MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 1. INTRODUCTION

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1 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION FABIO PUSATERI ABSTRACT We consider the question of scattering for the boson star equation in three space dimensions This is a semi-relativistic Klein-Gordon equation with a cubic nonlinearity of Hartree type We combine weighted estimates, obtained by exploiting a special null structure present in the equation, and a refined asymptotic analysis performed in Fourier space, to obtain global solutions evolving from small and localized Cauchy data We describe the behavior at infinity of such solutions by identifying a suitable nonlinear asymptotic correction to scattering As a byproduct of the weighted energy estimates alone, we also obtain global existence and linear) scattering for solutions of semi-relativistic Hartree equations with potentials decaying faster than Coulomb CONTENTS 1 Introduction 1 2 Main ideas 4 3 Strategy of the proof 6 4 Proof of Proposition Proof of Propositions 33 and Appendix A Subcritical Semi-relativistic Hartree equations 20 Appendix B Auxiliary Estimates 22 References 25 1 INTRODUCTION 11 The Equation We consider the semi-relativistic Klein-Gordon equation with a cubic Hartree-type nonlinearity i t u m 2 u = λ x 1 u 2) u, 11) with u : t, x) R C, m > 0, and λ R The operator m 2 is defined as usual by its symbol m 2 + ξ 2 in Fourier space, and denotes the convolution on In theoretical astrophysics, 11) is used to describe the dynamics of boson stars Chandrasekhar theory), and it is often referred to as the boson star equation In [10], Elgart and Schlein rigorously derived 11) via the mean field theory for quantum many-body systems of boson particles with Coulomb type gravitational) interaction In the past few years the semi-relativistic equation 11) has been analyzed by several authors with regard to various aspects of the PDE theory We will discuss some of the most relevant works on 11), and on some of its generalizations, in section 12 below In this paper we are interested in the asymptotic behavior as t of small solutions of the Cauchy problem associated to 11), and, in particular, in the question of scattering Our main result is the following: For any given u 0 x) = ut = 0, x) which is small enough in a suitable weighted Sobolev space, there exists a unique global solution of 11) which decays pointwise over time like a solution of the linear equation, but, as time goes to infinity, behaves The author was supported in part by a Simons Postdoctoral Fellowship and NSF Grant DMS

2 2 FABIO PUSATERI in a nonlinear fashion This phenomenon of nonlinear modified) scattering happens similarly for the standard Hartree equation [15, 23] i t u u = x 1 u 2) u, x R n, n 2, and, in its essence, it is the same type of asymptotic behavior that can be found in several other dispersive equations which are scattering-critical or L -critical) An additional result contained in the present paper concerns some generalizations of 11) with potentials decaying faster than the Coulomb potential x 1 We will prove regular) scattering for those models, closing some gaps in the existing literature 12 Background and known results As pointed out above, the semi-relativistic equation 11) can be rigorously derived as the mean field limit of an N-body system of interacting boson particles In the time independent case, the question of convergence and existence of solutions for the limiting equation had been studied earlier by Lieb and Yau [21] More recent investigations on the relation between the N-particle system and the limiting nonlinear equation 11) can be found in [26] The conserved energy associated to 11) is Eu) := 1 ū m 2 2 u dx + λ x 1 u 2) u 2 dx, 12) R 4 3 and therefore the energy space is H 1/2 Solutions of 11) also enjoy conservation of mass, ut) L 2 = u0) L 2, and the nonlinearity x 1 u 2 )u is critical with respect to L 2 in three dimensions Local existence of solutions for the Cauchy problem with data in H s ), s 1/2, was proved by Lenzmann in [22], also for more general models than 11), including a wide class of external potentials In the cited paper, using conservation of energy, global existence is obtained for any data in the defocusing case λ 0 In the focusing case λ < 0, one needs instead to restrict the size of the L 2 -norm of the initial data to be smaller than that of the ground state [11] It was shown by by Frölich and Lenzmann [12] that, in the focusing case, any radially symmetric smooth compactly supported initial data with negative energy leads to finite time blow-up Sharp low regularity wellposedness below the energy space was recently proven by Herr and Lenzmann [18], both in the radial s > 0) and non-radial case s 1/4) Without loss of generality we can normalize m = 1, and rescale λ to be 1 or 1 depending on its sign In this paper we will only consider small solutions, and therefore the sign of λ will not be relevant, and λ will be taken to be 1 for convenience To better put 11) into context in relation to the global well-posedness and scattering theory for the Cauchy problem, let us consider the following generalized model i t u 1 u = x γ u 2) u, x R n, 0 < γ < n 13) In [3, 4], Cho and Ozawa showed global existence of large solutions for 0 < γ < 2n/n + 1) for n 2, and small data global existence and scattering for γ > 2 in dimension n 3 They also proved the non-existence of asymptotically free solutions ie, converging to a solution of the linear equation as time goes to infinity) for the case 0 < γ 1 when n 3, and for 0 < γ < n/2 when n = 1 or 2 Our main result shows that indeed solutions of 13) with γ = 1 in 3d scatter to a nonlinear profile An additional result that we prove, namely Theorem A1, closes the gap in the small data scattering 1 2 for 1 < γ 2 In [5] the authors obtained scattering for radially symmetric small solutions when 3/2 < γ < 2 and n 3 Cho and Nakanishi [6] obtained several results in higher dimensions: in dimension n 4 they proved global existence with radial symmetry for 1 < γ < 2, and small data scattering also without data 1 At least for a class of initial data in a suitable weighted Sobolev space 2 Notice that there seem to be no global solutions in the literature, in the intermediate range 3/2 γ 2, for non-radial

3 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 3 symmetry) for γ = 2 We refer to [6] for a survey of some of the techniques employed in the above mentioned papers 13 Main Result We have seen that for certain values of γ 0, 2), large global solutions to 13) can be constructed combining conservation laws, and low regularity wellposedness or Strichartz estimates and Hardy s inequalities in order to estimate the nonlinearity) When the question of scattering is considered, even treating small data outside the energy space is quite challenging As mentioned above, in three dimensions, scattering is known if γ is large enough γ > 3/2 in the radial case, γ > 2 in the general case) Clearly, larger values of γ are easier to treat, since the time decay of the L 2 norm of the nonlinearity in 13), computed on a solution of the linear equation, is t γ Theorem 11 below shows scattering in a modified sense) for 11), that is 13) with γ = 1 We refer to this as the scatteringcritical case, because the decay of the nonlinearity is barely) non-integrable in time Moreover, our proof can be adapted to obtain scattering in the L -subcritical cases 1 < γ 2, which were left open so far See Theorem A1 for details This is our main result: Theorem 11 Let 3 N = 1000, and let u 0 : C be given such that u 0 H N + x 2 u 0 H ξ ) 10 û 0 L ε 0 14) Then there exists ε 0 such that for all ε 0 ε 0, the Cauchy problem { i t u + 1 u = x 1 u 2) u ut = 0, x) = u 0 x) has a unique global solution ut, x), such that sup 1 + t ) 3/2 ut) L ε 0 16) t R Moreover, the behavior of u as t can be described as follows Let Bt, ξ) := 1 2π) 3 t 0 ξ ξ σ σ 1 15) ûs, σ) 2 dσ ϕξs 1/300 ) ds s + 1, 17) where ϕ is a smooth compactly supported function Then, there exists an asymptotic state f +, such that for all t > ξ ) 10 [ e ibt,ξ) e it 1+ ξ 2 ût, ξ) f + ξ) ] L ξ ε t) p 1, 18) for some 0 < p 1 < 1/1000 A similar statement holds for t < 0 Solutions of 15) will be constructed through a priori estimates in the space given by the norm 22) We refer to section 2 for some explanation of the main ideas involved the proof of Theorem 11, and to section 3 for a detailed description of our strategy It would be possible to express the asymptotic behavior of a solution of 15) in physical coordinates rather than in Fourier space However, the asymptotic formula 17)-18) clearly emerges from our proof, which is performed in Fourier space, and can be seen from some heuristic considerations, see section 2 Therefore, we leave 17)-18) as a satisfactory description of modified scattering Before moving on to describe the difficulties and the tools involved in the proof of Theorem 11, let us mention some known results concerning modified scattering Famous examples of dispersive PDEs whose solutions exhibit a behavior which is qualitatively different from the behavior of a linear solution are the nonlinear Sch rodinger [28, 7, 15, 23], the Benjamin-Ono [1, 17], and the mkdv [8, 16] equations Besides these one-dimensional completely integrable examples, for which large data results are also 3 For convenience, and to simplify the proof a bit, we let N be comfortably large; however, it is certainly possible to reduce the value of N to a number between 10 and 100

