On the Cauchy problem in Besov spaces for a non-linear Schrödinger equation

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1 On he Cauchy problem in Besov spaces for a non-linear Schrödinger equaion Fabrice Planchon Absrac We prove ha he iniial value problem for a non-linear Schrödinger equaion is well-posed in he Besov space Ḃ n α, (R n ), where he nonlineariy is of ype u α u. This allows o obain self-similar soluions, and o recover previous resuls under weaker smallness assumpions on he daa. Inroducion We are ineresed in he following equaion: (1) { i u + u = ǫ u α u, u(x, 0) = u 0 (x), x R n, 0, where ǫ is eiher 1 or 1, and n. One imporan propery of (1) is is invariance by scaling: () { u0 (x) u 0,λ (x) = λ αu 0 (λx) u(x, ) u λ (x, ) = λ αu(λx, λ ).. Le s α be such ha s α n =. One herefore expecs he homogeneous Sobolev space α Ḣ sα o be he criical space for well-posedness as is norm is invarian by rescaling. Theory ([5]) indeed assers ha (1) is locally well-posed in ha Sobolev space provided s α 0. Tha is, here exiss a (weak) soluion of (1) which is C([0, T], ), unique under Ḣsα an addiional assumpion. Moreover, his soluion is global in ime if he norm of he Ḣsα iniial daa is small. For he sake of compleeness we also recall ([9]) ha (1) (wih ε = 1) is globally well-posed in he energy space H 1 hanks o he conservaion of he Hamilonian H(u) = 1 u + u α+, provided α < 4. Recen resuls by Bourgain ([3]) exend his n global well-posedness o H 1 η for an appropriae (small) value of η. Alhough mos of he previously resuls apply for non ineger values of α, under appropriae Laboraoire d Analyse Numérique, URA CNRS 189, Universié Pierre e Marie Curie, 4 place Jussieu BP 187, 75 5 Paris Cedex, fab@ann.jussieu.fr 1

2 condiions involving he dimension n, we will resric ourselves o α N \ {0}. In fac one could replace he nonlineariy by any homogeneous polynomial of u, ū, wih degree α + 1. Such resricions are mosly echnical, and geing around hem requires some lenghy compuaions which are no direcly relaed o he equaion and will be presened elsewhere. Our moivaions in he presen work are slighly differen from wha has previously been done. Firsly, we aim a a beer undersanding of he recen consrucion ([6, 7, 17, 1]) of self-similar soluions for (1). A self-similar soluion is by definiion invarian by he scaling (), and herefore canno be obained by hese aforemenioned resuls in Sobolev spaces. However, [6] shows how o obain soluions such ha (3) sup β u(x, ) α+ < where β is chosen o preserve he scaling invariance, provided (4) sup β e i u 0 (x) α+ < ε 0. Direc calculaions ([6]) prove ha ε 0 x α saisfies (4), hus giving a self-similar soluion u(x, ) = 1 U( x α ). More generally, ε 0 could be replaced by a small C n (S n ) funcion ([17]). I should be noed ha he profile U does no a priori conserve he same regulariy displayed by such iniial values. Bu a naural exension o is he homogeneous Besov Ḣsα space Ḃsα,, and we aim a considering iniial daa in such a space and soluions bounded in ime wih values in ha space. Le us recall how hese spaces can be defined: f(x) ξ sαj ˆf(ξ) dξ j+1 Ḣsα sαj ˆf(ξ) dξ < +, j j and one can relax his definiion o se (5) j+1 f(x) Ḃsα, sup sαj ˆf(ξ) dξ < +. j j 1 From his definiion, one can easily check ha x α Ḃsα,, and hus solving he Cauchy problem in ha space will allow self-similar soluions. Secondly, Bourgain noiced ([3]) how he small daa heory in Sobolev spaces can be improved in he following way. Consider M > 0; hen, for he equaion (1) in dimension, if u 0 Ḣsα < M, here exiss an ε(m) such ha u 0 Ḃsα, < ε(m) will be enough o ge a global soluion. Our aim is o exend such resuls in a unified framework wih he self-similar soluions. This will allow us o gain insigh for such soluions as well as for more regular soluions.

