Asymptotic analysis of a non-linear non-local integro-differential equation arising from bosonic quantum field dynamics

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1 Nonlinear Differ. Equ. Appl. 21 (214), c 213 Springer Basel /14/ published online May 8, 213 DOI 1.17/s y Nonlinear Differenial Equaions and Applicaions NoDEA Asympoic analysis of a non-linear non-local inegro-differenial equaion arising from bosonic quanum field dynamics Sébasien Breeaux Absrac. We inroduce a one parameer family of non-linear, non-local inegro-differenial equaions and is limi equaion. These equaions originae from a derivaion of he linear Bolzmann equaion using he framework of bosonic quanum field heory. We show he exisence and uniqueness of srong global soluions for hese equaions, and a resul of uniform convergence on every compac inerval of he soluions of he one parameer family owards he soluion of he limi equaion. Mahemaics Subjec Classificaion (21). Primary 34G2; Secondary 45J5. Keywords. Inegro-differenial equaion, Nonlinear equaion, Nonlocal equaion, Equaion wih memory. 1. Inroducion Le d 1 be an ineger, h a sricly posiive parameer, f a funcion in he Schwarz space S(R d ; C) andξ a vecor in R d \{}. The spaces of bounded and coninuous funcions from X R o Y R d are denoed by B(X; Y )andc(x; Y ). For a funcion u B(X; Y ) we wrie u,x =sup{ u(x), x X } for X X. The applicaions F (h) and F () from B(R + ; R d )oc(r + ; R d ) are defined for u B(R + ; R d )by F (h) ( u)() := 2R F () ( u)() := 2R /h R d where ffl b 1 b u(σ)dσ := a b a a u(σ)dσ. e ir( η.2 2 η. ffl hr uσdσ) η f( η) 2 drd η R d e ir( η.2 2 η. u ) η f( η) 2 d ηdr

2 52 S. Breeaux NoDEA Theorem 1. For every h>, he non-linear inegro-differenial equaion d dξ (h) = F (h) ( ξ (h) ) ξ (h) = = ξ, admis a unique soluion ξ (h) in C 1 (R + ; R d ). The limi equaion as h d dξ () = F () ( ξ () ) ξ () = = ξ. admis a unique maximal soluion ξ () in C 1 ([,T max ); R d ) wih T max >. The norm of ξ () decreases wih ime. If min{ f( η), η B(, 2 ξ () )} is sricly posiive hen T max =+, and ξ () as +. For any T (,T max ), ξ (h) converges uniformly o ξ () on [,T] as h. Remark 2. The non-localiy feaure of Eq. (1) disappears in he limi Eq. (2). Origin of he equaions. These equaions come from a problem of derivaion of he linear Bolzmann equaion in dimension d from a model wih a paricle in a Gaussian random field (cenered and invarian by ranslaion), in he weak densiy limi, as presened in [4]. Noe ha such a random field is characerized by is covariance, and we assume ha he Fourier ransform of he covariance is of he form f 2 for some funcion f in S(R d ; C). The parameer h has hen he inerpreaion of he raio of he microscopic ypical lengh over he macroscopic one, and i is hus naural o be ineresed in he behaviour of he equaions as h. Noe ha he same scaling is used o rescale he ime. I is no he Planck consan (here aken equal o one by a suiable choice of unis). I urns ou ha his sochasic problem can be ranslaed in a deerminisic one wih more geomeric feaures by using an isomorphism beween he L 2 (Ω) space associaed wih he Gaussian random field and he symmeric Fock space ΓL 2 = n= (L2 ) sn over L 2 := L 2 (R d ; C) (see[6] for informaion abou his isomorphism). Up o some isomorphisms he Schrödinger equaion can be rewrien as ih u = Q(z) Wick u, where The funcion u has values in ΓL 2. The polynomial Q in he variable z L 2 is defined by Q(z) = ξ.2 + z,( η.2 2 ξ. η)z + z, ηz.2 +2 hr z,f, for a fixed ξ in R d \{}, wih a.2 = d j=1 a2 j for every vecor a,, is he scalar produc in L 2, R denoes he real par. η denoes he muliplicaion operaor which o a funcion v of he variable η R d associaes he funcion wih d componens η ηv(η). We do here as if his operaor (1) (2)