4 4 FABIO PUSATERI available, the phenomenon of modified scattering for small solutions has been observed in several other equations Examples are given by Hartree equations [15, 23], Klein-Gordon equations [9], and, more recently, gravity water waves [20] see also [19] for a simpler fractional Schödinger model, and [2] for a similar result on the water waves system) Notation We define the Fourier transform by Fgξ) = ĝξ) := e ix ξ gx) dx = gx) = 1 2π) 3 e ix ξ ĝξ) dξ We fix ϕ : R [0, 1] an even smooth function supported in [ 8/5, 8/5] and equal to 1 in [ 5/4, 5/4], and let ϕ k x) := ϕ x /2 k ) ϕ x /2 k 1 ), k Z, x 19) For any interval I R we define ϕ I := ϕ k 110) k I Z More generally, for any m, k Z, m k, and x we define ϕ m) k x) := { ϕ x /2 k ) ϕ x /2 k 1 ), if k m + 1, ϕ x /2 k ), if k = m 111) We let P k, k Z, denote the operator on defined by the Fourier multiplier ξ ϕ k ξ) We will sometimes denote f k = P k f For an integer n Z we denote n + = max0, n) 2 MAIN IDEAS Let p 0 = 1/1000, N = 1000 and Λ ) := 1 Define ft, x) := e itλ u)t, x), 21) where ut) is a solution of 15) We will solve 15) in the space given by the norm [ sup 1 + t ) p 0 ut) H N t ) p 0 xft) H t ) 2p 0 x 2 ft) H 2 t R + ] 1 + ξ ) 10 ft) L If f is defined as in 21), we can write Duhamel s formula for 15) in Fourier space as follows: ft, ξ) = û 0 ξ) + t 0 Is, ξ) ds, Is, ξ) := ic 1 e is[ Λξ)+Λξ η)+λη+σ) Λσ)] η 2 fs, ξ η) fs, η + σ) fs, σ) dηdσ, c 1 := 22π) 5 Here we used F x 1 )ξ) = 4π ξ 2 Norms and decay When dealing with nonlinear equations which are scattering-critical in the sense explained in the introduction, one often has to resort to spaces which incorporate some strong decay information Even more so if modified scattering is expected In this case, one needs to extract very precise asymptotic information, and be able to prove some L t L p x bound on solutions Thanks to the decay estimate 33) below, one sees that solutions whose norm 22) is bounded, decay pointwise like a solution of the linear equation, ie, at the rate of t 3/2 It is important to underline the key role played by the F 1 L -norm Since the equation 15) is L -critical, one might not be able 22) 23)

5 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 5 to prove a bound on weighted L 2 -norms of f which is uniform in time Therefore, sharp time decay cannot be obtained as a consequence of standard weighted) L p L q linear estimates The idea, already exploited in several works on other critical models [15, 17, 16, 23, 19, 20], is to include a norm which guarantees decay, but is weaker than L 1, and as such can be controlled uniformly in time Our choice is the last norm appearing in 22) Weighted Estimates Weighted norms play an important role in the whole construction Firstly, they control remainders in the linear estimate 33) Secondly, and most importantly, they are used to control remainders in the asymptotic expansions which allow us to bound the F 1 L -norm see the next paragraph below for more on this) In the literature, a standard way of establishing weighted estimates is given by the use of vectorfields [24, 25] In the case of the boson star equation, the use of such a tool is limited by the lack of scaling and Lorentz invariance The quantities we shall control are xf and x 2 f in L 2 These correspond to Γu and Γ 2 u in L 2, for Γ = x itλ Despite the fact that Γ does not commute properly with the equation, we will be able to bound these weighted norms as follows We apply ξ and 2 ξ to f as given in 23) The worst term obtained by applying ξ to Is, ξ) is of the form e is[ Λξ)+Λξ η)+λη+σ) Λσ)] smξ, η) η 2 fs, ξ η) fs, η + σ) fs, σ) dηdσ, where mξ, η) = ξ Λξ)+Λξ η)) Now notice that m is a smooth function with mξ, 0) = 0 This is essentially a null condition satisfied by the equation 4 Thanks to this, we can think that the multiplier smξ, η) η 2 behaves, as far as estimates are concerned, like the original Coulomb potential η 2, so that the loss of the factor s can be recovered 5 We can then control xf and x 2 f in L 2, allowing a small growth in t Asymptotic analysis Let us change variables in 23) and write Is, ξ) := ic 1 e is[ Λξ)+Λξ+η)+Λξ+σ) Λξ+η+σ)] η 2 fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ Our goal is to identify the leading order term of the above expression in terms of powers of s, neglecting all contributions that decay faster than s 1 Let us assume that ξ 1 and we are integrating over a region η s l, with l < 0 small enough, but not so small that the integral of η 2 over this region is Os 1 ) We can then Taylor expand the oscillating phase, and approximate Is, ξ) by ic 1 e isη z η 2 fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ, where z = zξ, σ) := σ/ σ ξ/ ξ Using the bounds on f, ie on weighted norms, we can further approximate the expression above by ic 1 e isη z η 2 fs, ξ) fs, ξ + σ) fs, ξ + σ) dηdσ = ic 1 fs, ξ) F η 2 )sz) fs, ξ + σ) 2 dσ = i s fs, ξ)cs, ξ), 24) 4 This type of generalized null condition was used in [13] and [27], as an important aspect of the space-time resonance analysis 5 Another alternative possibility is to proceed similarly to [27] and [14], by exploiting an algebraic identity for the phase φξ, η, σ) = Λξ) + Λξ η) + Λη + σ) Λσ), of the form ξ φ = L η φ, σ φ, φ), where L denotes some linear combination with coefficients given by smooth functions of ξ, η, σ) One could use such an identity to integrate by parts in time and frequency and recover the loss of s

6 6 FABIO PUSATERI for some function Cs, ξ) which is real-valued, and uniformly bounded under suitable assumptions on f Thanks to the above we have obtained t ft, ξ) = it 1 ft, ξ)ct, ξ) + Ot 1 ), from which we can deduce a uniform bound on sup t,ξ ft, ξ) The estimates leading to this latter bound will also show the modified scattering property 18) In order to make the above intuition rigorous, we need to identify a suitable scale in η, say s l 0, such that the above asymptotics are true for η s l 0, and, at the same time, the integral 24) on the region η s l 0 is Os 1 ) Lemma 53 contains the derivation of the asymptotic correction term in the critical region The remaining contributions are estimated in section 52, using integration by parts in η 3 STRATEGY OF THE PROOF By time reversal symmetry, we can restrict to times t 0 Local-in-time solutions to 15) can be constructed by a standard fixed point argument Given a local solution u on a time interval [0, T ], we assume that the following norm is a priori small: u XT := sup t [0,T ] [ 1 + t) p 0 ut) H N t) p 0 xft) H t) 2p 0 x 2 ft) H ξ ) 10 ft, ξ) L ξ ] ε 1 To obtain the existence of a global solution which is bounded in the space X T it will suffice to show 31) u XT ε 0 + Cε 3 1, 32) where ε 0 is the size of the initial datum, see 14) In order to deduce sharp pointwise decay from the above a priori bounds we will use the following: Proposition 31 Refined Linear Decay Estimate) For any t R one has e it 1 f L t ) 3/2 1 + ξ ) 6 fξ) L ξ t ) 31/20 [ x 2 f L 2 + f H 50 ] 33) Proposition 31 is proven in section B1 As a consequence of 33) and the a priori assumptions 31) we have for t [0, T ] ut) W 2, ε t) 3/2 34) The proof of 32) will be done in two main steps given by the following Propositions Proposition 32 Weighted Energy Estimates) Assume that f C[0, T ] : H N ) satisfies the a priori assumptions 31), and let p 0 = 1/1000 Then, and sup 1 + t) p 0 ft) H N ε 0 + Cε 3 1, t [0,T ] sup 1 + t) p 0 x ft) H 2 ε 0 + Cε 3 1, t [0,T ] sup 1 + t) 2p 0 x 2 ft) H 2 ε 0 + Cε 3 1 t [0,T ] The proof of the above Proposition is contained in section 4

