Well-posedness for the surface quasi-geostrophic front equation

We consider the well-posedness of the surface quasi-geostrophic (SQG) front equation. Hunter–Shu–Zhang (2021 Pure Appl. Anal. 3 403–72) established well-posedness under a small data condition as well as a convergence condition on an expansion of the equation’s nonlinearity. In the present article, we establish unconditional large data local well-posedness of the SQG front equation, while also improving the low regularity threshold for the initial data. In addition, we establish global well-posedness theory in the rough data regime by using the testing by wave packet approach of Ifrim–Tataru.


Introduction
The surface quasi-geostrophic (SQG) equation takes the form (1.1) θ t + u • ∇θ = 0, u = (−∆) − 1 2 ∇ ⊥ θ where θ is a scalar evolution equation on R 2 , (−∆) − 1 2 denotes a fractional Laplacian, and ∇ ⊥ = (−∂ y , ∂ x ).The SQG equation arises from oceanic and atmospheric science as a model for quasi-geostrophic flows confined to a surface.This equation is also of interest due to similarities with the three dimensional incompressible Euler equation.In particular, the question of singularity formation remains open for both problems.
The SQG equation is one member in a family of two-dimensional active scalar equations parameterized by the transport term in (1.1), with The case α = 2 gives the two dimensional incompressible Euler equation, while the α = 1 case gives the SQG equation (1.1) above.
Front solutions to (1.1) refer to piecewise constant solutions taking the form where the front is modeled by the graph y = ϕ(t, x) with x ∈ R. Front solutions are closely related with patch solutions where Ω is a bounded, simply connected domain.
When α ∈ (0, 1), contour dynamics equations for patches and fronts may be derived and analyzed in a similar way.Global well-posedness for small and localized data was established by Córdoba-Gómez-Serrano-Ionescu in [3].
However, when α ∈ [1,2], the derivation of contour dynamics equations for fronts has complexities arising from the slow decay of Green's functions.The derivation in this range was addressed by Hunter-Shu [7] via a regularization procedure, and again by Hunter-Shu-Zhang in [9].
In the case of SQG patches, Gancedo-Nguyen-Patel proved in [4] that in a suitable parametrization, the SQG front equation is locally well-posed in H s (T), where s > 2. Local well-posedness for the generalized SQG family, where α ∈ (0, 2) and α = 1, was also considered by Gancedo-Patel in [5], establishing in particular local well-posedness in H 2 for α ∈ (0, 1).
In the case of nonperiodic SQG fronts, Hunter-Shu-Zhang studied the local well-posedness for a cubic approximation of (1.3) in [8].In [11], they considered the local well-posedness for the full equation (1.3) with initial data in H s , s ≥ 5, along with global well-posedness for small, localized, and essentially smooth (s ≥ 1200) initial data.However, these wellposedness results require a small data assumption to ensure the coercivity of the modified energies used in the energy estimates, along with a convergence condition on an expansion of the nonlinearity A ϕ ϕ appearing in (1.3).These results were extended to the range α ∈ (1,2] for the generalized SQG family in [10]. In the present article, our objective is to revisit and streamline the analysis of (1.3), while improving the established well-posedness results.Our contributions include: • offering substantial simplifications to the paradifferential analysis, • removing the convergence and small data assumptions in the local well-posedness result of [11], • establishing the local well-posedness in a significantly lower regularity setting at 1 + ǫ derivatives above scaling, corresponding to the classical threshold of Hughes-Kato-Marsden for nonlinear hyperbolic systems [6], and • establishing the global well-posedness in a low regularity setting, by applying the wave packet testing method of Ifrim-Tataru (see for instance [12,14]).We anticipate that our streamlined analysis will also open the way to substantial simplifications and improvements in the analysis of related equations, including the generalized SQG family (1.2).
Our main local well-posedness result is as follows: Theorem 1.1.Equation (1.3) is locally well-posed for initial data in H s with s > 5  2 .Precisely, for every R > 0, there exists T = T (R) > 0 such that for any ϕ 0 ∈ H s (R) with ϕ 0 H s < R, the Cauchy problem (1.3) has a unique solution ϕ ∈ C([0, T ], H s ).Moreover, the solution map ϕ 0 → ϕ from H s to C([0, T ], H s ) is continuous.
