = a = 2, in [5], Keel and Tao obtain a substitute for (1.4) with the LpxLrv norm(where p = 2dd−1 and r = 2dd+1 ) replaced by that of a certain real interpolation spacewhich is between Lp1x Lr1v and Lpx Lrv . See also work of Ovcharov [8] where adifferent substitute bound was given for velocities v belonging to a bounded subsetof d.AcknowledgmentsWe would like to thank José Cañizo for his contributions during the early stages ofthis project.FundingThis work was supported by the European Research Council (grant number 307617;Bennett); the Engineering and Physical Sciences Research Council (grant numbersEP/J021490/1, EP/J01155X/1; Bez, Gutiérrez); National Research Foundation ofKorea (grant number 2012008373; Lee).References[1] Castella, F., Perthame, B. (1996). Estimations de Strichartz pour les èquationsde transport cinétique [Strichartz estimates for kinetic transport equations]. C.R. Acad. Sci. Paris Sér. I Math. 332:535–540.[2] Christ, M. (1985). On the restriction of the Fourier transform to curves:endpoint results and the degenerate case. Trans. Amer. Math. Soc. 287:223–238.[3] Frank, R., Lewin, M., Lieb, E.H., Seiringer, R. A Strichartz inequality fororthonormal functions. J. Euro. Math. Soc., to appear.[4] Guo, Z., Peng, L. (2007). Endpoint Strichartz estimate for the kinetic transportequation in one dimension. C. R. Math. Acad. Sci. Paris 345:253–256.[5] Keel, M., Tao, T. (1998). Endpoint Strichartz estimates. Amer. J. Math.120:955–980.[6] Montgomery-Smith, S.J. (1998). Time decay for the bounded mean oscillationof solutions of the Schrödinger and wave equations. Duke Math. J. 91:393–408.[7] Ovcharov, E. (2011). Counterexamples to Strichartz estimates for the kinetictransport equation based on Besicovitch sets. Nonlinear Anal. 74:2515–2522.[8] Ovcharov, E. (2011). Strichartz estimates for the kinetic transport equation.SIAM J. Math. Anal. 43:1282–1310.[9] Perthame, B. (2004). Mathematical tools for kinetic equations. Bull. Amer. Math.Soc. 41:205–244.Downloaded by [University of Birmingham] at 06:41 12 September 2014 

On the Strichartz estimates for the kinetic transport equation

Jonathan Bennett

Neal Bez

Susana Gutiérrez

Sanghyuk Lee

Crossref

On the Strichartz Estimates for the Kinetic Transport Equation

  University of BirminghamOn the Strichartz estimates for the kinetic transportequationBennett, Jonathan; Bez, Neal; Gutierrez, Susana; Lee, SanghyukDOI:10.1080/03605302.2013.850880License:Creative Commons: Attribution (CC BY)Document VersionPublisher's PDF, also known as Version of recordCitation for published version (Harvard):Bennett, J, Bez, N, Gutierrez, S & Lee, S 2014, 'On the Strichartz estimates for the kinetic transport equation',Communications in Partial Differential Equations, vol. 39, no. 10, pp. 1821-1826.https://doi.org/10.1080/03605302.2013.850880Link to publication on Research at Birmingham portalPublisher Rights Statement:Eligibility for repository : checked 12/09/2014General rightsUnless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or thecopyright holders. The express permission of the copyright holder must be obtained for any use of this material other than for purposespermitted by law.•Users may freely distribute the URL that is used to identify this publication.•Users may download and/or print one copy of the publication from the University of Birmingham research portal for the purpose of privatestudy or non-commercial research.•User may use extracts from the document in line with the concept of ‘fair dealing’ under the Copyright, Designs and Patents Act 1988 (?)