3 1 Definiions and heorems For sake of compleeness, we firs recall one (of he many) definiions of homogeneous Besov spaces ([13],[0]). In order o consider more general indices, one has o replace he rough Fourier cu-off from (5) by a smooher one. Definiion 1 Le φ S(R n ) such ha φ 1 in B(0, 1) and φ 0 in B(0, ) c, φ j (x) = nj φ( j x), S j = φ j, j = S j+1 S j. Le f be in S (R n ). If s < n, or if s = n and q = 1, f belongs o Ḃs,q p p p condiions are saisfied if and only if he following wo The parial sum m m j(f) converge o f as a empered disribuion. The sequence ǫ j = js j (f) L p belongs o l q. If s > n p, or s = n p and q = 1, le us denoe m he greaes ineger less han s n p. Then Ḃs,q p is he space of disribuions f, modulo polynomials of degree less han m + 1, such ha We have f = j(f) for he quoien opology. The sequence ǫ j = js j (f) L p belongs o l q. Laer on, we acually work wih an easy exension of hese spaces where he space L p is replaced by he (more general) Lorenz space L p,r. We will denoe such a modified space as Ḃs,q (p,r). We refer o [1, 13] for definiion and deailed properies of Lorenz spaces. For our purposes, i suffices o know hey behave like Lebesgue spaces, while providing beer accuracy. In he proofs, we will deal wih dimension n = and hen briefly indicae he (easier) case n 3 (noe ha n = 1 could be deal he same way as n = as well). The main difference beween hese wo cases is he he end-poin Sricharz esimaes proved in [10], which allow us o carry a shorer and somewha simpler proof in dimensions where i holds. In eiher case, a resricion on α will appear, namely α > 4. The meaning of he resricion n on α will be clear from he proof of our main resul. Theorem 1 Le α > 4, α N \ {0}, u n 0 Ḃsα,, such ha u 0 Ḃsα, < C 0 (α, n). Then here exiss a global soluion of (1) such ha (6) u(x, ) L (Ḃsα, ), (7) u(x, ) 0 u 0 (x) weakly (in he sense ofσ(s, S )). Moreover, his soluion is unique under an addiional assumpion, o be explained laer. 3

4 The uniqueness condiion can be formulaed easily in dimension (or when α < 4 n ), via weak Lebesgue spaces (which are a paricular case of Lorenz spaces). We hen have uniqueness in a ball, defined as follows: (8) u(x, ) L β, (L α+, < C 1, β = α(α + ). Remark ha uniqueness holds wihou specifying condiion (6), which comes as an addiional propery of he soluions. While i should be possible o obain a similar uniqueness assumpion in bigger dimensions (when α 4 ), he proof given below gives only uniqueness in (a ball of) a space defined as he inersecion of (6) and an addiional space used n o carry a fixed-poin argumen. Noe ha (8) is a relaxed version of (3), as 1 L β, The resricion on a ball for uniqueness and condiion (7) have more o do wih he naure of he spaces under consideraion han wih he equaion iself (see [4, 14] for oher insances of such siuaions). Indeed we canno have srong coninuiy a = 0, and herefore we obain a somewha weaker resul han wha is usually mean for well-posedness. The soluions obained in heorem 1 verify various addiional space-ime esimaes, which ranslae nicely in erm of esimaes on he profile for a self-similar soluion: Theorem Under he assumpions of heorem 1, if moreover u 0 is homogeneous of degree α, he soluion is self-similar, 1 β. (9) u(x, ) = 1 U(, α and is profile is such ha (10) U(x) Ḃsα,q p, wih + n = n, q, (q, p) (, ). p q We will see how o recover and exend he known resuls for Sobolev spaces, o obain Theorem 3 Le u 0 Ḣsα verify he hypohesis of Theorem 1. Then he global soluion obained by Theorem 1 is such ha (11) u(x, ) C (Ḣsα ). The ineresing poin here is he exisence of sequences of funcions for which he Ḣsα norm is growing, while he Ḃsα, norm remains bounded. A simple example would be f m (x) = ( 1 + m x ) ( 1 α + 1 m ), which ends o 1 for large m. x α 4