We also consider global well-posedness for small and localized data.To describe localized solutions, we define the operator which commutes with the linear flow ∂ t − 2 log |D x |∂ x , and at time t = 0 is simply multiplication by x.Then we define the time-dependent weighted energy space where s > 4. To track the dispersive decay of solutions, we define the pointwise control norm Then the solution ϕ to (1.3) with initial data ϕ 0 exists globally in time, with energy bounds Further, the solution ϕ exhibits a modified scattering behavior, with an asymptotic profile W , in a sense that will be made precise in Section 7. There, we also observe that (1.3) has a conserved L 2 mass.We remark that Hunter-Shu-Zhang [10] makes a similar observation regarding mass conservation for the generalized SQG in the range α ∈ (1, 2].
Our paper is organized as follows.In Section 2, we establish notation and preliminaries used through the rest of the paper.We define a parameter-dependent paradifferential quantization which will ensure coercivity in our energy estimates, and which is key in removing the small data assumption in the local well-posedness theory.We also establish Moser estimates to be applied toward the paralinearization of A ϕ ϕ, as well as a more general linearized counterpart A ϕ v. Lastly, we record some elementary lemmas involving difference quotients.
In Section 3, we paralinearize the operator A ϕ v, up to a perturbative error.We then apply this result to reduce the analysis of both the equation (1.3) and its linearization to the analysis of a paradifferential flow with perturbative source.
In Section 4, we prove energy estimates for the paradifferential flow of Section 3, by defining modified energies that are comparable to the classical Sobolev energies in the spaces H s (R).This part crucially uses the quantization established in Section 2 to remove the small data assumption made in the local well-posedness result from [11].
In Section 5, we prove Theorem 1.1, the local well-posedness result for (1.3).We do this by first using an iterative scheme to construct smooth solutions, and then using the method of frequency envelopes to lower the regularity exponent for the solutions.This method was introduced by Tao in [16] in order to better track the evolution of energy distribution between dyadic frequencies.A systematic presentation of the use of frequency envelopes in the study of local well-posedness theory for quasilinear problems can be found in the expository paper [13].
In Section 6 we use the wave packet testing method of Ifrim-Tataru to prove the globalwellposedness part of Theorem 1.2, along with the dispersive bounds for the resulting solution.This method is systematically presented in [14].
Finally, in Section 7 we provide a short proof for the mass conservation for the solutions of (1.3).Then we discuss the modified scattering behavior of the global solutions constructed in Section 6.
1.1.Acknowledgements.The first author was supported by the NSF grant DMS-2220519 and the RTG in Analysis and Partial Differential equations grant DMS-2037851.The second author was supported by the NSF grant DMS-2054975, as well as by the Simons Foundation.The authors would like to thank Mihaela Ifrim and Daniel Tataru for many helpful discussions.

Notation and preliminaries
To facilitate the analysis of the operator A ϕ , we define the smooth function which in particular vanishes to second order at s = 0, satisfying F (0) = F ′ (0) = 0. Using this notation, we define the following generalization of (1.4): This generalized operator will be useful when we study the linearized equation.We let 0 < δ < s − 5 2 refer to a small positive exponent throughout.Implicit constants may depend on our choice of δ.
Given a symbol a(x, η), we use the above cutoff symbol χ to define an M dependent paradifferential quantization of a by (see also [2]) where the Fourier transform of the symbol a = a(x, η) is taken with respect to the first argument.We make the following remarks: • The Fourier transform, and in particular T a and Littlewood-Paley projections, will all operate with respect to the x variable throughout.• On the Fourier side, the support conditions on P>M and χ imply that |η| ≈ |ξ|.