•Users may not further distribute the material nor use it for the purposes of commercial gain.Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.When citing, please reference the published version.Take down policyWhile the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has beenuploaded in error or has been deemed to be commercially or otherwise sensitive.If you believe that this is the case for this document, please contact UBIRA@lists.bham.ac.uk providing details and we will remove access tothe work immediately and investigate.Download date: 26. Dec. 2025This article was downloaded by: [University of Birmingham]On: 12 September 2014, At: 06:41Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UKCommunications in Partial Differential EquationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lpde20On the Strichartz Estimates for the Kinetic TransportEquationJonathan Bennett a , Neal Bez b , Susana Gutiérrez a & Sanghyuk Lee ca School of Mathematics , University of Birmingham , Birmingham , Englandb Department of Mathematics , Saitama University , Saitama , Japanc Department of Mathematical Sciences , Seoul National University , Seoul , KoreaPublished online: 19 Aug 2014.To cite this article: Jonathan Bennett , Neal Bez , Susana Gutiérrez & Sanghyuk Lee (2014) On the StrichartzEstimates for the Kinetic Transport Equation, Communications in Partial Differential Equations, 39:10, 1821-1826, DOI:10.1080/03605302.2013.850880To link to this article:  http://dx.doi.org/10.1080/03605302.2013.850880PLEASE SCROLL DOWN FOR ARTICLETaylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained inthe publications on our platform. Taylor & Francis, our agents, and our licensors make no representations orwarranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Versionsof published Taylor & Francis and Routledge Open articles and Taylor & Francis and Routledge Open Selectarticles posted to institutional or subject repositories or any other third-party website are without warrantyfrom Taylor & Francis of any kind, either expressed or implied, including, but not limited to, warranties ofmerchantability, fitness for a particular purpose, or non-infringement. Any opinions and views expressed in thisarticle are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. Theaccuracy of the Content should not be relied upon and should be independently verified with primary sourcesof information. Taylor & Francis shall not be liable for any losses, actions, claims, proceedings, demands,costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly inconnection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Terms & Conditions of access anduse can be found at http://www.tandfonline.com/page/terms-and-conditions It is essential that you check the license status of any given Open and Open Select article to confirmconditions of access and use.Communications in Partial Differential Equations, 39: 1821–1826, 2014Published with license by Taylor & FrancisISSN 0360-5302 print/1532-4133 onlineDOI: 10.1080/03605302.2013.850880On the Strichartz Estimates for the KineticTransport EquationJONATHAN BENNETT1, NEAL BEZ2, SUSANAGUTIÉRREZ1, AND SANGHYUK LEE31School of Mathematics, University of Birmingham, Birmingham, England2Department of Mathematics, Saitama University, Saitama, Japan3Department of Mathematical Sciences, Seoul National University,Seoul, KoreaWe show that the endpoint Strichartz estimate for the kinetic transport equation isfalse in all dimensions. We also present an alternative approach to proving the non-endpoint