7 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 7 Proposition 33 Estimate of the L ξ -norm) Assume that f C[0, T ] : HN ) satisfies the a priori bounds 31) Then sup 1 + ξ ) 10 ft, ξ) L ε 0 + Cε 3 1 t [0,T ] ξ As a corollary of the proof of Proposition 33 we will obtain the main result of our paper concerning modified scattering of small solution of 15): Proposition 34 Modified Scattering) Assume that u C[0, T ] : H N ) is a solution of 15) with u 0 small enough as in 14) Define t Bt, ξ) := 2π) 3 ξ ξ σ 1 σ ûs, σ) 2 dσ ϕξs 1/300 ) ds s + 1, 35) 0 where ϕ is a smooth compactly supported function as the one described before 19) Then, there exists f + L ξ, and δ > 0, such that 1 + ξ ) 10 e ibt,ξ) ft, ξ) f+ ξ) ) L ε 3 ξ 11 + t) δ 36) We refer to section 5 for the proof of Propositions 33 and 34 In particular, Proposition 51 implies both Propositions 33 and 34, through the estimate 55) Let us define 4 PROOF OF PROPOSITION 32 N h 1, h 2, h 3 ) := x 1 h 1 h 2 ) h3 41) A priori energy estimates for the Sobolev norms of solutions to 15) are straightforward In particular, using the Hardy-Littlewood-Sobolev inequality, it is not hard to show that, given a solution u : [0, T ], for all t [0, T ] one has ut) H N u 0 H N + C t 0 us) L 2 us) L 6 us) H N ds 42) for all integers N 0, and some constant C > 0 Interpolating the a priori decay assumption in 31) and the bounds on the L ξ -norm which controls the L2 x-norm), one has us) L 6 ε s) 1 Using the a priori assumptions 31) it follows from 42) that ut) H N ε 0 + Cε t) p 0 43) To obtain Proposition 32, we then aim to prove that under the a priori assumptions 31) one has x ft) H 2 ε 0 + Cε t) p 0, 44) x 2 ft) H 2 ε 0 + Cε t) 2p 0 45) 41 Proof of 44) Since we already have control on the L 2 -norm of f we just need to estimate xf in H 2 Recall the integral equation satisfied by f: ft, ξ) = û 0 ξ) + t 0 Is, ξ) ds, Is, ξ) := ic 1 e R isφξ,η,σ) η 2 fs, ξ η) fs, η + σ) fs, σ) dηdσ, 3 φξ, η, σ) := Λξ) + Λξ η) + Λη + σ) Λσ) 46)

8 8 FABIO PUSATERI For 44) it suffices to prove that under the a priori assumptions 31) we have ξ ξ Is) ε 3 L 11 + s) 1+p 0 47) 2 Applying ξ to I we have ξ Is, ξ) = ic 1 I1 s, ξ) + I 2 s, ξ) ), 48) I 1 s, ξ) = e isφξ,η,σ) η 2 ξ fs, ξ η) fs, η + σ) fs, σ)dηdσ, 49) I 2 s, ξ) = is e isφξ,η,σ) mξ, η) η 2 fs, ξ η) fs, η + σ) fs, σ) dηdσ, 410) where we have denoted ) mξ, η) := ξ Λξ) + Λξ η) = Λ ξ) ξ ξ + Λ ξ η) ξ η ξ η 411) We will crucially use the fact that mξ, η) η, for small η In particular, this will allow us to cancel part of the singularity given by the transform of the Coulomb potential η 2, so to compensate for the growing factor s present in I Estimate of 49) This term can be directly estimated using the Hardy-Littlewood-Sobolev inequality and the a priori assumptions 31): ξ 2 I 1 s) L 2 N us), us), e isλ xfs) ) H 2 xfs) H 2 us) H 2 us) L 6 ε s) 1+p Estimate of 410) With the notation 19), we perform dyadic decomposition in the variables ξ, η and ξ η, and write ξ 2 I 2 s, ξ) = ξ I k,k 1,k 2 2 s, ξ) k,k 1,k 2 Z I k,k 1,k 2 2 s, ξ) := is e is Λξ)+Λξ η)) m 1 ξ, η) f k1 s, ξ η) P k2 u 2 s, η) dη, R 3 m 1 ξ, η) := ξ Λ ξ) ξ ξ + Λ ξ η) ξ η ) η 2 ϕ k ξ)ϕ [k2 2,k ξ η 2 +2]η) 412) In what follows we shall always work under the assumption that the integral above is not zero, and, in particular the sums are taken over those indexes k, k 1, k 2 ) satisfying either k max{k 1, k 2 } 10 or k 1 k 2 10 Using m 1 ξ, η) 2 k 2 ϕ k ξ), we see that I k,k 1,k 2 2 s) L 2 s 2 3k/2 f k1 s) L 22 k 2 P k2 us) 2 L 2 413) From the a priori assumptions 31) we know that f k1 s) L 2 2 3k1/2 2 10k 1) + ε 1 414) { } P k2 us) 2 L 2 min 2 3k2/2, 2 5k 2 ε ) Using 414) and 415) in 413), we see that the sum over those indexes k such that 2 k 1 + s) 2 can be easily dealt with: I k,k 1,k 2 2 s) L 2 ε s) 2 k,k,k 2 Z 2 k 1+s) 2

9 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 9 From the definition in 412) one can verify that m 1 satisfies the hypothesis of Lemma B1 with A 2 k 2 Applying B13) we obtain ξ I k,k 1,k 2 2 s) s 2 k + P k1 fs) L 2 L 22 k 2 P k2 us) 2 L 416) k,k,k 2 Z 2 k 1+s) 2 Using 34) we see that P k2 us) 2 L min k,k,k 2 Z 2 k 1+s) 2 { 2 3k 2, k 2 ) s) 3 } ε ) Therefore, in view of 414) and 417), we can bound the sum in 416) for 2 k s) 1 by 2 k + 2 3k1/2 2 10k 1) + 2 2k 2 ε 3 1 ε s) 2+p 0 2 k 1+s) 2 2 k 2 1+s) 1 To estimate the sum in 416) for 2 k s) 1 we use again 414) and 417) to obtain 2 k + 2 3k1/2 2 10k 1) + 2 k ) 2k s) 3 ε 3 1 ε s) 2+p 0 2 k 1+s) 2 2 k 2 1+s) 1 The last two inequalities imply the desired bound 47) for the term I 2, and give us 44) 42 Proof of 45) In order to estimate x 2 f in H 2 we compute the contributions from ξ 2 I Applying ξ to the terms I 1 and I 2 as they appear in 48)-411), and with a slight abuse of notation, we can write ξ I1 s, ξ) + I 2 s, ξ) ) = J 1 s, ξ) + 2J 2 s, ξ) + J 3 s, ξ) + J 4 s, ξ), 418) J 1 s, ξ) := e isφξ,η,σ) η 2 ξ 2 fs, ξ η) fs, η + σ) fs, σ)dηdσ, 419) J 2 s, ξ) := is e isφξ,η,σ) mξ, η) η 2 ξ fs, ξ η) fs, η + σ) fs, σ) dηdσ, 420) J 3 s, ξ) := is e isφξ,η,σ) ξ mξ, η) η 2 fs, ξ η) fs, η + σ) fs, σ) dηdσ, 421) J 4 s, ξ) := s 2 e isφξ,η,σ) [mξ, η)] 2 η 2 fs, ξ η) fs, η + σ) fs, σ)dηdσ, 422) where m is defined in 411) To obtain 45) it is then enough to show that for i = 1,, 4, one has ξ 2 J i s) L 2 ε s) 1+2p 0 423) 421 Estimate of 419) This is the easiest term and can be directly estimated as follows: ξ 2 J 1 s) L 2 N us), us), e isλ x 2 fs) ) H 2 x 2 fs) H 2 us) H 2 us) L 6 ε s) 1+2p Estimate of 420) Similarly to what has been done above for I 2 in 412), we can write ξ 2 J 2 s, ξ) = is ξ e is Λξ)+Λξ η)) m 2 ξ, η) P k1 xfs, ξ η) P k2 u 2 s, η) dη, 424) k,k 1,k 2 Z where m 2 ξ, η) := ξ m 1 ξ, η) = ξ Λ ξ) ξ ξ + Λ ξ η) ξ η ) η 2 ϕ k ξ)ϕ [k2 2,k ξ η 2 +2]η) 425)