• When a is independent of η, the above definition coincides with the Weyl quantization, which means that the operator T a will be self-adjoint if a is real valued.This will be useful when proving energy estimates in Section 4. Implicit constants may depend on M ≫ 1, which we view as a constant parameter except in Section 5. There, we will choose M using the following lemma, which will be key for establishing local well-posedness for large data: Proof.On |ξ − η| ≤ M 2 , we have χ(ξ − η, ξ + η) = 1.Thus we may write and conclude We have by Sobolev embedding Further, using the form of F combined with Sobolev embedding, we have a L ∞ < 1 − δ uniformly over u H s ≤ R, for some δ > 0 depending on R. Choosing M sufficiently large depending on R, we obtain 2.2.Classical estimates.We recall the following Moser-Schauder estimates.Note that in the present article, we typically take F to be given by (2.1).
Theorem 2.2 (Moser-Schauder).Let s ≥ 0 and F : R → R be a smooth function satisfying x f Ḣs .c) Under the hypotheses of b), in the case s = 0, we have For a proof of a), see [17,Lemma A.9]. Here, we prove parts b) and c).In the following, denote Then write We consider B 2 .Write We estimate the summand by By the chain rule, for each j ≥ −4, Thus, In this case of the frequencies where N has been chosen such that N + 1 > s.In particular, when s = 0, we may choose For the low frequency terms k ≤ 0 in B 2 , using the fact that F ′ (0) = 0, We conclude in the case s > 0 that We now exchange B 1 for The summand of the difference so that summing orthogonally, we have the desired bound.It remains to exchange (2.6) with T F ′ (f ) f .Write The second term contributes x f k Ḣs which sums orthogonally over k.The third term is estimated similarly.For the first, using cancellation with which can be done by noting that x f k Ḣs which is estimated in a similar way as in the frequency-balanced case.
We will use the following logarithmic commutator estimate: Lemma 2.3.We have for s ≥ 0, Proof.We have , and observing that (η − ξ)/ξ remains away from −1 on the support of χ and P>M , where |η| ≥ M and |ξ| |η|, we see that χ1 is bounded.Further, since χ1 satisfies the same kind of bounds as χ, we can apply paraproduct Coifman-Meyer estimates to deduce the desired result.
We also recall the following variant of [1, Lemma 2.5]. When 2.3.Difference quotients.We denote difference quotients by We also use the following estimates which we will apply to averages over difference quotients: Lemma 2.5.We have Proof.We decompose the integral Then sgn(y) Proof.We write sgn(y) For the former integral, we have by Hölder The latter integral is treated similarly.

Paralinearizations and the linearized equation
The objective of this section is to paralinearize the operator A ϕ v defined in (2.2).With this paralinearization, we reduce the analysis of (1.3) and its linearization to the analysis of a paradifferential flow with perturbative source.
The analysis in this section is time independent.We will use the notation B 0 to denote the zeroth order coefficient of the paralinearization, Proposition 3.1.We have the paralinearization where Proof.We decompose We estimate A 1 using Lemma 2.5, For the first term, we have For the second term, we have so that the contribution from A 1 may be absorbed into R(ϕ, v).A similar analysis applies to absorb A 2 into R(ϕ, v).
We proceed with A 3 , which we may write as Since F (δ y ϕ)(1 − e iηy ) vanishes at y = 0 and F (δ y ϕ) satisfies the decay in y, we may express the symbol of the paradifferential operator in terms of p.v.|y| −1 : Using this, we write Since the first two terms on the right hand side of (3.1) form the main terms of the paralinearization of A ϕ , it remains to absorb the contribution of the last term in (3.1) into R. From Lemma 6.2 in [15], we have iηe iηy dy v, where T ′ is another low-high paraproduct.
Integrating the paradifferential symbol by parts, It is immediate to see that the contribution from the first term on the right in (3.3) may be absorbed into R.It remains to consider the contribution from the second term, which we write y 2 so that the integral over {|y| > 1} converges with the appropriate bound.On the other hand, for the integral over {|y| ≤ 1}, we use instead which also suffices.
Next, we paralinearize the nonlinear term A ϕ ϕ in (1.3), evaluating the H s higher regularity of the errors: Proposition 3.2.We have the paralinearization where for any s ≥ 0, We remark that the R difference estimate is not optimal with respect to the pointwise control coefficients.However, this estimate is only used in the construction of smooth solutions, and will play no role in the refined low regularity analysis.