cases using multilinear analysis.Keywords Endpoint estimates; Kinetic transport equations; Multilinear analysis;Strichartz estimates.Mathematics Subject Classification 42B37.1. IntroductionThe solution of the kinetic transport equationtft x v+ v · xft x v = 0 f0 x v = f 0x vfor t x v ∈ ×d ×d, satisfies the Strichartz estimatesfLqt LpxLrv  f 0Laxv  (1.1)where2q= d(1r− 1p)1a= 12(1r+ 1p) q > a p ≥ a (1.2)(We write X  Y and Y  X if X ≤ CY for some finite constant C dependingat most on the parameters q p r a d, and X ∼ Y if X  Y and X  Y .) With© Jonathan Bennett, Neal Bez, Susana Gutiérrez and Sanghyuk LeeReceived August 28, 2013; Accepted September 23, 2013Address correspondence to Neal Bez, Department of Mathematics, Graduate Schoolof Science and Engineering, Saitama University, Saitama 338-8570, Japan; E-mail:nealbez@mail.saitama-u.ac.jp1821Downloaded by [University of Birmingham] at 06:41 12 September 2014 1822 Bennett et al.q p r a satisfying (1.2), but with the further condition q > 2 ≥ a, this was provedby Castella and Perthame [1], and it was observed by Keel and Tao [5] that thislatter condition can be relaxed to q > a and hence (1.2) suffices. In [5] it is tentativelyconjectured that the Strichartz estimate (1.1) holds at the endpoint q = a, at leastfor d > 1. Using the invariance under the transformationsf 0 ↔ f 0 f ↔ f q p r a ↔(qpra)(1.3)this conjectured endpoint can be (and usually is) stated for initial data in L2xv asfL2t L2dd−1x L2dd+1v f 0L2xv  (1.4)The main purpose of this paper is to disprove this conjecture.Theorem 1. The endpoint Strichartz estimate (1.4) for the kinetic transport equationfails for all d ≥ 1.The case d = 1 of Theorem 1 was proved by Guo and Peng [4] and later byOvcharov [7] using different arguments (where the norm in x on the left-hand sideis Lx ). The argument we present uses key ideas from recent work of Frank et al. [3].We remark that for the free Schrödinger propagator, the endpoint Strichartzestimate fails in the case d = 2 (see [6]) but is true for all d > 2, as shown in thelandmark paper of Keel and Tao [5]. Thus, Theorem 1 highlights a fundamentaldifference in the Strichartz estimates for these related equations.In the next section, we prove Theorem 1. In the final section, we provide analternative proof of the Strichartz estimates (1.1) in all non-endpoint cases usingmultilinear analysis.2. Proof of Theorem 1Using the invariance under the transformation (1.3) with  = 2dd+1 , estimate (1.4) isequivalent tofLd+1dt Ld+1d−1x L1v f 0Ld+1dxv (2.1)Since ft x v = f 0x − tv v, it is clear that (2.1) impliesf 0Ld+1dt Ld+1d−1x f 0Ld+1dxv (2.2)where f 0 is the macroscopic density defined by the linear mappingf 0t x =∫df 0x − tv v dvHence, by duality, (2.2) implies∗gLd+1xv  gLd+1t L d+12x  (2.3)Downloaded by [University of Birmingham] at 06:41 12 September 2014 On the Strichartz Estimates 1823where the adjoint ∗ is given by∗gx v =∫gt x + tv dtFrom here, the argument strongly uses ideas from the paper of Frank et al.[3] concerning refined Strichartz estimates for the free Schrödinger propagatorassociated with orthonormal initial data.Suppose g ∈  ×d \ 	0 is nonnegative and such that ĝ ∈ Cc ×d isalso nonnegative. Here, we use ĝ to denote the space-time Fourier transform of ggiven byĝ  =∫×dgt xe−it+x· dtdxIn this proof, we shall also use c to denote a constant depending on at most d, whichmay change from line to line.Proceeding formally, using Fourier inversion we get∗gd+1Ld+1xv=∫ d+1∏j=1gtj x + tjv d	tdxdv= c∫ d+1∏j=1ĝj jd+1∏k=1eitkk+v·keix·∑d+1=1  d	d	d	tdxdv= c∫ d+1∏j=1ĝj