10 10 FABIO PUSATERI It is not hard to verify that m 2 satisfies the hypothesis of Lemma B1 with A 2 k 2 We can then apply B13) and obtain ξ 2 J 2 s) L 2 s 2 k+ P k1 xfs) L 22 k2 P k2 us) 2 L 426) 2 k 1+s) 2 + s 2 3k/2 P k1 xfs) L 2 2 k2 P k2 us) 2 L 2 427) 2 k 1+s) 2 Using Bernstein s inequality and interpolating weighted norms we can estimate P k1 xfs) L 2 2 k1/2 xfs) L 3/2 2 k1/2 xfs) 1/2 L x 2 fs) 1/2 2 L 2 This and the a priori bounds 31) give us 2 k 1/2 1 + s) 3p 0/2 ε 1 { 1 P k1 xfs) L 2 min + 2 2k 1 ) 11 } + s) p 0, 2 k1/2 1 + s) 3p 0/2 ε 1 428) { P k2 us) 2 L min 2 3k 2, ) } 2k s) 3 ε ) Using 428) and 429), we can estimate the sum in 426) for 2 k s) 1 as follows: { 2 k + 1 min + 2 2k 1 ) 11 } + s) p 0, 2 k1/2 1 + s) 3p 0/2 2 2k 2 ε 3 1 ε s) 2+2p 0 430) 2 k 1+s) 2 2 k 2 1+s) 1 Using again 428) and 429) we see that { 2 k + 1 min + 2 2k 1 ) 11 } + s) p 0, 2 k1/2 1 + s) 3p 0/2 2 k s) 3 ε 3 1 ε s) 2+2p 0 2 k 1+s) 2 2 k 2 1+s) 1 Eventually, we estimate similarly the sum in 427): { 1 2 3k/2 min + 2 2k 1 ) 11 } + s) p 0, 2 k1/2 1 + s) 3p 0/2 min{2 2k 2, 2 k 2 }ε 3 1 ε s) 2 2 k 1+s) 2 The last three inequalities give us the desired bound 423) for the term J Estimate of 421) We write ξ 2 J 3 s, ξ) = is ξ e is Λξ)+Λξ η)) m 3 ξ, η) f k1 s, ξ η) P k2 u 2 s, η) dη, 431) k,k 1,k 2 Z where m 3 ξ, η) = ξ ξ Λ ξ) ξ ξ + Λ ξ η) ξ η ) η 2 ϕ k ξ)ϕ [k2 2,k ξ η 2 +2]η) Proceeding analogously to the estimates in the above two paragraphs, we can easily bound by ε s) 2 the summation in 431) over k Z such that 2 k 1 + s) 2 For the remaining contribution we apply

11 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 11 once again Lemma B1 to the symbol m 3, with A 2 k 2, and use the a priori bounds to deduce ξ 2 J 3 s) L 2 s 2 k+ P k1 fs) L 22 k2 P k2 us) 2 L + ε s) 2 2 k 1+s) 2 { s 2 k+ 2 3k1/2 2 10k1)+ 2 k2 min 2 3k 2, ) 2k s) 3 } ε ε s) 2 2 k 1+s) 2 ε s) 1+p 0 The last inequality has been deduced once again by separately analyzing the two cases 2 k s) 1 and 2 k s) Estimate of 422) With a slight abuse of notation, we can write ξ 2 J 4 s, ξ) = s 2 ξ e is Λξ)+Λξ η)) m 4 ξ, η) f k1 s, ξ η) P k2 u 2 s, η) dη, where m 4 ξ, η) := k,k 1,k 2 Z Λ ξ) ξ ξ + Λ ξ η) ξ η )2 η 2 ϕ k ξ)ϕ [k2 2,k ξ η 2 +2]η) For k with 2 k 1 + s) 2 a bound of ε s) 1 can be obtained as before To estimate the remaining contribution we notice that we can apply Lemma B1 to m 4 with A 1, and obtain ξ 2 J 4 s) L 2 s 2 2 k+ P k1 fs) L 2 P k2 us) 2 L + ε s) 1 2 k 1+s) 2 { s 2 2 k+ 2 3k1/2 2 10k1)+ min 2 3k 2, ) 2k s) 3 } ε ε s) 1 2 k 1+s) 2 ε s) 1+p 0 This concludes the proof of 423), which together with 47) gives us Proposition 32 5 PROOF OF PROPOSITIONS 33 AND 34 The aim of this section is to control uniformly in time the key norm 1 + ξ ) 10 ft) and show, as a byproduct of the proof, modified scattering as stated in 36) in Proposition 34 Given a solution u of 15), satisfying the a priori bounds 31), we define for any t [0, T ] and ξ t Bt, ξ) := c 0 ξ ξ σ 1 σ ûs, σ) 2 dσ ϕ s ξ) ds s + 1, c 0 := 2π) 3, 51) where 0 ϕ s ξ) := ϕξs 1/300 ), 52) for a smooth compactly supported function ϕ as the one described before 19) We also define the modified profile gξ, t) := e ibt,ξ) ft, ξ) 53)

12 12 FABIO PUSATERI Notice that B is a well-defined and real-valued function In particular f and g have the same L ξ -norm We are going to prove the following: Proposition 51 Assume that f C[0, T ] : H N ) satisfies the a priori bounds 31), that is [ 1 + t) p 0 ut) H N t) p 0 xft) H t) 2p 0 x 2 ft) H 2 sup t [0,T ] + ] 1 + ξ ) 10 ft) L ε 1 54) Then, for some small enough δ > 0, In particular sup 1 + t 1 ) δ 1 + ξ ) 10 gt 2, ξ) gt 1, ξ) ) L ε ) t 1 t 2 [0,T ] ξ sup 1 + ξ ) 10 ft, ξ) L ε 0 + Cε ) t [0,T ] ξ It is clear that the above statement implies Propositions 33 and 34: 56) gives Proposition 33, while 55) implies 36) once we define f + ξ) := lim t gt, ξ), where the limit is taken in 1 + ξ ) 10 L ξ For 55) it suffices to prove that if t 1 t 2 [2 m 2, 2 m+1 ] [0, T ], for some m {1, 2 }, then 1 + ξ ) 10 gt 2, ξ) gt 1, ξ)) L ξ ε 3 12 p 1m, 57) for some p 1 > 0 To prove this we start by looking at the nonlinear term I in 23), and after a change of variables we write it as Is, ξ) := ic 1 e isφξ,η,σ) η 2 fs, ξ + η + σ) fs, ξ + η) fs, ξ + σ) dηdσ φξ, η, σ) := Λξ) + Λη + ξ) + Λσ + ξ) Λξ + η + σ) Let l 0 be the smallest integer larger than 29m/40, [ l 0 := 29 ] 40 m + 1, 59) using the notation 111), we write Is, ξ) = I 0 s, ξ) + I l1 s, ξ), Z, >l 0 I 0 s, ξ) := ic 1 e isφξ,η,σ) η 2 ϕ l 0) l 0 η) fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ, 510) R 3 I l1 s, ξ) := ic 1 e isφξ,η,σ) η 2 ϕ l 0) η) fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ The term I 0 is the one responsible for the correction to the scattering, whereas I I 0 is a remainder term, under the a priori assumptions 54) The profile f verifies t ft, ξ) = I 0 s, ξ) + >l 0 I l1 s, ξ) 58)