Proof.We write The analysis of A 3 closely follows the analysis of the counterpart A 3 in the proof of Proposition 3.1, with ϕ in the place of v and H s in the place of L 2 .This contributes both terms in the paralinearization.The balanced frequency term A 2 is also similar to its counterpart and may be absorbed into R(ϕ).

It remains to absorb A 1 into R(ϕ). We further decompose
The first difference is estimated using the Moser estimates of Theorem 2.2, while the second may be estimated using the para-product estimates of Lemma 2.4.
We now prove the difference estimate.We rewrite We consider the components of the remainders arising from the terms of the form B 3 .One such contribution is We analyze the last term, by splitting the integral over the regions {|y| ≤ 1} and {|y| > 1}, respectively.We first consider the former.We claim that For this purpose, we write the low frequency component as which can in turn be written as We can now move the derivatives in the first three terms from the low frequencies to the high frequencies and obtain the claimed estimates.
The estimate for this term now follows using ideas similar to the ones for the region {|y| ≤ 1} and from the proof of 3.1.
From the contribution corresponding to A 1 , we also have As F has a zero of order 2 at 0, Lemma 2.6 implies that |y|≤1 T |δ| y vx F (δ y ϕ) dy The other bounds follow by reasoning similarly as before.
The other terms in R(ϕ) − R(ψ) can be treated similarly.The proof of the H s x -bound is analogous.
Recall that the linearized equation (1.5) corresponding to (1.3) may be written Using the paralinearization Proposition 3.1, we obtain Proposition 3.3.The linearized equation (1.5) admits the paralinearization where

Energy estimates
The goal of this section is to prove energy estimates for the paradifferential flow (3.3), which will in turn be used to derive energy estimates for solutions of (1.3), along with its linearization (1.5).
We define the modified energies Due to our construction of the paraproduct, these modified energies are comparable to the classical energies in the Sobolev spaces H s (R).
We prove the following: ).Then the initial value problem has a unique solution v ∈ C([0, T ]; H s x ), satisfying the estimate Applying T (1−F (ϕx)) s |D x | s , we obtain an equation for g t , where R 1 consists of commutators which we will record and estimate momentarily.Using this, we have From the first integral on the right hand side, we rewrite the contribution where as before, R 2 consists of commutators which we record and estimate momentarily.
Observe that the first term on the right is a divergence which thus vanishes.Similarly, since ∂ x log |D x | is skew-adjoint, the corresponding contribution to (4.3) vanishes and we conclude where The second commutator of R 1 may be estimated using the log |D x | commutator Lemma 2.3.For the third commutator of R 1 , applying again Lemma 2.3, we may commute log |D x | to the front, so this reduces to The third commutator on the right may be estimated using Lemma 2.4.The first may be written log v, while the principal term of the second is x T F ′ (ϕx)ϕxx v, which cancel up to commutators and paraproducts estimated via Lemma 2.4.
Returning to R 1 , it remains to consider the first commutator.For the commutator with respect to ∂ t , we have T s(1−F (ϕx)) s−1 F ′ (ϕx)ϕtx |D x | s v which contributes to the right hand side of (4.2).For the commutator with respect to ∂ x T B 0 (ϕ) , and in particular to estimate B 0 (ϕ), we apply Lemma 2.5 to estimate using the decay of F for the first term.The same estimate for B 0 (ϕ) then suffices to estimate each term of R 2 .
We conclude, using Lemma 2.1 to pass between v H s and g L 2 , which along with the similar case s = 0 gives the estimate of the proposition.By using the adjoint method, it follows that the equation has a unique solution.