jd+1∏k=1k + v · k ( d+1∑=1)d	d	dv= c∫ d+1∏j=1ĝ−v · j j ( d+1∑k=1k)d	dvand hence∗gd+1Ld+1xv= c∫ d∏j=1ĝ−v · j j ̂g(v ·d∑k=1k−d∑=1)d	dv (2.4)We remark that by appropriately truncating the integrals in the above identitiesand limiting arguments, (2.4) makes sense in 0 for the class of g underconsideration.Define K to be the d by d matrix whose consecutive rows are −1    −d.Using the change of variables w = Kv, so that wj = −j · v for each 1 ≤ j ≤ d, weobtain∗gd+1Ld+1xv= c∫ d∏j=1ĝwj j ̂g(−d∑k=1wk−d∑=1)1 detKd 	wd	Writing each j = rjj in polar coordinates, we have detK =( d∏j=1rj) det1 · · · dDownloaded by [University of Birmingham] at 06:41 12 September 2014 1824 Bennett et al.Since ĝ0 0 > 0 and ĝ is continuous, it follows that∗gd+1Ld+1xv∫r1∫d−1d( d∏j=1rd−2j)1 det1 · · · dd	rd	 (2.5)For d = 1 the radial integral is infinite, and for d ≥ 2,∫d−1d1 det1 · · · dd	 = so the angular integral is infinite. Hence, for all d ≥ 1 we have shown that (2.3), andconsequently (1.4), cannot hold.Remark. The above argument shows that the endpoint estimate (2.3) fails rathergenerically. For example, the space-velocity norm ∗gd+1 is infinite whenever g ∈ ×d \ 	0 is nonnegative and such that ĝ ∈ Cc ×d is also nonnegative.3. A Multilinear Approach to the Non-Endpoint CasesFix  > 1. In this section, the notation  allows, in addition, the implicit constantto depend on .We shall prove∗gLd+1xv gLqt Ld+12x (3.1)for all g ∈ Lqt Ld+12x , where the exponent q satisfies1q+ dd + 1 = 1Our argument is a multilinear variant of that in [1] and [9] (see also [8]), which dealswith all non-endpoint cases directly. Similar multilinear arguments have been usedrecently in [3].Using the invariance under transformations in (1.3), to prove the full range ofnon-endpoint Strichartz estimates, it suffices to consider q p a satisfyingq > a p ≥ a 2q= d(1− 1p)1a= 12(1+ 1p)(3.2)and show that (1.1) holds with r = 1, or equivalently, thatf 0Lqt Lpx  f 0Laxvholds for all f 0 ∈ Laxv. By duality, this is equivalent to∗gLa′xv  gLq′t Lp′x (3.3)Downloaded by [University of Birmingham] at 06:41 12 September 2014 On the Strichartz Estimates 1825for all g ∈ Lq′t Lp′x . Note that (3.2) implies that a′ = 2p′ and 1q′ + da′ = 1, in which case(3.3) reads∗gLa′xv  gLq′t La′2xand the condition q > a is equivalent to a′ > d + 1. Therefore, (3.1) with  = a′d+1 >1 implies the full range of non-endpoint Strichartz estimates (1.1).Proof of (3.1). Without loss of generality, suppose g is nonnegative. By multiplyingout and using Minkowski’s integral inequality, we get∗gd+1Ld+1xv=( ∫ ( ∫ d+1∏j=1gtj x + tjvd	t)dxdv)1/≤∫ ( ∫ d+1∏j=1gtj x + tjvdxdv)1/d	tNow fix t1     td+1 and consider the multilinear form∫ d+1∏j=1gjtj x + tjvdxdvA straightforward estimate via the change of variables x v → x + tiv x + tjvgives∫ d+1∏j=1gjtj x + tjvdxdv 1ti − tjdgiti ·L1xgjtj ·L1x∏k =ijgktk ·Lxfor each 1 ≤ i < j ≤ d. A multilinear interpolation argument yields∫ d+1∏j=1gjtj x + tjvdxdv ∏1≤i<j≤dti − tj− 2d+1d+1∏k=1gktk ·Ld+12xApplying this with g for each gj we get∗gd+1Ld+1xv∫d+1∏1≤i<j≤dti − tj−2d+1d+1∏k=1gtk ·Ld+12xd	t gd+1Lqt Ld+12xThe last inequality is a consequence of the multilinear Hardy–Littlewood–Sobolevinequality due to Christ [2]. Remark. It seems natural to conjecture that ∗ satisfies the weak-type estimate∗gLd+1xv  gLd+1t L d+12x Downloaded by [University of Birmingham] at 06:41 12 September 2014 1826 Bennett et al.A similar comment was made in an operator-theoretic context in [3].Certain replacements for the endpoint are already known. For example, withq = a = 2, in [5], Keel and Tao obtain a