13 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 13 and, according to the definition of g above, we have t gt, ξ) = e ibt,ξ) [ I 0 t, ξ) + >l 0 I l1 s, ξ) i t Bt, ξ) ft, ξ) ] 511) Therefore, to prove 57) it suffices to show that if k Z, m {1, 2, }, ξ [2 k, 2 k+1 ], and t 1 t 2 [2 m 1, 2 m+1 ] [0, T ] the following two bounds are true: t2 [ e ibs,ξ) I 0 s, ξ) i s Bs, ξ) fs, ] ξ) ds ε3 12 p1m 2 10k +, 512) t 1 t2 e ibs,ξ) I l1 s, ξ) ds t 1 ε3 12 p1m 2 10k + 513) >l 0 From the definition of Bs, ξ) in 51), we see that 512) can be reduced to the following two bounds: I 0 s, ξ) 1 ϕ s ξ)) ε p1)m 2 10k +, I 0 s, ξ)ϕ s ξ) ic 0 ξ ξ σ 1 σ fs, σ) 2 dσ ϕ sξ) fs, ξ) 514) ε 3 s p1)m 2 10k + For 513) instead, it suffices to show >l 0 I l1 s, ξ) ds ε p1)m 2 10k+ 515) The estimates 514) are proven in the next section, while 515) is proven in section Proof of 514) In view of the definition of ϕ s in 52), the first estimate in 514) is a consequence of the following Lemma: Lemma 52 High frequency output) Assume that 31) holds Then, for all m {0, 1, }, k Z [m/300, ), s [2 m 1, 2 m+1 ] and ξ [2 k, 2 k+1 ] I0 s, ξ) ε m/2 2 10k 516) Proof We decompose I 0 s, ξ) = ic 1 I 0 s, ξ) := l I 0 s, ξ) e isφξ,η,σ) η 2 ϕ l1 η)ϕ l 0) l 0 η) fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ Using the a priori bounds 54) on the L ξ -norm and on Sobolev norms, we can estimate I 0 s, ξ) k f 2 H ξ ) 10 f L 2 2 2mp k ε ) On the other hand, using only the bounds on high Sobolev norms one can see that I 0 s, ξ) 2 /2 2 Nk f 3 H N 2 /2 2 3mp 0 2 Nk ε ) Using 517) in the case 2m, and 517) for 2m, together with k m/300 and N = 1000, we obtain the desired conclusion The next Lemma shows how to derive the correction term to the scattering, and proves the validity of the second inequality in 514)

14 14 FABIO PUSATERI Lemma 53 Critical frequencies) Assume that 31) holds Then, for all m {0, 1, }, s [2 m 1, 2 m ], and k Z, m/300], ξ [2 k, 2 k+1 ], we have I 0 s, ξ) ic 0 ξ ξ σ 1 σ fs, σ) 2 dσ fs, ξ) ε 3 s m/20 519) Proof Let us recall the definition of I 0 : I 0 s, ξ) := ic 1 e isφ η 2 ϕ l 0) l 0 η) fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ, φξ, η, σ) := Λξ) + Λη + ξ) + Λσ + ξ) Λξ + η + σ), where ϕ l 0) l 0 is defined in 111), and 2 l m/40 520) Phase approximation In the support of the integral in 520) we can approximate the phase φ by a simpler expression Defining ξ φ 0 ξ, η, σ) := η ξ ξ + σ ), 521) ξ + σ we compute and conclude φξ, η, σ) = η 2 + 2η ξ ξ + ξ + η + η 2 2η ξ + σ) ξ + σ + ξ + η + σ = η ξ η ξ + σ) + + O η 2), ξ ξ + σ Thus, if we let I 0,1 s, ξ) := ic 1 it follows that for s [2 m 1, 2 m ] φξ, η, σ) φ 0 ξ, η, σ) η 2 522) e isφ 0ξ,η,σ) η 2 ϕ l 0) l 0 η) fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ, I 0 s, ξ) I 0,1 s, ξ) s φξ, η, σ) φ 0 ξ, η, σ) η 2 ϕ l 0) l 0 η) fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) dηdσ 2 m 2 3l 0 fs) 2 L 2 fs) L ε m/10, having used 3l 0 21m/10 Profiles approximation We now want to further approximate I 0,1 by the expression I 0,2 s, ξ) := ic 1 e isφ0ξ,η,σ) η 2 ϕ l 0) l 0 η) fs, ξ) fs, ξ + σ) fs, ξ + σ) dηdσ 523) In order to do this, we use the notation 111) and define for J 1 f J x) := ϕ J) J x)fx) and f Jx) := fx) f J x) The function f J is the restriction of f to a ball centered at the origin and radius 2 J in real space f J is the portion of f which lies at a distance greater than 2 J from the origin Using the a priori weighted bounds on f, we see that for η 2 l 0 one has fρ + η) fρ) f J ρ + η) f J ρ) + f J ρ + η) f J ρ) f J L + f J L 2 l 0 2 J/2 x 2 f L J/2 x 2 f L 22 l 0 ε 1 2 2mp 0 2 J/2 + 2 J/2 2 l 0 )

15 Choosing J = l 0 we obtain From this, for η 2 l 0, we see that MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 15 fρ + η) fρ) ε 1 2 2mp 0 2 l 0/2 fs, ξ + η) fs, ξ + σ) fs, ξ + η + σ) fs, ξ) fs, ξ + σ) fs, ξ + σ) dσ ε l 0/2 2 2mp 0 As a consequence, for all s [2 m 1, 2 m ], we obtain since p 0 1/1000 I 0,1 s, ξ) I 0,2 s, ξ) 2 3l 0/2 ε mp 0 ε m/10, Final approximation To conclude the proof we need to show I 0,2s, ξ) ic 0 ξ ξ σ 1 σ ûs, σ) 2 dσ fs, ξ) s + 1 ε m/20, 524) where I 0,2 is given by 523) and 521) After a change of variables, 524) can be reduced to c 1 e isη z η 2 ϕ l 0) l 0 η) fs, σ) 2 dηdσ c 0 z 1 s fs, σ) 2 dσ ε m/20 525) where we have defined z := ξ ξ σ σ 526) Observe that since F η 2 )x) = 2π 2 x 1, the following general formula holds for x : e iη x η 2 ϕη2 l ) dη 2π 2 x 1 x 2 2 l 527) Applying this with x = sz, s [2 m 1, 2 m+1 ], and l = l 0 29m/40, gives us c 1 e isη z η 2 ϕ l 0) l 0 η) dη c 0 s z 1 sz 2 2 l 0 2 5m/4 z 2, having used c 0 /c 1 = 2π 2 Thus we can see that the left-hand side of 525) is bounded by 2 5m/4 z 2 fs, σ) 2 dσ, 528) where z is defined in 526) Since z min{1, σ, ξ σ σ 3 }, and 1 + ξ ) 10 f ε1, we see that z 2 fs, σ) 2 dσ ε2 1 Plugging this bound into 528) we obtain 525) and conclude the proof of the lemma