3 Vol. 21 (214) Analysis of a non-linear non-local inegro-differenial 53 was bounded on L 2 since he small improvemens needed o handle his case are irrelevan for his aricle. The Wick quanizaion is defined for a monomial b(z) = z q, bz p wih variable in z L 2 and b L((L 2 ) sp, (L 2 ) sq ) by he formula b Wick (n + p)!(n + q)! L = b s Id 2 (R d ) sp+n n! L 2 (R d ) sn. In his framework he Hamilon equaions associaed wih he polynomial Q(z) (see [1 3]) are ih z (h) = z Q(z (h) ) wih z Q(z) =( η.2 2ξ. η)z +2 z, ηz. ηz+ hf which can again be wrien as ih z (h) once we defined a new impulsion ξ (h) =( η.2 2 ξ. η)z (h) = ξ z (h) he soluion of his equaion wih iniial daa z (h) = z (h) = i h e i h s + hf ( η.2 2ξ (h). η)dσ hf ds., ηz (h). We are ineresed in = which is This soluion is fully deermined by ξ (h) which saisfies Eq. (1). The sudy of equaion of ξ (h) is hus jusified by he sudy of he hamilonian equaion associaed wih he polynomial Q(z) which is iself useful o give an approximae soluion o he Schrödinger equaion above in erms of coheren saes (see also he Hepp mehod, inroduced in [5] and presened in [1] whose viewpoin we use in his aricle). In [4] (whose resuls are independen of hose of his aricle) he quaric par z, ηz.2 of Q(z) is negleced and hus he erm z, ηz appearing in z Q(z) is negleced. Consequenly he erm ξ (h) in he expression of z is jus equal o is iniial value ξ and he expression of z (h) is oally explici. We waned o improve he resuls of [4] by aking ino accoun he quaric par of Q(z), and were hus led o he problem sudied in his aricle. The parameer dependen Eq. (1) is difficul o handle because of he non-localiy and he h-dependence, as h goes o hoping i would preserve he imporan informaion. Inrisic significance of he equaions. Evenhough his approach did no unil now allow us o improve our resuls on he derivaion of he linear Bolzmann equaion, hese equaions are ineresing in hemselves since i is a case of nonlinear, non-local equaions and hus an already non rival problem for which a horough sudy is possible. Organisaion of he paper. We show in Sec. 2 he exisence and uniqueness of a global soluion o he parameer dependen Eq. (1), along wih some esimaes abou he soluions. Secion 3 is devoed o he Eq. (2). We firs compue a more geomeric form of he applicaion F () and hen use i o prove he exisence and uniqueness of he soluion o Eq. (2) hrough he usual heory of differenial equaions. Then we compare boh soluions in Sec. 4 using a Grönwall ype argumen. we considered he limi of he parameer ξ (h)

4 54 S. Breeaux NoDEA Noaion. 1. For any ime T>he applicaions F (h) and F () induce applicaions from B([,T]; R d )oc([,t]; R d ). 2. We someime wrie F () ( u) wih u a fixed vecor in R d \{} for 2R.2 2 η. u) η f( η) 2 d ηdr. R d e ir( η 3. We inroduce he funcion g : η R d η f( η) 2 o simplify he noaions. 2. Parameer-dependen equaion Proposiion 3. In dimension d 1, for any h > Eq.(1) has a unique soluion in C 1 (R + ; R d ). Proof. Suppose we know ha Eq. (1) has a unique soluion ξ (h) on an inerval [,T] and wan o exend his soluion o a larger inerval [,T + δ]. We hus consider he applicaion Φ from C 1 ([T,T + δ]; R d ) o iself defined by Φ( u)() = ξ (h) (T )+ T F (h) ( ũ)(s)ds, where he funcion ũ is defined on [,T + δ] and coincide wih ξ (h) on [,T] and ũ = u on [T,T + δ]. We wan o prove ha for δ small enough Φ is a conracion. We hen obain Φ( u) () Φ( v) () = F (h) ( u)() F (h) ( v)() /h 2 e ir2 η ffl ũ σdσ hr e ir2η ffl ṽ hr σdσ g( η) drd η 4 R d R d /h r η. hr /h 4C g r T hr dr u v 4 h 2 ( T )C g u v ( ũ σ ṽ σ )dσ dr g( η) d η 4 T + δ h 2 ( T )C g u v wih C g = R η g( η) d η. Inegraing Φ( u) () Φ( v) () in ime and aking d he modulus we also ge he esimae Φ( u) Φ( v) 2 T + δ h 2 δ 2 C g u v. We can conrol he norm of Φ( u) Φ( v) byusing u v u v C 1 Φ( u) Φ( v) C 1 (4δ +2δ 2 ) T + δ h 2 C g u v C 1,