substitute for (1.4) with the LpxLrv norm(where p = 2dd−1 and r = 2dd+1 ) replaced by that of a certain real interpolation spacewhich is between Lp1x Lr1v and Lpx Lrv . See also work of Ovcharov [8] where adifferent substitute bound was given for velocities v belonging to a bounded subsetof d.AcknowledgmentsWe would like to thank José Cañizo for his contributions during the early stages ofthis project.FundingThis work was supported by the European Research Council (grant number 307617;Bennett); the Engineering and Physical Sciences Research Council (grant numbersEP/J021490/1, EP/J01155X/1; Bez, Gutiérrez); National Research Foundation ofKorea (grant number 2012008373; Lee).References[1] Castella, F., Perthame, B. (1996). Estimations de Strichartz pour les èquationsde transport cinétique [Strichartz estimates for kinetic transport equations]. C.R. Acad. Sci. Paris Sér. I Math. 332:535–540.[2] Christ, M. (1985). On the restriction of the Fourier transform to curves:endpoint results and the degenerate case. Trans. Amer. Math. Soc. 287:223–238.[3] Frank, R., Lewin, M., Lieb, E.H., Seiringer, R. A Strichartz inequality fororthonormal functions. J. Euro. Math. Soc., to appear.[4] Guo, Z., Peng, L. (2007). Endpoint Strichartz estimate for the kinetic transportequation in one dimension. C. R. Math. Acad. Sci. Paris 345:253–256.[5] Keel, M., Tao, T. (1998). Endpoint Strichartz estimates. Amer. J. Math.120:955–980.[6] Montgomery-Smith, S.J. (1998). Time decay for the bounded mean oscillationof solutions of the Schrödinger and wave equations. Duke Math. J. 91:393–408.[7] Ovcharov, E. (2011). Counterexamples to Strichartz estimates for the kinetictransport equation based on Besicovitch sets. Nonlinear Anal. 74:2515–2522.[8] Ovcharov, E. (2011). Strichartz estimates for the kinetic transport equation.SIAM J. Math. Anal. 43:1282–1310.[9] Perthame, B. (2004). Mathematical tools for kinetic equations. Bull. Amer. Math.Soc. 41:205–244.Downloaded by [University of Birmingham] at 06:41 12 September 2014 

  University of BirminghamOn the Strichartz estimates for the kinetic transportequationBennett, Jonathan; Bez, Neal; Gutierrez, Susana; Lee, SanghyukDOI:10.1080/03605302.2013.850880License:Creative Commons: Attribution (CC BY)Document VersionPublisher's PDF, also known as Version of recordCitation for published version (Harvard):Bennett, J, Bez, N, Gutierrez, S & Lee, S 2014, 'On the Strichartz estimates for the kinetic transport equation',Communications in Partial Differential Equations, vol. 39, no. 10, pp. 1821-1826.https://doi.org/10.1080/03605302.2013.850880Link to publication on Research at Birmingham portalPublisher Rights Statement:Eligibility for repository : checked 12/09/2014General rightsUnless a licence is specified above, all rights (including copyright and moral rights) in this document are retained by the authors and/or thecopyright holders. 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Mar. 2020This article was downloaded by: [University of Birmingham]On: 12 September 2014, At: 06:41Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UKCommunications in Partial Differential EquationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lpde20On the Strichartz Estimates for the Kinetic TransportEquationJonathan Bennett a , Neal Bez b , Susana Gutiérrez a & Sanghyuk Lee ca School of Mathematics , University of Birmingham , Birmingham , Englandb Department of Mathematics , Saitama University , Saitama , Japanc Department of Mathematical Sciences , Seoul National University , Seoul , KoreaPublished online: 19 Aug 2014.To cite this article: Jonathan Bennett , Neal Bez , Susana Gutiérrez & Sanghyuk Lee (2014) On the StrichartzEstimates for the Kinetic Transport Equation, Communications in Partial Differential Equations, 39:10, 1821-1826, DOI:10.1080/03605302.2013.850880To link to this article:  http://dx.doi.org/10.1080/03605302.2013.850880PLEASE SCROLL DOWN FOR ARTICLETaylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained inthe publications on our platform. 