5 More generally, one could exend he previous resuls, eiher o obain ha more regulariy on he iniial daa is conserved, or o consruc a local in ime heory for Ḃ sα,q, where q <. In addiion, i should be said ha he same kind of resuls apply o he semilinear wave equaion [15]. We inend o prove our resuls by a sandard fixed poin mehod, applied o an inegral version of (1). In order o do so we need linear esimaes ha lead o a good choice of a funcional seing. This will be deal wih in he firs par devoed o he (so-called) Sricharz esimaes. In he second par we will see how o handle he non-lineariy via Lilewood-Paley heory, which allows o carry on he fixed poin argumen. Sricharz esimaes in Lorenz spaces Sricharz ype esimaes have a long hisory, which in our conex goes back o [19]. They have been shown o be an essenial ool for sudying (1). In [10] Keel and Tao somehow pu an end o he sory by esablishing he full range of hese esimaes. Here we provide a slighly differen version, which does no claim novely, for i is in fac implici in he work of Ginibre and Velo, and conneced o he usual proof by a dualiy argumen. We briefly skech he proof for he sake of compleeness. Theorem 4 Le S() = e i, q, r R such ha q <, (q, r) (, ) and (1) 1 q + n r = n 4. Then (13) (14) (15) s< s< S()u 0 (x) L q, (L r,. u 0, S( s)f(x, s)ds L (L. f(x, ) L q, (L r, S( s)f(x, s)ds L q, (L r,. f(x, ) L q, (L r, where. denoes he presence of a consan (dependen on q and r), and p is he dual exponen of p. Excep for he endpoins (q = ) for which we refer o [10], he proof proceeds in he usual way: (13) is he dual esimae of (14); (14) is in urn a consequence of (15) by he so-called TT argumen: 5

6 S( s)f(x, s)ds = S( s)f(x, s), S() F(x, ) dsd, = S( s)f(x, s), F(x, ) dsd,. S( s)f(x, s) L r, x L q, F(x, ) L q., (L r, s Then S( s) sends L r, x ino L r, 1 x wih weigh, 0 < γ < 1 such ha he convoluion ( s) γ in ime in urn sends L q, ino L q,. This concludes he proof. Now we inend o prove he following Proposiion, which we only sae in dimension n =, for he paricular values of indices needed laer. Obvious generalizaions however hold in all dimensions for a large range of exponens. Proposiion 1 Le u 0 Ḃsα,, n = and s α > 0. Then (16) where β = α(α+). S()u 0 L β, (L α+ x. ) u 0 Ḃsα,, In order o obain his esimae, we would like o inerpolae he sandard classical Sricharz esimae. However, given wo spaces A and B in he x variable here is no reasonable way o perform real inerpolaion on he couple (L p 1 (A), L p (B)), unless A = B ([8]). In his paricular case, real inerpolaion gives L p,q (A) as one would expec if A = R. Recall ha Lorenz spaces could be defined via he real inerpolaion of Lebesgue spaces, (17) L p,q (R n ) = [L p 1 (R n ), L p (R n )] (θ,q), where he brackes denoe real inerpolaion ([1]), and 1 p = θ p θ p. On one end we have S()u 0 L β. u (Ḣsα η 0 Ḣsα η rη ) which gives via a Sobolev injecion (under he appropriae choice of η and r η, β being given from r η via (1)) (18) S()u 0 L β On he oher end we have, similarly so ha (L α+ x ). u 0 Ḣsα η. S()u 0. β u L + (Ḣsα+η 0 Ḣsα+η rη + ) (19) S()u 0 L β + (L α+ x ). u 0 Ḣsα+η. 6