16 16 FABIO PUSATERI 52 Proof of 515) We aim to prove: Z, >l 0 I l1 s, ξ) ds ε p1)m 2 10k+ 529) for I l1 defined in 510), s [2 m 1, 2 m+1 ] with m {0, 1, }, ξ 2 k with k Z, and l 0 given by 59) We decompose in dyadic pieces all the profiles and write I l1 = I k 1,k 2,k 3 s, ξ) k 1,k 2,k 3 Z I k 1,k 2,k 3 s, ξ) := We then aim to show e isφξ,η,σ) η 2 ϕ l 0) η) f k1 s, ξ + η) f k2 s, ξ + η + σ) f k3 s, ξ + σ) dηdσ I k 1,k 2,k 3 >l 0,k 1,k 2,k 3 Z For all s [2 m 1, 2 m+1 ], we can estimate Using the a priori bounds 54) we know that 530) s, ξ) ds ε p1)m 2 10k + 531) I k 1,k 2,k 3 s, ξ) 2 /2 f k1 L 2 f k2 L 2 f k3 L 2 532) f k L 2 min { 2 3k/2 2 10k +, 2 Nk + 2 mp 0 } 533) Since > l 0 3m/4, and N = 1000, the last two estimates above suffice to show that the sum in 531) over those indexes k 1, k 2, k 3 ) with max{k 1, k 2, k 3 } m/600 or min{k 1, k 2, k 3 } m, satisfies the desired bound Since max{k 2, k 3 } + 10, the remaining indexes in the sum are Om 4 ) The bound 531) can then be reduced to showing that for s [2 m 1, 2 m+1 ] with m {0, 1, }, and ξ 2 k with k Z, m/ ], one has for fixed triples k 1, k 2, k 3 ) with Let us further decompose I k 1,k 2,k 3 s, ξ) = I k 1,k 2,k 3,l 2 s, ξ) l 2 Z I k 1,k 2,k 3,l 2 s, ξ) := I k 1,k 2,k 3 s, ξ) ε p 1)m 2 10k +, 534) m k 1, k 2, k 3 m/600 e isφξ,η,σ) η 2 ϕ l 0) η)ϕ l2 σ) f k1 s, ξ + η) f k2 s, ξ + η + σ) f k3 s, ξ + σ) dηdσ The above terms are zero if l 2 m/ Moreover, we can estimate I k 1,k 2,k 3,l 2 s, ξ) max{k 1,k 2,k 3 } + fs) 3 L ϕ l1 η)ϕ l2 σ) dηdσ ε l k + This shows that l 2 : 3l 2 101m/100 I k 1,k 2,k 3,l 2 s, ξ) ds ε p1)m 2 10k + 535) We are then left again with a summation over l 2 with only Om) terms Therefore, we see that 529), and hence 515), will be a consequence of the following:

17 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 17 Proposition 54 Let I k 1,k 2,k 3,l 2 be defined as in 535), and assume 54) holds Then, for all s [2 m 1, 2 m+1 ] with m {0, 1, }, ξ [2 k, 2 k+1 ] with k Z, m/ ], one has I k 1,k 2,k 3,l 2 s, ξ) ε m/50, 536) whenever m k 1, k 2, k 3 m/600, 29m/40 and + 3l 2 101m/ ) The above Proposition is proven in a few steps, through Lemma 55, 56 and 57 below We will always be under the assumption that 54) holds, and all the indexes verify 537) Lemma 55 The bound 536) holds if Proof We write where I k 1,k 2,k 3,l 2 s, ξ) = max{k 1, k 2 } 538) e isφξ,η,σ) m 1 η, σ) f k1 s, ξ + η) f k2 s, ξ + η + σ) f k3 s, ξ + σ) dηdσ, m 1 ησ) := η 2 ϕ l 0) η)ϕ l2 σ)ϕ [k1 2,k 1 +2]ξ + η)ϕ [k2 2,k 2 +2]ξ + η + σ) 539) Since m 1 verifies the assumption of Lemma B1 with A = 2 2, we can estimate I k 1,k 2,k 3,l 2 s, ξ) 2 2 f k1 s) L 2 f k2 s) L 2 u k3 s) L k 1 ε 1 2 k 2 ε 1 2 3m/2 ε 1 ε m/2, having used the a priori bounds on the L norm of f and the hypothesis k 1, k 2 Lemma 56 Under the same assumptions of Proposition 54, the bound 536) holds if max{k 1, k 2 } and k 1 k ) Proof In this case we want to integrate by parts in η in the expression 535) for I k 1,k 2,k 3,l 2 s, ξ), using the identity e isφξ,η,σ) = 1 s m 0ξ, η, σ) η e isφξ,η,σ), m 0 ξ, η, σ) := ηφξ, η, σ) i η φξ, η, σ) 2 541) In particular, up to irrelevant constants, and with a slight abuse of notation, we can write I k 1,k 2,k 3,l 2 s, ξ) = I 1 s, ξ) + I 2 s, ξ), I 1 s, ξ) = e isφξ,η,σ) 1 s m 2η, σ) η fk1 s, ξ + η) f k2 s, ξ + η + σ)) fk3 s, ξ + σ) dηdσ, 542) I 2 s, ξ) = e isφξ,η,σ) 1 s m 2η, σ) f k1 s, ξ + η) f k2 s, ξ + η + σ) f k3 s, ξ + σ) dηdσ, 543) where omitting the variable ξ) we have denoted m 2 η, σ) := m 0 ξ, η, σ) ϕl 0) η) η 2 ϕ l2 σ)ϕ [k1 2,k 1 +2]ξ + η)ϕ [k2 2,k 2 +2]ξ + η + σ), 544) m 2η, σ) := η m 0 ξ, η, σ) ϕl 0) η)) η 2 ϕ l2 σ)ϕ [k1 2,k 1 +2]ξ + η)ϕ [k2 2,k 2 +2]ξ + η + σ) 545)

18 18 FABIO PUSATERI One can then verify that m 2 satisfies the hypothesis of Lemma B1 with A = max{k 1,k 2 }, so that we can apply B12) to estimate the term in 542) as follows: I1 s, ξ) [ 2 m max{k 1,k 2 } f k1 s) L 2 f k2 s) L 2 + f k1 s) L 2 f ] k2 s) L 2 u k3 s) L 2 m mp 0 2 3m/2 ε 3 1 ε m/50, having used 2 29m/20, and p 0 1/1000 For I 2 in 543) we perform an additional integration by parts and write again up to irrelevant constants) I 2 s, ξ) = J 1 s, ξ) + J 2 s, ξ), J 1 s, ξ) = e isφξ,η,σ) 1 s m 3η, σ) 2 η fk1 s, ξ + η) f k2 s, ξ + η + σ)) fk3 s, ξ + σ) dηdσ, 546) J 2 s, ξ) = e isφξ,η,σ) 1 s 2 m 3η, σ) f k1 s, ξ + η) f k2 s, ξ + η + σ) f k3 s, ξ + σ) dηdσ, 547) where m 3 η, σ) := m 0 ξ, η, σ)m 2η, σ), 548) [ m 3η, σ) := η m 0 ξ, η, σ) η m 0 ξ, η, σ) ϕl 0) η)) ] η 2 ϕ l2 σ) 549) ϕ [k1 2,k 1 +2]ξ + η)ϕ [k2 2,k 2 +2]ξ + η + σ) From the definition of m 0 and m 2 in 541) and 545), we see that the symbol m 3 satisfies the hypothesis B11) in Lemma B1 with A = max{k 1,k 2 } We then obtain J 1 s, ξ) [ 2 2m max{k 1,k 2 } f k1 s) L 2 f k2 s) L 2 + f k1 s) L 2 f ] k2 s) L 2 u k3 s) L 2 2m max{k 1,k 2 }/2 2 mp 0 2 3m/2 ε 3 1 From the hypothesis 537) we see that 3 9m/4, and l 2 2m/5 This latter implies and therefore max{k 1, k 2 }/2 l 2 / m/5 + 10, J1 s, ξ) 2 m/4 2 max{k 1,k 2 }/2 2 mp 0 2 3m/2 ε m/50 ε 3 1, as desired To estimate J 2 in 547) we only use the pointwise bound m 3η, σ) max{k 1,k 2 } and the a priori bounds 54) to deduce J2 s, ξ) 2 2m max{k 1,k 2 } f k1 s) L 2 3 f k2 s) L 2 f k3 s) L 2 550) 2 2m 2 2 max{k 1,k 2 }/2 ε m/50 ε 3 1, having used once again 3m/4, and max{k 1, k 2 }/2 m/ Lemma 57 The bound 536) holds if k 1 k 2 10 and max{k 1, k 2 } 551)