Corollary 4.2.Let R > 0. If ϕ is a solution of (1.3) on an interval [0, T ] on which ϕ C 0 t H s x R, where s > 5  2 , then we have the energy estimate Moreover, if v is a solution of the linearized equation (1.5) on an interval [0, T ] where the solution ϕ satisfies the previous conditions, then Proof.From Proposition 4.1, we have the energy estimate In both situations, ϕ is a solution of (1.3), so in order to control ϕ tx L ∞ x , we write From Lemma 2.6 and the product rule, we have We also have In the first case, v = ϕ solves the (1.3) equation, so by Proposition 3.2, we know that , and an application of Grönwall's lemma, along with the coercivity bounds implies the claimed estimate.When v solves the linearized equation (1.5), Proposition 3.1 similarly implies that

Local well-posedness
In this section we establish Theorem 1.1, our main local well-posedness result in Sobolev spaces for the SQG equation (1.3).We do this by first constructing smooth solutions using an iterative scheme, and then we employ frequency envelopes in order to construct rough solutions as limits of smooth ones and to prove continuous dependence on the initial data.
We first consider smoother data ϕ 0 ∈ H s with s ≥ 4. Fix R > 0 and choose M as in Lemma 2.1.We construct the sequence where we define the operator G(ϕ) = v using Proposition 4.1 with R = R(ϕ), the paralinearization error from Proposition 3.2.We have An application of Grönwall's inequality with Proposition 4.1 shows that along with an easy induction on n and the terms of the sequence ( ϕ x ) n≥0 show that for sufficiently small T > 0 depending on R, ϕ (n) has uniform H s x bounds.Moreover, we show that for sufficiently small T > 0, ϕ (n) is Cauchy.Let ϕ (m) and ϕ (n) be two terms of the sequence.Denoting v = ϕ (m) − ϕ (n) , we have (n) .We now apply the energy estimate of Proposition 4.1 with x .An application of Grönwall's inequality shows that the sequence is Cauchy in L 2 x , for T > 0 small enough.This settles the existence.Uniqueness follows from the energy estimate for the difference.
To establish the local well-posedness result at low regularity, we follow the approach outlined in [13].We consider ϕ 0 ∈ H s with s > 5  2 .Let ϕ h 0 = (ϕ 0 ) ≤h , where h ∈ Z. Since ϕ h 0 → u 0 in H s x , we may assume that ϕ h 0 H s x < R for all h.We construct a uniform H s x frequency envelope {c k } k∈Z for ϕ 0 having the following properties: a) Uniform bounds: Let ϕ h be the solutions with initial data ϕ h 0 .Using the energy estimate for the solution ϕ of (1.3) from Corollary 4.2, we deduce that there exists T = T ( ϕ 0 H s x ) > 0 on which all of these solutions are defined, with high frequency bounds Further, by using the energy estimates for the solution of the linearized from Corollary 4.2, we have By interpolation, we infer that As in [13], we get and that for every k ≥ 1.Thus, ϕ h converges to an element ϕ belonging to C 0 t H s x ([0, T ]×R).Moreover, we also obtain (5.1) .
We now prove continuity with respect to the initial data.We consider a sequence and an associated sequence of H s x -frequency envelopes {c j k } k∈Z , each satisfying the analogous properties enumerated above for c k , and further such that c j k → c k in l 2 (Z).In particular, (5.2) .
Using the triangle inequality with (5.1) and (5.2), we write To address the third term, we observe that for every fixed h, ϕ h j → ϕ h in H s x .We conclude

Global well-posedness
In this section we prove global well-posedness for the SQG equation (1.3) with small and localized initial data.We do this by using the wave packet method of Ifrim-Tataru, which is systematically described in [14].
which when frequency localized may be written We partition the frequency space into dyadic intervals I λ localized at dyadic frequencies λ ∈ 2 Z , and consider the associated partition of velocities which form a covering of the real line, and have equal lengths.To these intervals J λ we select reference points v λ ∈ J λ , and consider an associated spatial partition of unity where J λ is a slight enlargement of J λ , of comparable length, uniformly in λ.
Lastly, we consider the related spatial intervals, tJ λ , with reference points x λ = tv λ ∈ tJ λ .
6.2.Overview of the proof.We provide a brief overview of the proof.
1. We make the bootstrap assumption for the pointwise bound (6.1) where C is a large constant, in a time interval t ∈ [0, T ] where T > 1.
2. The energy estimates for (1.3) and the linearized equation will imply 3. We aim to improve the bootstrap estimate (6.1) to (6.3) ϕ(t) Y ǫ t − 1 2 .We use vector field inequalities to derive bounds of the form (6.4) which is the desired bound but with an extra t Cǫ 2 loss.