7 Then we do real inerpolaion wih he pair ( 1, ) for which and [Ḣsα+η, Ḣsα η ] (θ, ) = Ḃsα, (0) [L β + (L α+ x ), L β (L α+ x )] (θ, ) = L β, (L α+, which concludes he proof. Noe we didn explicily wrie all he involved exponens merely for noaional convenience. Scaling consideraions show ha our only freedom is he choice of η, and all oher exponens are consrained by he seing, and follow in a unique way. 3 Non-linear esimaes 3.1 The wo dimensional case We are now in posiion o se up a fixed poin in he space (1) F = {u(x, ) u L β, (Lx α+, )} and we chose Lorenz spaces in ime and in space in order o ge a class as large as possible. Noe he close resemblance beween (1) and (3). Indeed, for homogeneous iniial daa he wo classes coincide if we keep L α+ x. Thus Proposiion 1 provides admissible iniial (homogeneous) daa such ha heir resricion o S n are in H sα (S n ), which improves noably on he previous resuls, by allowing funcions on he sphere which are no necessarily coninuous or even bounded. We hen perform a fixed poin argumen in F on he inegral equaion () u(x, ) = S()u 0 (x) + ε 0 S( s) u α u(x, s)ds. Indeed, F α+1 = L β α+1, (L α+ α+1, and via he generalized Young inequaliy he non-lineariy Γ sends F o F, and is a conracion on a small ball, wih Γ defined as (3) Γu(x, ) = 0 S( s)u α+1 (x, s)ds. Now we aim a preserving he regulariy of he iniial daa. This we accomplish by using he Sricharz esimaes from he previous secion, localized in frequency. If j is he usual operaor localizing around ξ = j, from (13) we have (4) sup jsα j S()u 0 j (α+) L α, (L α+,. u 0 Ḃsα,. 7

8 Le us define (5) E = {u(x, ) sup jsα j u(x, ) j (α+) L α, (L α+, < }). We inend o prove Lemma 1 The operaor Γ is bounded from E F α E and from E F α L (Ḃsα, ). In order o ake advanage of he frequency localizaion, we wrie u α+1 as u(u(u...(u.u))..). For each produc, we use a paraproduc ([]) decomposiion, in is simples form: u.v = j 1,j j1 u j v = j j us j v + j j vs j 1 u. Each of hese wo sums has is generic erm localized in a frequency ball of size roughly j. We will ake advanage of his propery, using a characerizaion lemma. For our purposes, le E be any L r (Lp ) (or heir Lorenz counerpar). If we consider a funcion f(x, ), definiion 1 can be easily exended o handle Besov-like spaces, which we noe Ḃs,q E. Then we have Lemma Le s > 0, E a Banach funcional space, q [0, + ], and define f Ḃs,q E js j f E = ε j l q Then, if f = j f j, where supp ˆf j B(0, j ) and ( js f j E ) j l q, we have f Ḃs,q E and < f Ḃs,q E C(s) js f j E l q. We omi is easy proof, and jus emphasize ha he hypohesis s > 0 plays a fundamenal role and canno be removed. One noices ha he wors erm in he sums is S j u, for which no a priori informaion is known. However we know ha u L β, (Lx α+, ), so S j u (and j u) verify he same propery, uniformly in j. The same remark applies as well o any power of u and is localized versions. Combining his informaion wih (5), we obain, for u (6) sup jsα j us j 1 u. q L, j (L r x, u ) E u F, where we used he generalized Hölder inequaliy from O Neil ([11]), so ha 1 q = and 1 r = By Lemma we ge ha α+ α+ u Ḃsα, L q, (L r,. 8 α (α+) + 1 β