5 Vol. 21 (214) Analysis of a non-linear non-local inegro-differenial 55 for δ(t + δ) h2 8C g we ge a conracion, and we can apply a classical fixed poin heorem. Thus o prove a global exisence resul i is enough o observe ha, for any posiive consan C, he sequence ( n ) n such ha =and( n+1 n ) 2 + n ( n+1 n )=C wih posiive incremens is such ha 2 n+1 n = n +4C n 2 = 2C C 2 n +4C + n n + C. using a + b a + b for a and b non-negaive. A sum over n gives N+1 N C n= and his shows ha ( n+ C n) is necessarily unbounded. We hus obain he exisence and he uniqueness of a soluion on R +. Proposiion 4. In dimension d 3, he family ( ξ (h) ) h> is uniformly bounded, i.e. ξ (h),[,+ ) 2C 1 g, wih Cg 1 = g L 1 +( d 2 1) 1 π d/2 ĝ L 1 û(x) = 1 d e ix.η u( η)d η). 2π R d (The Fourier ransform is defined by Proof. I is enough o show ha F (h) is bounded. Indeed, for d 3, using Parseval equaliy we ge /h ffl F (h) (u)() 2 e ir( η.2 2 η u σdσ) hr dr g( η)d η R d /h ( ) 2 g L 1 +2 e ir η.2 g η + u σ dσ d η dr 1 R d hr /h ( π ) d/2 2 g L 1 +2 ĝ L 1 dr 1 r 2 g L 1 +(d 2)π d/2 ĝ L Limi equaion Proposiion 5. For any u in R d \{}, wih u = u F () ( u) = π u d 1 [,2] S d 2 u, (ρ u + 2ρ ρ 2 ω ) ( f u (ρû + 2ρ ρ2 ω ) ) 2 dh d 2 ( ω ) 2ρ ρ 2d 2 dρ, where ω is in he orhogonal of u.

6 56 S. Breeaux NoDEA Proof. We inroduce a parameer ε o permue he inegrals + F () ( u) = lim 2R e rε e ir( η.2 2 η. u) g( η)d ηdr ε + R d η + = lim 2R e ir( η.2 2 η. u) rε g( η)d ηdr ε + R d η + = lim 2R e ir( η.2 2 η. u) rε drg( η)d η ε + R d η ( + ) = lim 2 R e 2 η. u)+ε] r[i( η.2 dr g( η)d η. ε + R d η The ime inegral can be explicily compued + e 2 η. u)+ε] ε i( η.2 2 η. u) r[i( η.2 dr = ( η.2 2 η. u) 2 + ε 2 and aking he real par and he limi as ε + we obain Thus lim R ε + + F () ( u) = 2π e r[i( η.2 2 η. u)+ε] dr = πδ( η.2 2 η. u). = 2π R d η δ( η.2 2 η. u)g( η)d η u+ u S d 1 g( η) dh d 1 ( η) ( η.2 2 η. u) = π u 1 u+ u S d 1 g( η)dh d 1 ( η) wih dh d 1 he (d 1)-dimensional Hausdorff measure. Then, using u = u/ u, F () ( u) = π u 1 u+ u S d 1 η f( η) 2 dh d 1 ( η) = π u 1 u+s d 1 u η f( u η ) 2 u d 1 dh d 1 ( η ) = π u d 1 u+s d 1 η f( u η) 2 dh d 1 ( η) (3) we can now paramerize he sphera in a fashion which shows he paricular imporance of he direcion of u. We define he change of variable [, 2] S d 2 u + S d 1 whose inverse ransformaion is given by (ρ, ω ) ρ u + 2ρ ρ 2 ω ρ = η. u and ω = η ( η. u) u η ( η. u) u.

7 Vol. 21 (214) Analysis of a non-linear non-local inegro-differenial 57 The Jacobian under his change of variable is 2ρ ρ 2d 2 as η ( η. u) u = 2ρ ρ2 and hus we ge he resul. Proposiion 6. In dimension d 3 he limi differenial Eq. (2) has a unique maximal soluion ξ (). The norm of his soluion decreases wih he ime. If min{ f( η), η B(, 2 ξ () )} is sricly posiive hen his soluion is defined on R +,andξ () converges o as +. Proof. We firs show ha on any compac subse of R d \{} he map F () is Lipschiz so ha he heorem of Cauchy Lipschiz applies. Indeed on C(r, R) B(, 2), wih C(r, R) ={ x R d,r x R} and <r<r, he applicaion ( u, η) η f( u η) 2 u d 1 is smooh and hus Lipschiz wih a consan L, and F () ( u) F () ( v) = π u+s u d 1 η f( u η) 2 dh d 1 ( η) d 1 v v+s d 1 η f( v η) 2 dh d 1 ( η) d 1 πl( u v + u v ) L u v where we used an equivalen formulas for F () derived in Eq. (3) in he proof of Proposiion 5. Thus here exiss a unique soluion ξ () defined on a maximal inerval [,T max ). We hen prove ha he norm of his soluion is decreasing, indeed d 1 d 2 ξ () 2 = ξ (). ξ () = ξ ().F ()( ξ () = π () ξ d 1 ) [,2] ρ ( ξ () + 2ρ ρ 2 ω ). ξ () S d 2 ( f ξ () (ρ u + 2ρ ρ 2 ω ) ) 2 dh d 2 ( ω ) 2ρ ρ 2d 2 dρ = π () ξ d ρ ( f ξ () (ρ u + 2ρ ρ 2 ω ) ) 2 [,2] S d 2 dh d 2 ( ω ) 2ρ ρ 2d 2 dρ. Thus ξ () is necessarily bounded and canno blow up. The maximal inerval is hus necessarily [, + ).