4.
In order to rectify the extra loss, we use the wave packet testing method define a suitable asymptotic profile γ, which is then shown to be an approximate solution for an ordinary differential equation.This enables us to obtain suitable bounds for the asymptotic profile without the aforementioned loss, which can then be transferred back to the solution ϕ.
6.3.Energy estimates.From Corollary 4.2, and by using the fact that ǫ ≪ 1, which satisfies the linearized equation.From Corollary 4.2, along with Grönwall's lemma and the fact that ǫ ≪ 1, we have Along with the bootstrap assumptions, these readily imply that 6.4.Vector field bounds.Proposition 2.1 from [14] implies that and when λ > 1, By dyadic summation and Bernstein's inequality, we deduce the bound By the localized dispersive estimate [14, Proposition 5.1], To end this section we record the following elliptic bounds: Lemma 6.1.We have Moreover, the difference quotient satisfies the bounds and Proof.We use the bounds From 6.7 applied for The first two bounds immediately follow from 6.7, and the L 2 elliptic estimate similarly follows from [14, Proposition 5.1].
For the bounds involving the difference quotient, from 6.7 applied for δ y ϕ, we have The other bound is proved similarly.
6.5.Wave packets.We construct wave packets as follows.Given the dispersion relation a(ξ), the group velocity v satisfies Then we define the linear wave packet u v associated with velocity v by , where the phase φ is given by φ(v) = vξ v − a(ξ v ), and χ is a unit bump function, such that χ(y) dy = 1.
We remark that we will typically apply frequency localization u v λ = P λ u v with v ∈ J λ .
We observe that since we may write (6.11) and u v,II has a similar wave packet form.We also recall from [14, Lemmas 4.4, 5.10] the sense in which u v is a good approximate solution: Lemma 6.2.The wave packet u v solves an equation of the form The asymptotic profile at frequency λ is meaningful when the associated spatial region tJ λ dominates the wave packet scale at frequency λ: 6.6.Wave packet testing.In this section we establish estimates on the asymptotic profile function x .We will see that γ λ essentially has support v ∈ J λ .
We will also use the following crude bounds involving the higher regularity of γ λ : Lemma 6.3.We have Proof.Using the second form of ∂ v u v in (6.11), we have where the t 1 2 loss in front arises from the L 1 norm of the wave packet.Higher derivatives are obtained similarly, along with the L 2 estimates.
For the last bound, we use the first form of ∂ v u v in (6.11).The contribution from the wave packet u v,II is easily estimated as above.For the remaining bound, Lemma 2.3 from [14] implies that which finishes the proof.
6.6.1.Approximate profile.We recall from [14] that γ λ provides a good approximation for the profile of ϕ.In our setting, we will also need to compare the profile with the differentiated flow ∂ x ϕ.Define Lemma 6.4.Let t ≥ 1.Then we have Proof.The first estimate may be obtained from the proof of [14,Proposition 4.7].For the latter, we use the first representation in (6.11) to write (6.12) To address the first term, we see that we may apply the undifferentiated estimate with ∂ x ϕ λ in place of ϕ λ .Precisely, we may apply the first estimate on We estimate the third term of (6.12) via It remains to estimate the middle term, We also observe that on the wave packet scale, we may replace γ(t, v) with γ(t, x/t) up to acceptable errors.Denote ))e itφ(x/t) , Lemma 6.5.Let v ∈ J λ , and (t, v) ∈ D.Then, for every y = 0 and x such that |x − vt| δx = t 1/2 λ −1/2 , we have the bound Proof.We have δ y β v = −t −1/2 δ y (γ(t, •/t))χ λ ((x+y)/t)e itφ((x+y)/t) +t −1/2 (γ(t, v)−γ(t, x/t))δ y (χ λ (•/t)e itφ(•/t) ).