9 we proceed o ierae he compuaion, and herefore ge (7) u α+1 Ḃsα, (α+) α+4 L, α+ α+1 (L, Then using he Sricharz esimae (14) we ge Γu Ḃsα, (α+) L α, (L α+, where he relaionship beween all he exponens is beer undersood as, for he ime norm and, for he space norm, 4 + α (α + ) = α (α + ) + α α(α + ) = 1 α (α + ) 1 + α + α = 1 α + + α 1 α + = 1 1 α +. More precisely, we have he following conrol in erm of norm (8) Γu E. u α F u E, and knowing (7) wih he same norm conrol as (8), via (14) we ge Γu L (Ḃsα, ). Noe ha his is precisely where Theorem 4 is needed, as wihou he Lorenz space in he ime variable one would ge u α+1 L α(α+) α+4, (α+) α α(α+) α+4. usual Sricharz esimae which require L These esimaes all ogeher lead o Theorem 1, under he condiion, and i would be impossible o apply he (9) S()u 0 F = S()u 0 L β, (L α+, < C 0 which is in view of proposiion 1, implied by a small Ḃ sα, norm (bu no equivalen). The uniqueness condiion is simply he one implied by he fixed poin argumen in F. One may ask wheher resricing he weak Lebesgue space in ime, say o an inerval (0, T), wouldn lead o uniqueness in F, as well as o local in ime soluions, consruced on F T wih obvious noaions. Looking a homogeneous iniial daa provides he answer: for such daa u 0, resricing he ime inerval doesn change he norm u FT. If we were o add some decay assumpion on u 0, say lim j + jsα j u 0 = 0, hen we could recover uniqueness in F as well as local in ime resuls. Remark ha we have lef aside he issue of weak coninuiy a = 0. We pospone i for he ime being, and proceed o prove Theorem 3. In he proof of Theorem 1, we use (8) o verify ha he ieraes of he fixed poin remain in our space E. For Theorem 3 we simply replace l by l in he definiion of E, and observe ha by Minkowski inequaliy his new space is included in he more familiar L (α+) α (Ḃsα, (α+,)). Having u in ha space, we hen 9

10 recover u C (Ḣsα ) by he sandard Sricharz esimae, which concludes he proof. Now we go back o he weak coninuiy a = 0 saed in Theorem 1. Combining (7) and Bernsein s inequaliy, which gives α+1 sαj ( α+ 1 ) j (u 1+α ) L x (). sαj j (u 1+α ) α+ α+1 L, x (), Therefore, j 4 α(+α) j (Γu). 0 j 4 α(+α) j (u α+1 ) (s)ds. 1 [0,] L α+. α (α+). Γu(x, ) Ḃ 4 α(+α), u α+1 α, Ḃsα, 4+α (α+), α+ L (L α+1,. α (α+), which proves he weak coninuiy of u(x, ) = S()u 0 + Γu a he origin, and concludes he resul for n =. A ha poin, we observe ha hree resricions arise on α in he proof. These resricions would arise as well for non-ineger values: firs of all, coming from inegrabiliy issues relaed o he dispersion of he Schrödinger operaor, α < 4, n which of course is no an issue if n =. The wo oher resricions are echnical. One is he aforemenioned s α > 0, which leads o α > 4. Therefore we canno deal wih he cubic n equaion in dimension wo. The las resricion comes from he use of a paraproduc. Proper use of various oher definiions of Besov spaces permis o ge around i, bu goes beyond he scope of his paper, and will be addressed elsewhere. For he sake of compleness, i should be said ha we recover and exend resuls from [6] only in he range α > 4, while heir argumen (or our fixed-poin in he space F) is valid in a range which n exends up o α 0 < 4. For such α, no classical resul for well-posedness in a criical Sobolev n space (of negaive regulariy) is known, excep for non-lineariies exhibing a special form ([18]). 3. Higher dimensions For n 3, we are able o boh overcome he resricion α < 4 and o provide a somewha n more sraighforward argumen, using he endpoin Sricharz esimae. Namely, we will se up a fixed poin argumen in he inersecion of wo spaces, and F = {u(x, ) sup j E = L (Ḃsα, ), jsα j n < + }. u L (L n, 10