8 58 S. Breeaux NoDEA If f does no vanish we have even more informaion, we know ha C M,f ξ () d d () ξ 2 C m,f ξ () d where C M,f = C d max B(,2 ξ () ) f and C m,f = C d min B(,2 ξ () ) f wih C d = 2π [,2] ρ 2ρ ρ 2d 2 dρh d 2 (S d 2 ). By an inegraion we conclude ha () ξ 2 is in he inerval bounded by he quaniies ( ( ) ) 2 ξ d 1 d/ d 2 C for C = C M,f and C m,f.thusξ () does no exi from any compac se of R d \{} in finie ime, and his a priori esimae implies T max =+. Remark 7. We have he conrol, for <hr<, ξ (h) (σ)dσ ξ (h) () 1 ξ (h) 2,[ hr,] hr Cg 1 hr. hr Proof. Indeed d hr ξ (h) (σ)dσ ξ (h) () = σ hr ξ (h) (v)dvdσ, and by aking he norm we ge ξ (h) (σ)dσ ξ (h) () ξ (h),[ hr,] hr hr ( σ)dσ which gives he resul. 4. Conrol of he difference beween he wo equaions We se he applicaion F (h) from B(R + ; R d )oc(r + ; R d ), defined for u B(R + ; R d )by /h F (h) ( u)() = 2R e ir( η.2 2 η. u ) η f( η) 2 drd η. R d Lemma 8. In dimension d 3, F (h) ( ξ (h) ) F (h) ( ξ (h) ) Cg 2 h ν(d) δ wih <δ<ν(d) = d 2 4 and C 2 g =4 g L 1 +2π d/2 (C 1 g ĝ L 1 + ĝ L 1).

9 Vol. 21 (214) Analysis of a non-linear non-local inegro-differenial 59 Proof. Le be in R + and Λ >, by a change of variable F (h) ( ξ (h) )() F (h) ( ξ (h) )() Λ 4 g( η) d ηdr R d +2 /h Λ 4Λ g L 1 /h +2 Λ R d e ir( η R d e ir η.2 2 η. ξ (h) ) ( e 2ir η.(.2 ( e 2ir( η + ξ (h) ffl hr ffl ).( hr ξ (h) σ dσ ξ (h) ξ (h) σ dσ ξ (h) ) 1 ) g( η)d η dr ) 1 )g( η + ξ (h) )d η dr as e ir ξ (h) 2 = 1. We hen conrol he inernal inegral using Parseval s equaliy. Indeed we have he Fourier ransforms and F(e ir η.2 )(y) = F ( g( η + ξ (h) F ( e 2ir( η + ξ (h) ).( ffl = e i ξ (h). yĝ ( π ir ) d/2e i y2 4r, ) ) (y) =e i ξ (h). yĝ( y) ξ (h) hr σ dσ ξ (h) ( ( y 2r hr ) g( η + ξ (h) ) ) ( y) )) ξ σ (h) dσ ξ (h). The difference beween he wo las Fourier ransforms can be esimaed using Remark 7, and hus ( ( )) ĝ y 2r ξ σ (h) dσ ξ (h) ĝ( y) L 1 2 min { Cg 1 ĝ L 1r 2 h, ĝ L 1}. hr The inernal inegral is hen conrolled as.2 ( e 2ir( η + ξ ffl (h) ).( hr R d e ir η ξ (h) σ dσ ξ (h) ) 1 )g( η + ξ (h) 2π d/2( C 1 g ĝ L 1 + ĝ L 1) min(r 2 h, 1) r d/2. )d η By inerpolaion, min(r 2 h, 1) r d/2 h θ r 2θ d 2, and we hen obain F (h) ( ξ (h) ) F (h) ( ξ (h) ) 4Λ g L 1 +2π d/2( Cg 1 ĝ L 1 + ĝ L 1) h θ Λ 2θ d An opimizaion of he parameer θ and of μ in Λ = h μ shows ha we can ake μ as high as d 2 4 δ for any δ> (We hen ge θ = μd 2(1+2μ) ). This concludes he proof. Lemma 9. In dimension d 3, forh, >, wih C 3 g = F (h) ( ξ (h) ) F () ( ξ (h) ),[T,+ ) C 3 g ( d 2 1 ) 1π d/2 ĝ L 1. ( ) d h 2 1, T