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Terms & Conditions of access anduse can be found at http://www.tandfonline.com/page/terms-and-conditions It is essential that you check the license status of any given Open and Open Select article to confirmconditions of access and use.Communications in Partial Differential Equations, 39: 1821–1826, 2014Published with license by Taylor & FrancisISSN 0360-5302 print/1532-4133 onlineDOI: 10.1080/03605302.2013.850880On the Strichartz Estimates for the KineticTransport EquationJONATHAN BENNETT1, NEAL BEZ2, SUSANAGUTIÉRREZ1, AND SANGHYUK LEE31School of Mathematics, University of Birmingham, Birmingham, England2Department of Mathematics, Saitama University, Saitama, Japan3Department of Mathematical Sciences, Seoul National University,Seoul, KoreaWe show that the endpoint Strichartz estimate for the kinetic transport equation isfalse in all dimensions. We also present an alternative approach to proving the non-endpoint cases using multilinear analysis.Keywords Endpoint estimates; Kinetic transport equations; Multilinear analysis;Strichartz estimates.Mathematics Subject Classiﬁcation 42B37.1. IntroductionThe solution of the kinetic transport equationtft x v+ v · xft x v = 0 f0 x v = f 0x vfor t x v ∈ ×d ×d, satisﬁes the Strichartz estimatesfLqt LpxLrv  f 0Laxv  (1.1)where2q= d(1r− 1p)1a= 12(1r+ 1p) q > a p ≥ a (1.2)(We write X  Y and Y  X if X ≤ CY for some ﬁnite constant C dependingat most on the parameters q p r a d, and X ∼ Y if X  Y and X  Y .) With© Jonathan Bennett, Neal Bez, Susana Gutiérrez and Sanghyuk LeeReceived August 28, 2013; Accepted September 23, 2013Address correspondence to Neal Bez, Department of Mathematics, Graduate Schoolof Science and Engineering, Saitama University, Saitama 338-8570, Japan; E-mail:nealbez@mail.saitama-u.ac.jp1821Downloaded by [University of Birmingham] at 06:41 12 September 2014 1822 Bennett et al.q p r a satisfying (1.2), but with the further condition q > 2 ≥ a, this was provedby Castella and Perthame [1], and it was observed by Keel and Tao [5] that thislatter condition can be relaxed to q > a and hence (1.2) sufﬁces. In [5] it is tentativelyconjectured that the Strichartz estimate (1.1) holds at the endpoint q = a, at leastfor d > 1. Using the invariance under the transformationsf 0 ↔ f 0 f ↔ f q p r a ↔(qpra)(1.3)this conjectured endpoint can be (and usually is) stated for initial data in L2xv asfL2t L2dd−1x L2dd+1v f 0L2xv  (1.4)The main purpose of this paper is to disprove this conjecture.Theorem 1. The endpoint Strichartz estimate (1.4) for the kinetic transport equationfails for all d ≥ 1.The case d = 1 of Theorem 1 was proved by Guo and Peng [4] and later byOvcharov [7] using different arguments (where the norm in x on the left-hand sideis Lx ). The argument we present uses key ideas from recent work of Frank et al. [3].We remark that for the free Schrödinger propagator, the endpoint Strichartzestimate fails in the case d = 2 (see [6]) but is true for all d > 2, as shown in thelandmark paper of Keel and Tao [5]. Thus, Theorem 1 highlights a