11 We only skech he proof as he echnical ools are exacly he same as in he previous par. We have o evaluae u α+1 : for his, we use he Sobolev embedding This gives u α L (L n,, and hus u α+1 F = {v(x, ) sup j Ḃ sα, L nα,. jsα j n < + }. v L (L n+, Noice ha n is he dual exponen of n, and herefore he operaor Γ as defined in he n+ n previous secion will send E F back o F as well as o E. Seing up he fixed poin is hen essenially sraighforward, and will be omied, as well as he needed modificaions of he argumen required o obain he weak coninuiy and heorem 3. We simply remark ha indeed he soluion obained by his mehod verify a very large range of esimaes, namely u E q,r where E q,r = {u(x, ) sup jsα j u L q, (L r, < + }, j for all admissible pairs from he Sricharz esimaes. When we consider daa u 0 which have he righ homogeneiy, we can ge rid he ime variable, aking advanage of he scaling invariance, and hus ge esimaes on he profile U, given in heorem. We refer o [16] for a deailed derivaion of esimaes on he profile from E q,r esimaes. The same siuaion occurs of course in dimension n =, excep for he end-poin (, ). An ineresing. Then applying Sobolev embedding (or for a Besov space wih q close o in dimension ) gives U C sα. consequence of such esimaes arises whenever α > n n heorem for Ḃsα, n n Thus, in a sense, self-similar soluions have more regulariy han he iniial daa, as a fixed 0 he profile U exhibis more regulariy han u 0. This of course is already rue for he linear equaion. Acknowledgmens The auhor wishes o hank T. Cazenave, S. Tahvildar-Zadeh and F. Weissler for useful discussions, and he Program in Applied and Compuaional Mahemaics a Princeon Universiy where mos of his work was conduced, under ONR gran N and AFOSR gran F References [1] J. Bergh and J. Löfsrom. Inerpolaion Spaces, An Inroducion. Springer-Verlag, [] J.-M. Bony. Calcul symbolique e propagaion des singulariés dans les équaions aux dérivées parielles non linéaires. Ann. Sci. Ecole Norm. Sup., 14:09 46,

12 [3] J. Bourgain. Refinemens of Sricharz inequaliy and applicaions o -D NLS wih criical nonlineariy. I.M.R.N., 5:53 83, [4] M. Cannone. Ondelees, Paraproduis e Navier-Sokes. Didero Edieurs, Paris, [5] T. Cazenave and F. Weissler. The Cauchy problem for he criical nonlinear Schrödinger equaion in H s. Nonlinear Anal. T.M.A., 14: , [6] T. Cazenave and F. Weissler. Asympoically self-similar global soluions of he non linear Schrödinger and hea equaions. Mah. Zei., 8:83 10, [7] T. Cazenave and F. Weissler. More self-similar soluions of he nonlinear Schrödinger equaion. No D.E.A., 5: , [8] M. Cwickel. On (L p 0 (A 0 ), L p 1 (A 1 )) θ,q. Proc. Am. Mah. Soc., 44():86 9, [9] J. Ginibre and Velo. The global Cauchy problem for he NLS equaion revisied. Ann. IHP, An. non-linéaire, :309 37, [10] M. Keel and T. Tao. Enpoin Sricharz esimaes. Amer. Journal Mah., 10: , [11] R. O Neil. Convoluion operaors and L(p, q) spaces. Duke Mahemaical Journal, 30:19 14, [1] F. Oru. Rôle des oscillaions dans quelques problèmes d analyse non-linéaire. PhD hesis, ENS Cachan, [13] J. Peere. New houghs on Besov Spaces. Duke Univ. Mah. Series. Duke Universiy, Durham, [14] F. Planchon. Asympoic Behavior of Global Soluions o he Navier-Sokes Equaions. Rev. Ma. Iberoamericana, 14(1), [15] F. Planchon. Self-similar soluions and Besov spaces for semi-linear Schrödinger and wave equaions. In Journées Equaions aux dérivées parielles, S-Jean de Mons, [16] F. Planchon. Soluions auosimilaires e espaces de données iniiales pour l équaion de Schrödinger. Compes-Rendus de l Acad. Sci., 38, [17] F. Ribaud and A. Youssfi. Regular and self-similar soluions of nonlinear Schrödinger equaions. preprin. [18] G. Saffilani. Quadraic forms for a -d semilinear Schrödinger equaion. Duke Mah. J., 86(1):79 107,

13 [19] R. Sricharz. Resricion of Fourier ransform o quadraic surfaces and decay of soluions of he wave equaions. Duke Mahemaical Journal, 44: , [0] H. Triebel. Theory of Funcion Spaces, volume 78 of Monographs in Mahemaics. Birkhauser,