19 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 19 Proof Recall the notation 541), 544), 545), and 549) for the symbols m 0, m 2, m 2, and m 3 from the proof of Lemma 56 above: m 0 η, σ) = ηφξ, η, σ) i η φξ, η, σ) 2, m 2 η, σ) = m 0 η, σ) ϕl 0) η) η 2 ϕ l2 σ)ϕ [k1 2,k 1 +2]ξ + η)ϕ [k2 2,k 2 +2]ξ + η + σ), m 2η, σ) := η m 0 η, σ) ϕl 0) η)) η 2 ϕ l2 σ)ϕ [k1 2,k 1 +2]ξ + η)ϕ [k2 2,k 2 +2]ξ + η + σ) [ m 3η, σ) := η m 0 η, σ) η m 0 η, σ) ϕl 0) η)) ] η 2 ϕ l2 σ) ϕ [k1 2,k 1 +2]ξ + η)ϕ [k2 2,k 2 +2]ξ + η + σ) Integrating by parts twice in the expression for I k 1,k 2,k 3,l 2 in 535), or once more in 542)-543), one can see that I k 1,k 2,k 3,l 2 s, ξ) K1 s, ξ) + K 2 s, ξ) + K 3 s, ξ) K 1 s, ξ) = e isφξ,η,σ) 1 s q 1η, σ) 2 2 η fk1 s, ξ + η) f k2 s, ξ + η + σ)) fk3 s, ξ + σ) dηdσ, K 2 s, ξ) = e isφξ,η,σ) 1 s q 552) 2η, σ) 2 η fk1 s, ξ + η) f k2 s, ξ + η + σ)) fk3 s, ξ + σ) dηdσ, K 3 s, ξ) = e isφξ,η,σ) 1 s q 3η, σ) f 2 k1 s, ξ + η) f k2 s, ξ + η + σ) f k3 s, ξ + σ) dηdσ, where q 1 η, σ) := m 0 η, σ)m 2 η, σ), q 2 η, σ) := m 0 η, σ)m 2η, σ), q 3 η, σ) := m 3η, σ) 553) We now proceed to estimate the three integrals above First let us notice that for ξ + η 2 k 1 and ξ + η + σ 2 k 2 with k 1 k 2, η 2 and σ 2 l 2, one has for a, b Z 3 + with a, b 10 As a consequence for a, b Z 3 + with a, b 10 It then follows that a η b σm 0 η, σ) 2 l max{k 1,k 2 } 2 a 2 b l 2, 554) a η b σm 2 η, σ) l max{k 1,k 2 } 2 a 2 b l 2, 555) F 1 q 1 L l max{k 1,k 2 } 556) We then apply Lemma B1 and obtain K1 s, ξ) 2 2m l max{k 1,k 2 } x 2 f k1 s) L 2 x 2 f k2 s) L 2 u k3 s) L 2 2m l 2 2 4mp 0 2 2k 1 2 3m/2 ε /50 ε 3 1, having used 2 29m/20, 2l 2 4m/5, k 1 m/600 and p 0 1/1000 We can estimate similarly the term K 2 in 552) From the definition of q 2 in 553), and the estimates 554) and 555) for m 0 and m 2, we see that F 1 q 2 L l max{k 1,k 2 } 557)

20 20 FABIO PUSATERI Using 557) and Lemma B1 we can obtain the bound K2 s, ξ) 2 2m l max{k 1,k 2 } x f k1 s) L 2 x f k2 s) L 2 u k3 s) L 2 2m l 2 2 4mp 0 2 2k 1 2 3m/2 ε 3 1 Now observe that the second constraint in 537) gives 2 3m/2 m/20 Moreover, the second and third inequalities in 537) imply 2l 2 m, as it can be seen, for instance, by considering the two cases l 2 m/16 and l 2 m/16 From the chain of inequalities above we can then conclude that K2 s, ξ) 2 51m/50 ε 3 1 Eventually we come to K 3 In this case we only use the pointwise bound for q 3 q 3 η, σ) l max{k 1,k 2 }, and estimate K3 s, ξ) 2 2m l max{k 1,k 2 } f k1 s) L f k2 s) L f k3 s) L l 2 2 2m 2 2 l max{k 1,k 2 } ε m/50 ε 3 1, having used once again the lower bound on in 537), and l 2, k 1, k 2 m/ APPENDIX A SUBCRITICAL SEMI-RELATIVISTIC HARTREE EQUATIONS As already discussed in the introduction, some generalized models related to the boson star equation 11) have also been studied recently, and, in particular, the class of semi-relativistic Hartree equations i t u Λu = x γ u 2) u, Λ = 1, x R n, 0 < γ < n A1) We are interested here in constructing small scattering solutions when γ > 1 For γ > 2, and γ > 3/2 in the radial case, such solutions have been obtained in [3] and [5] Our proof of the weighted bounds in Proposition 32, done for the case γ = 1, can be adapted to prove the following: Theorem A1 Let u 0 : C be given such that u 0 H 10 + x 2 u 0 H 3 ε 0 There exists ε 0 such that for all ε 0 ε 0, the Cauchy problem associated to A1) with 1 < γ < 3, with initial datum ut = 0, x) = u 0 x), has a unique global solution satisfying [ sup ut) H 10 + x 2 e itλ ut) ] H 3 ε0 A2) t Furthermore, there exist δ > 0, and f + L 2 x 4 dx), such that e itλ ut, x) f + L 2 x 4 dx) ε 01 + t) δ, for all t > 0 A similar statement holds for t < 0 A3) Since the above result follows from arguments similar to those in section 4, we will just provide some ideas of its proof below Sketch of the proof of Theorem A1 Let us start by defining N γ h 1, h 2, h 3 ) := x γ h 1 h 2 ) h3, A4)

21 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 21 for 1 < γ < 3 The Hausdorff-Young inequality then gives N γ h 1, h 2, h 3 ) L 2 h 1 L p 1 h 2 L p 2 h 3 L p 3, A5) for any p 1, p 2, p 3 [2, ] and 1/p 1 + 1/p 2 + 1/p 3 = 3/2 γ/3 A6) We want to construct a global solution such that A2) holds Let us assume that we are given a local solution on [0, T ] which is a priori bounded as follows: [ sup ut) H 10 + x 2 e itλ ut) ] H 3 ε1, A7) t [0,T ] for some ε 1 > 0 Duhamel s formula for ft) = e itλ ut) reads: ut) = e itλ u 0 + e itλ Nt), Nt) := t 0 e iλs N γ us), us), us)) ds A8) To obtain a global solution it suffices to show that under the a priori assumptions A7) one has [ sup Nt) H 10 + x 2 Nt) ] H 3 ε 3 1, A9) t [0,T ] for some C > 0 Notice that A7) implies, via the standard L p L q estimates sup 1 + t) 3/2 3/p ut) L p ε 1, A10) t [0,T ] for all p 2 Also notice than any global solution ut) which is bounded as in A2), automatically scatters to a linear solution in L 2, because N γ ut), ut), ut)) L 2 ut) 2 L 6/3 γ) ut) L 2 ε t) γ, which is an integrable function of time The first term in A9) can be bounded directly by A5) and A10): Nt) H 10 t 0 u 2 L 6/3 γ) u H 10 ds ε 3 1 t s) γ ds ε 3 1 To bound the second norm in the left-hand side of A9) let us write Nt) in Fourier space as t Nt, ξ) = ic Is, ξ) ds, 0 Is, ξ) = e isφξ,η,σ) η 3+γ fs, ξ η) fs, η + σ) fs, σ) dηdσ, φξ, η, σ) := Λξ) + Λξ η) + Λη + σ) Λσ) A11) Here c = cγ) denotes an appropriate positive constant which is irrelevant for the proof Then the idea is to proceed as in section 42, applying ξ 3 2 ξ to I and estimating the resulting terms in L2 Applying