fundamentaldifference in the Strichartz estimates for these related equations.In the next section, we prove Theorem 1. In the ﬁnal section, we provide analternative proof of the Strichartz estimates (1.1) in all non-endpoint cases usingmultilinear analysis.2. Proof of Theorem 1Using the invariance under the transformation (1.3) with  = 2dd+1 , estimate (1.4) isequivalent tofLd+1dt Ld+1d−1x L1v f 0Ld+1dxv (2.1)Since ft x v = f 0x − tv v, it is clear that (2.1) impliesf 0Ld+1dt Ld+1d−1x f 0Ld+1dxv (2.2)where f 0 is the macroscopic density deﬁned by the linear mappingf 0t x =∫df 0x − tv v dvHence, by duality, (2.2) implies∗gLd+1xv  gLd+1t L d+12x  (2.3)Downloaded by [University of Birmingham] at 06:41 12 September 2014 On the Strichartz Estimates 1823where the adjoint ∗ is given by∗gx v =∫gt x + tv dtFrom here, the argument strongly uses ideas from the paper of Frank et al.[3] concerning reﬁned Strichartz estimates for the free Schrödinger propagatorassociated with orthonormal initial data.Suppose g ∈  ×d \ 	0 is nonnegative and such that ĝ ∈ Cc ×d isalso nonnegative. Here, we use ĝ to denote the space-time Fourier transform of ggiven byĝ  =∫×dgt xe−it+x· dtdxIn this proof, we shall also use c to denote a constant depending on at most d, whichmay change from line to line.Proceeding formally, using Fourier inversion we get∗gd+1Ld+1xv=∫ d+1∏j=1gtj x + tjv d	tdxdv= c∫ d+1∏j=1ĝj jd+1∏k=1eitkk+v·keix·∑d+1=1  d	d	d	tdxdv= c∫ d+1∏j=1ĝj jd+1∏k=1k + v · k ( d+1∑=1)d	d	dv= c∫ d+1∏j=1ĝ−v · j j ( d+1∑k=1k)d	dvand hence∗gd+1Ld+1xv= c∫ d∏j=1ĝ−v · j j ̂g(v ·d∑k=1k−d∑=1)d	dv (2.4)We remark that by appropriately truncating the integrals in the above identitiesand limiting arguments, (2.4) makes sense in 0 for the class of g underconsideration.Deﬁne K to be the d by d matrix whose consecutive rows are −1    −d.Using the change of variables w = Kv, so that wj = −j · v for each 1 ≤ j ≤ d, weobtain∗gd+1Ld+1xv= c∫ d∏j=1ĝwj j ̂g(−d∑k=1wk−d∑=1)1 detKd 	wd	Writing each j = rjj in polar coordinates, we have detK =( d∏j=1rj) det1 · · · dDownloaded by [University of Birmingham] at 06:41 12 September 2014 1824 Bennett et al.Since ĝ0 0 > 0 and ĝ is continuous, it follows that∗gd+1Ld+1xv∫r1∫d−1d( d∏j=1rd−2j)1 det1 · · · dd	rd	 (2.5)For d = 1 the radial integral is inﬁnite, and for d ≥ 2,∫d−1d1 det1 · · · dd	 = so the angular integral is inﬁnite. Hence, for all d ≥ 1 we have shown that (2.3), andconsequently (1.4), cannot hold.Remark. The above argument shows that the endpoint estimate (2.3) fails rathergenerically. For example, the space-velocity norm ∗gd+1 is inﬁnite whenever g ∈ ×d \ 	0 is nonnegative and such that ĝ ∈ Cc ×d is also nonnegative.3. A Multilinear Approach to the Non-Endpoint CasesFix  > 1. In this section, the notation  allows, in addition, the implicit constantto depend on .We shall prove∗gLd+1xv gLqt Ld+12x (3.1)for all g ∈ Lqt Ld+12x , where the exponent q satisﬁes1q+ dd + 1 = 1Our argument is a multilinear variant of that in [1] and [9] (see also [8]), which dealswith all non-endpoint cases directly. Similar multilinear arguments have been usedrecently in [3].Using the invariance under transformations in (1.3), to prove the full range ofnon-endpoint Strichartz estimates, it sufﬁces to consider q p a satisfyingq > a p ≥ a 2q= d(1− 1p)1a= 12(1+ 1p)(3.2)and show that (1.1) holds with r = 1, or equivalently, thatf 0Lqt Lpx  f 0Laxvholds for all f 0 ∈ Laxv. By duality, this is equivalent to∗gLa′xv  gLq′t Lp′x (3.3)Downloaded by [University of Birmingham] at 06:41 12 September 2014 On the Strichartz Estimates 1825for all g ∈ Lq′t