10 6 S. Breeaux NoDEA Proof. Indeed F (h) ( ξ (h) )() F () ( ξ (h) )() π d/2 ĝ L 1 which gives he resul. + /h dr r d/2 As a corollary we ge: Lemma 1. In dimension d 3, forh, T > and <δ< d 2 4 F () ( ξ (h) ) ξ (h),[ h,+ ) Cg 2 h d 2 4 δ + Cg 3 h d 2 4 δ(h), wih δ(h) as h. Γ (h) = We now compare ξ () and ξ (h). F () ( ξ s (h) ) ξ s (h) ds h + δ(h) =δ 1 (h). Proposiion 11. In dimension d 3, let (,T max ) hen ( ξ (h) ) [,T ] converges uniformly o ( ξ () ) [,T ] as h. Proof. Le r, R > andε> be such ha r< ξ () ε and ξ () + ε<r for all [,T]. Thus he open ubular neighbourhood A ε = ( ) B ξ (),ε [,T ] of he rajecory ( ξ () ) [,T ] is included in he compac ring C(r, R) ={ x R d,r x R}. On his ring he applicaion F () is Lipschiz wih he Lipschiz consan L. We show ha for h small enough ξ (h) remains in A ε. We firs observe ha by coninuiy, for small, ξ (h) remains in A ε. And ha if, on an inerval [, ), ξ (h) remains in C(r, R) hen we can compue a Gronwall ype esimae. Le G() = () ξ s ξ s (h) ds, hen G () = () ξ ξ (h) = ( F () ( ξ s () ) F () ( ξ s (h) )+F () ( ξ s (h) ) ξ s (h) ) ds LG()+Γ (h). As G() = we ge G() e L( s) Γ s (h) ds e L δ 1 (h)ds e L δ 1 (h)

11 Vol. 21 (214) Analysis of a non-linear non-local inegro-differenial 61 and () ξ ξ (h) = G () LG()+Γ (h) (Le L +1)δ 1 (h). for [, ). Le h such ha, for h smaller han h,(lt e LT +1)δ 1 (h) < ε 2. Suppose here exiss a ime in [,T] such ha ξ (h) / A ε we define he inferior bound of such imes. As A ε is open and ξ () and ξ (h) are coninuous hen ξ (h) / A ε.buas ξ () ξ (h) (LT e LT +1)δ 1 (h) ε 2 on [, ), by coninuiy ξ () ξ (h) ε 2 and so ξ (h) A ε we hus ge a conradicion and, for any [,T]andh h, ξ (h) remains in A ε. And hus we ge he uniform convergence of ( ξ (h) ) [,T ] o ( ξ ) [,T ] as h. Acknowledgmens The auhor would like o hank Francis Nier for proposing he problem which led o he sudy of hose equaions. References [1] Ammari, Z., Nier, F.: Mean field limi for bosons and infinie dimensional phase space analysis. Ann. Henri Poincaré 9(8), (28) [2] Ammari, Z., Nier, F.: Mean field limi for bosons and propagaion of Wigner measures. J. Mah. Phys. 5(4), (29) [3] Ammari, Z., Nier, F.: Mean field propagaion of Wigner measures and BBGKY hierarchies for general bosonic saes. J. Mah. Pures Appl. (9) 95(6), (211) [4] Breeaux, S.: A geomeric derivaion of he linear Bolzmann equaion for a paricle ineracing wih a gaussian random field. arxiv: (212) [5] Hepp, K.: The classical limi for quanum mechanical correlaion funcions. Comm. Mah. Phys. 35, (1974) [6] Simon, B.: The P (φ) 2 Euclidean (quanum) field heory. Princeon Universiy Press, Princeon (1974) (Princeon Series in Physics) Sébasien Breeaux Insiu für Analysis und Algebra Technische Universiä Braunschweig Rebenring 31, A Braunschweig Germany sebasien.breeaux@ens-cachan.org Received: 18 December 212. Acceped: 17 April 213.