22 22 FABIO PUSATERI 2 ξ to I we obtain four terms: ξ 2 Is, ξ) = J γ 1 s, ξ) + 2J γ 2 s, ξ) + J γ 3 s, ξ) + J γ 4 s, ξ), J γ 1 s, ξ) := e isφξ,η,σ) η 3+γ ξ 2 fs, ξ η) fs, η + σ) fs, σ)dηdσ, J γ 2 s, ξ) := is e isφξ,η,σ) mξ, η) η 3+γ ξ fs, ξ η) fs, η + σ) fs, σ) dηdσ, J γ 3 s, ξ) := is e isφξ,η,σ) ξ mξ, η) η 3+γ fs, ξ η) fs, η + σ) fs, σ) dηdσ, J γ 4 s, ξ) := s 2 e isφξ,η,σ) [mξ, η)] 2 η 3+γ fs, ξ η) fs, η + σ) fs, σ)dηdσ, A12) A13) A14) A15) where the symbol m is defined as in 411): mξ, η) := ξ Λξ) + Λξ η) ) The terms J γ i in A12)-A15), for i = 1,, 4, look exactly like the terms J i in 419)-422), with the exception that the power on η is now 3 + γ > 2 To obtain A9), it would be sufficient to prove ξ 3 J γ i s) L 2 ε s) 1 γ 1)/4 A16) To see this, one should proceed as in sections and use the following two facts: 1) Under the a priori assumptions A7) one can set p 0 = 0 in all the estimates in ) Let 2 k 2 denote the size of η as in the estimates of sections Since A12)-A15) have a factor η 3+γ instead of η 2 as in 419)-422), one can obtain estimates for A12)-A15) which are a factor 2 γ 1)k 2 better than those for 419)-422) Thanks to these observations, one can verify that the bounds 423) for the L 2 norms of J 1,, J 4, can be improved to the bounds A16) for J γ 1,, J γ 4 This gives A9) The scattering statement A3) follows from the bounds A16) and the integrable time decay of N γ This concludes the proof the Theorem APPENDIX B AUXILIARY ESTIMATES B1 Proof of Proposition 31: Refined Linear Estimates In this section we give the proof of Proposition 31 by showing e it 1 1 f 1 + ξ ) 6 fξ) L 1 [ x L 1 + t ) 3/2 + 2 f ] ξ 1 + t ) 31/20 L 2 + f H 50 B1) for any t R Estimate B1) is a simple but crucial ingredient in deriving the modified scattering behavior for solutions of 11) It identifies the leading order norm that needs to be controlled in order to obtain the necessary sharp pointwise decay of t 3/2, and dictates what expression needs to be analyzed in order to capture the asymptotic behavior of solution of 11) Similar estimates, as well as some variants, have been used when dealing with other L critical equations and not only), see for example [15, 9, 16, 17, 23, 19] Our proof is in the same spirit of the proof of Lemma 23 of [19], where the author and Ionescu treated the linear propagator expit x 1/2 ) The analogous estimate for this propagator was then used to obtain global solutions to the gravity water waves problem in the case of one dimensional interfaces [20]

23 MODIFIED SCATTERING FOR THE BOSON STAR EQUATION 23 Proof of B1) Set Λ ) := 1 = Using the notation 19), we write e itλ ) fx, t) = e itφξ) fξ)ϕk ξ) dξ, φξ; x, t) := Λξ) + ξ x k Z R t 3 For B1) it then suffices to prove that e itφξ) fξ)ϕk ξ) dξ 1, k Z for any t R, x, and any function f satisfying B2) B3) 1 + t ) 3/2 1 + ξ ) 6 f L ξ t ) 31/20[ x 2 f L 2 + f H 50] 1 B4) High and low frequencies Using only the bound f L 1+ t ) 3/2, we estimate first the contribution of small frequencies, e itφξ) fξ)ϕk ξ) dξ 2 3k P k f L 1 2 k 1+ t ) 1/2 2 k 1+ t ) 1/2 Using instead the bound f H t ) 31/20, we can control the contribution of large frequencies: e itφξ) fξ)ϕk ξ) dξ 2 3k/2 P k f L k f H k 1+ t ) 1/30 2 k 1+ t ) 1/30 2 k 1+ t ) 1/30 Non-stationary frequencies From above we see that for B3) it suffices to prove e itφξ) fξ)ϕk ξ) dξ 1 1+ t ) 1/2 2 k 1+ t ) 1/30 In proving B5) we may assume that t 1 Notice that for x < t x ξ φξ) = 0 ξ = ξ 0 := t2 x, 2 while ξ φ Λξ) 2, for x t We estimate first the non-stationary contributions when ξ is away from ξ 0, and more precisely when 2 k 2 4 ξ 0 or 2 k 2 4 ξ 0 In these cases we have r φ ξ ξ k ) 1, where r = ξ/ ξ ξ φ denotes the radial derivative We can integrate by parts twice in B2) and write: e itφξ) fξ)ϕk ξ) dξ = I 1) k + I 2) k + I 3) k, I 1) k := 1 t 2 e itφξ) r φ) 2 2 r fξ)ϕk ξ) ) dξ, I 2) k := 3 1 t 2 e itφξ) r φ) 1 r r φ) 1 r fξ)ϕk ξ) ) dξ, I 3) k := 1 t 2 e itφξ) r r φ) 1 r r φ) 1) fξ)ϕk ξ) dξ For ξ [2 k 2, 2 k+2 ] with 2 k 2 4 ξ 0 or 2 k 2 4 ξ 0, one has r φ 2 k k ) 1 Therefore, using B4) we can estimate 1) I k t 2 2 2k k ) 2 ) r 2 fϕk L 1 t 2 2 2k k ) 2 3k/2 x 2 f L k f ) k L 1 t k ) 2 2 k/2 t 31/ k t 3/2), B5) B6) B7)

24 24 FABIO PUSATERI and deduce that k I1) k 1, from the fact that we are only summing over those k such that t 1/2 2 k t 1/30, To estimate I 2) k in B7), we first notice that r φ) 1 r r φ) 1 2 3k k ) 2 Moreover one has r fϕk ) L 1 2 2k f k L + 2 5k/2 r f L 6 2 2k f k L + 2 5k/2 x 2 f L 2 Therefore, we see that t 2 2 3k k ) 2 ) r fϕk L 1 t 2 2 k f L k )2 k/2 x 2 f L 2) I 2) k Using again B4) and the restrictions t 1/2 2 k t 1/30, we get k I2) k 1 We can deal similarly with I 3 Since r r φ) 1 r r φ) 1) 2 4k k ) 2, we obtain k I 3) k t 2 2 k t 1/2 2 4k k ) 2 fϕ k L 1 t 2 2 k t 1/2 2 k k ) f k L 1 Stationary contributions To eventually conclude the proof of B5) it suffices to show e itφξ) fξ)ϕk ξ) dξ 1, provided that t 1, 1 + t ) 1/2 2 k 1 + t ) 1/30, and 2 k [2 4 ξ 0, 2 4 ξ 0 ] Let l 0 denote the smallest integer with the property that 2 l 0 t 1/2 and estimate the left-hand side of B8) by k+100 e itφξ) fξ)ϕk ξ) dξ J l, B9) l=l 0 where, with the notation 111), for any l l 0 we have defined J l := e itφξ) fk ξ)ϕ l 0) l ξ ξ 0 ) dξ From B4) it immediately follows J l0 2 3l 0 f k L t 3/2 f L 1 For l > l 0 we integrate by parts in the expression for J l above, relying on the fact that ξ ξ 0 2 l t 1/2 on the support of the integral Two integration by parts, like the ones performed in the previous paragraph, give J l = J 1) l + J 2) l + J 3) l, J 1) l := 1 e t R itφξ) 2 r φ) 2 r 2 3 := 3 1 t 2 J 2) l J 3) l := 1 t 2 fk ξ)ϕ l 0) l ξ ξ 0 ) ) dξ, e itφξ) r φ) 1 r r φ) 1 r fk ξ)ϕ l 0) l ξ ξ 0 ) ) dξ, B8) B10) e itφξ) r r φ) 1 r r φ) 1) fk ξ)ϕ l 0) l ξ ξ 0 ) dξ Most of the above contributions can be estimated in exactly the same way as we have estimated the terms in B7), using the fact that r 2 φξ) k ) 3 and ξ ξ 0 2 l t 1/2, which imply r φξ) k ) 3 2 l in the support of J l The term J 3) l, for example, verifies the exact same bound as I 3) k : J 3) l t k ) 2 f k )ϕ l ξ 0 ) L 1 t k ) f k L 2 l 1,