The Mean Value Theorem ensures that and that Bounds for Q.Write, slightly abusing notation, Lemma 6.6.For 0 < δ ≪ 1, we have the difference estimates where the integrand may be written sgn(y)(|δ The first integral contributes to the two estimates respectively, For the second, using Sobolev embedding, We will be considering separately the balanced and unbalanced components of Q. Precisely, we denote the diagonal set of frequencies by D and write The unbalanced portion of Q satisfies the better bound as follows: Lemma 6.7.Q unbal satisfies the bounds , where χ 1 λ is a cut-off widening χ λ .Proof.We shall denote and consider two cases in the frequency sum for First we consider the case in which we have two low separated frequencies.We assume without loss of generality that λ 3 = λ and λ 1 < λ 2 ≪ λ.In this case, the elliptic estimates will be applied for the factor ϕ λ 1 .Precisely, from Lemma 2.6 and estimates 6.6, 6.7, and 6.10, we get that . By using dyadic summation in λ 1 and λ 2 , we deduce that Similarly, we deduce that We now analyze the situation in which λ 1 , λ 2 λ, and λ 1 and λ 2 are comparable and both separated from λ.Thus, we will be able to use λ 1 and λ 2 interchangeably.We replace χ 1 λ by χλ , which has double support, and equals 1 on a comparably-sized neighbourhood of the support of χ 1 λ .We write χ For the first term, using Lemma 2.6, along with estimates 6.6, 6.7, 6.10, we get the bounds . By using dyadic summation in λ 1 , λ 2 , and λ 3 (and by using the fact that λ 1 and λ 2 are close), we deduce the bound We look at the second term.For every N, we know that . By carrying out a similar analysis as above, along with Lemma 2.6 and dyadic summation, we deduce that the contributions corresponding to these terms are also acceptable.Lemma 6.8.We have where for every a ∈ (0, 1) Proof.We write e −itφ(x/t) δ y e itφ(x/t) = e iyφ ′ (x/t) (e it/2φ ′′ (cx,y/t)y 2 /t 2 − 1) y + e iyφ ′ (x/t) − 1 y =: a + b, where c x,y is between x and x + y.We now use the fact that x/t belongs to the support of χ λ .We have We note that T 1 = (χ λ (x/t)) 3 e itφ(x/t) |b| 2 b dy = (χ λ (x/t)) 3 e itφ(x/t) q(φ ′ (x/t)), so we only need to analyze T 2 .
We first bound the contribution over the region |y| ≤ t a , which we shall denote by T 1 2 .We denote the contribution over the region |y| > t a by T 2 2 .We have 6.8.The asymptotic equation for γ.Here we prove the following: Proposition 6.9.Let v ∈ J λ .Under the assumption (t, v) ∈ D, we have We first analyze I 2 .We use Lemma 6.2 to write (we have used the condition (t, v) ∈ D.) In the remaining part of this section we shall analyze the term I 1 .We first exchange F for its principal quadratic term, expanding From Moser's estimate (the nonlinear version, as well as the one for products), Lemma 2.6, and Sobolev embedding, we get that We first replace ϕ λ by χ λ ϕ λ .From Lemma 6.6, we have By interpolation, along with Lemma 6.1 and the condition (t, v) ∈ D, it follows that the errors are acceptable.The L 2 x -bound is similar.
We now denote ψ(t, x) = t − 1 2 χ λ (x/t)γ(t, x/t)e itφ(x/t) and replace χ λ ϕ λ by ψ.From Lemma 6.6, we have By interpolation, along with Lemmas 6.4 and 6.3, and the condition (t, v) ∈ D, it follows that the errors are acceptable.The L 2 x -bound is similar.We now denote θ(t, x) = t − 1 2 χ λ (x/t)γ(t, v)e itφ(x/t) and replace ψ by θ.We evaluate The support condition of χ implies that x is in the region |x − vt| δx = t 1/2 λ −1/2 .From Lemma 6.5 we now get that From Lemma 6.3 along with the condition (t, v) ∈ D, it follows that this error is acceptable.The L 2 x -bound is similar.

Modified scattering
In this section we discuss the modified scattering behaviour of the global solutions constructed in Section 6.We begin by proving the conservation of mass for the solutions of (1. λ −1/4 t −1/5+C 2 ǫ 2 ǫ.
Proof.This is an immediate consequence of Proposition 6.9.