Lp′x . Note that (3.2) implies that a′ = 2p′ and 1q′ + da′ = 1, in which case(3.3) reads∗gLa′xv  gLq′t La′2xand the condition q > a is equivalent to a′ > d + 1. Therefore, (3.1) with  = a′d+1 >1 implies the full range of non-endpoint Strichartz estimates (1.1).Proof of (3.1). Without loss of generality, suppose g is nonnegative. By multiplyingout and using Minkowski’s integral inequality, we get∗gd+1Ld+1xv=( ∫ ( ∫ d+1∏j=1gtj x + tjvd	t)dxdv)1/≤∫ ( ∫ d+1∏j=1gtj x + tjvdxdv)1/d	tNow ﬁx t1     td+1 and consider the multilinear form∫ d+1∏j=1gjtj x + tjvdxdvA straightforward estimate via the change of variables x v → x + tiv x + tjvgives∫ d+1∏j=1gjtj x + tjvdxdv 1ti − tjdgiti ·L1xgjtj ·L1x∏k =ijgktk ·Lxfor each 1 ≤ i < j ≤ d. A multilinear interpolation argument yields∫ d+1∏j=1gjtj x + tjvdxdv ∏1≤i<j≤dti − tj− 2d+1d+1∏k=1gktk ·Ld+12xApplying this with g for each gj we get∗gd+1Ld+1xv∫d+1∏1≤i<j≤dti − tj−2d+1d+1∏k=1gtk ·Ld+12xd	t gd+1Lqt Ld+12xThe last inequality is a consequence of the multilinear Hardy–Littlewood–Sobolevinequality due to Christ [2]. Remark. It seems natural to conjecture that ∗ satisﬁes the weak-type estimate∗gLd+1xv  gLd+1t L d+12x Downloaded by [University of Birmingham] at 06:41 12 September 2014 1826 Bennett et al.A similar comment was made in an operator-theoretic context in [3].Certain replacements for the endpoint are already known. For example, withq = a = 2, in [5], Keel and Tao obtain a substitute for (1.4) with the LpxLrv norm(where p = 2dd−1 and r = 2dd+1 ) replaced by that of a certain real interpolation spacewhich is between Lp1x Lr1v and Lpx Lrv . See also work of Ovcharov [8] where adifferent substitute bound was given for velocities v belonging to a bounded subsetof d.AcknowledgmentsWe would like to thank José Cañizo for his contributions during the early stages ofthis project.FundingThis work was supported by the European Research Council (grant number 307617;Bennett); the Engineering and Physical Sciences Research Council (grant numbersEP/J021490/1, EP/J01155X/1; Bez, Gutiérrez); National Research Foundation ofKorea (grant number 2012008373; Lee).References[1] Castella, F., Perthame, B. (1996). Estimations de Strichartz pour les èquationsde transport cinétique [Strichartz estimates for kinetic transport equations]. C.R. Acad. Sci. Paris Sér. I Math. 332:535–540.[2] Christ, M. (1985). On the restriction of the Fourier transform to curves:endpoint results and the degenerate case. Trans. Amer. Math. Soc. 287:223–238.[3] Frank, R., Lewin, M., Lieb, E.H., Seiringer, R. A Strichartz inequality fororthonormal functions. J. Euro. Math. Soc., to appear.[4] Guo, Z., Peng, L. (2007). Endpoint Strichartz estimate for the kinetic transportequation in one dimension. C. R. Math. Acad. Sci. Paris 345:253–256.[5] Keel, M., Tao, T. (1998). Endpoint Strichartz estimates. Amer. J. Math.120:955–980.[6] Montgomery-Smith, S.J. (1998). Time decay for the bounded mean oscillationof solutions of the Schrödinger and wave equations. Duke Math. J. 91:393–408.[7] Ovcharov, E. (2011). Counterexamples to Strichartz estimates for the kinetictransport equation based on Besicovitch sets. Nonlinear Anal. 74:2515–2522.[8] Ovcharov, E. (2011). Strichartz estimates for the kinetic transport equation.SIAM J. Math. Anal. 43:1282–1310.[9] Perthame, B. (2004). Mathematical tools for kinetic equations. Bull. Amer. Math.Soc. 41:205–244.Downloaded by [University of Birmingham] at 06:41 12 September 2014 

On the Strichartz estimates for the kinetic transport equation

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