Hs BOUNDS FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION

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Hs BOUNDS FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION

Hajer Bahouri, Trevor Leslie, Galina Perelman

To cite this version:

Hajer Bahouri, Trevor Leslie, Galina Perelman. Hs BOUNDS FOR THE DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION. 2021. �hal-03300023�

HAJER BAHOURI, TREVOR M. LESLIE, AND GALINA PERELMAN

ABSTRACT. We study the derivative nonlinear Schr¨odinger equation on the real line and obtain global-in- time bounds on high order Sobolev norms.

1. INTRODUCTION

We consider the Cauchy problem for the derivative nonlinear Schr¨odinger equation (DNLS) on the real lineR:

(1)







i∂_tu+∂_x²u=−i∂_x(|u|²u), u

_t=0 =u₀ ∈H^s(R), s≥ ¹₂.

We remark right away that the DNLS isL² critical, as it is invariant under the scaling (2) u(t, x)7→u_µ(t, x) := √

µu(µ²t, µx), µ >0.

The DNLS equation was introduced by Mio-Ogino-Minami-Takeda and Mjølhus [20, 21] as a model for studying magnetohydrodynamics, and it has received a great deal of attention from the mathematics community after being shown to be completely integrable by Kaup-Newell [13]. The infinitely many conserved quantities admitted by the DNLS equation play an important role in the wellposedness theory.

The first three—the mass, momentum, and energy—are as follows.

M(u) :=

Z

R

|u|²dx, (3)

P(u) := Im Z

R

uu_xdx+1 2

Z

R

|u|⁴dx, (4)

E(u) :=

Z

R

|u_x|²− 3

2Im(|u|²uu_x) + 1 2|u|⁶

dx.

(5)

Before stating our main result, let us give a very brief review of what is known about the wellposedness of the DNLS equation. More detailed overviews can be found, for example, in the introductions of [2]

and [14]. Local wellposedness in H^s(R) for s ≥ ¹₂ was proven by Takaoka [25], improving earlier work [22] by Ozawa. On the other hand, fors < ¹₂, the uniform continuity of the data-to-solution map fails in H^s(R) [3, 26]. One can, however, close the ¹₂-derivative gap between the H¹² threshold and the critical space L²(R) by working in more general Fourier-Lebesgue spaces, c.f. Gr¨unrock [6] and references therein.

Date: July 22, 2021.

This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the authors participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2021 semester.

1

A line of results, due to Hayashi-Ozawa [8], Colliander-Keel-Staffilani-Takaoka-Tao [4], Wu [28], and Guo-Wu [7], establishes global well-posedness of the DNLS equation in H^s(R)fors ≥ ¹₂, for ini- tial data having mass less than4π. Another line (Pelinovsky-Saalmann-Shimabukuro [23], Pelinovsky- Shimabukuro [24], and Jenkins-Liu-Perry-Sulem [12,11, 10]) uses inverse scattering techniques to es- tablish global wellposedness under stronger regularity and decay assumptions on the initial data, but without a smallness requirement on the mass.

The first and third authors proved in [2] that the DNLS equation is globally well-posed in H^s(R) fors≥ ¹₂ and that solutions generated fromH¹² initial data remain bounded inH¹²(R)for all time. There have also been some recent works below the aforementioneds = ¹₂ threshold of uniformH^s continuity with respect to initial data [3, 26]. Klaus-Schippa [17] gave H^s a priori estimates for 0 < s < ¹₂ in the case of small mass, Killip-Ntekoume-Vis¸an [14] improved the small mass assumption to 4π and furthermore proved a global wellposedness result in H^s(R), ¹₆ ≤ s < ¹₂, for initial data with mass less than4π. Very recently, Harrop-Griffiths, Killip, and Vis¸an [9] have removed the small mass assumption both from theirH^sa priori bounds,0< s < ¹₂, as well as from their global wellposedness result inH^s(R) with ¹₆ ≤s < ¹₂.

In this paper, we are concerned with the global-in-time boundedness of solutions to the DNLS equation inH^sspaces. We prove that a uniform-in-time bound inH^s(R)holds for alls≥ ¹₂.

Theorem 1.1. Supposeuis a solution to the DNLS equation with initial datau₀ ∈H^s(R),s ≥ ¹₂. There exists a finite positive constantC =C(s,ku0k_H^s_(R)), such that

sup

t∈R

ku(t)k_H^s_(R) ≤C(s,ku₀k_H^s_(R)).

The main idea is to build off of theH^sbounds with0< s < ¹₂ from [9] and to take advantage of the complete integrability of the equation. As in [2], [9], the present work relies heavily on the conservation of the transmission coefficient for the spectral problem associated to the DNLS equation. This property has already been used in many other works; of particular relevance to us are the papers of G´erard [5], Killip-Vis¸an-Zhang [16], Killip-Vis¸an [15], and Koch-Tataru [18], on the cubic NLS and KdV equations.

Note that by continuity of the flow, and the preservation of the Schwartz class under the flow, we lose nothing by restricting attention to the Schwartz class; we will thus work exclusively with Schwartz functions for the remainder of the manuscript. We will also suppress the time dependence when it does not play a role.

One can easily prove Theorem 1.1 in the special case s = 1, using the conserved quantity E(u).

Indeed, simply rearranging (5) yields

kuk²_H˙1(R)=E(u)−1

2kuk⁶_L6(R)+3 2

Z

R

Im(|u|²uu_x) dx.

Clearly, the last term can be bounded above in absolute value by ¹₂kuk²_H˙1(

R) +Ckuk⁶_L6(R), whence the desired bound follows by Sobolev embedding.

The higher-order Sobolev norms of integer order can be dealt with similarly, once we have a formula for the corresponding higher-order conserved quantities. We will show that for any nonnegative integer`, one of the conserved quantities is equal to a constant multiple ofkuk<sup>2</sup><sub>H˙`(</sub>

R), plus terms which are of lower order. For nonintegers, we will use a sort of ‘generalized energy’, comparable tokuk²_H˙s(

R), that will be defined in terms of the transmission coefficient of the DNLS spectral problem. We sketch presently the background necessary to define these objects precisely; for more details, see, for example, [1,12,11,10, 13,19,24,27].

The DNLS equation can be obtained as a compatibility condition of the following system [13]:

∂_xψ =U(λ)ψ,

∂_tψ = Υ(λ)ψ.

(6)

Here λ ∈ C is a spectral parameter, independent of t and x, and ψ = ψ(t, x, λ) is C²-valued. The operatorsU(λ)andΥ(λ)are defined by

U(λ) = −iσ₃(λ²+iλU), Υ(λ) = −i(2λ⁴−λ²|u|²)σ3+

0 2λ³u−λ|u|²u+iλu_x

−2λ³u+λ|u|²u+iλu_x 0

(7) ,

where

σ3 =

1 0 0 −1

, U =

0 u u 0

.

To be more specific about the sense in which the DNLS is a compatibility condition, we note that u satisfies the DNLS equation if and only ifU andΥsatisfy the so-called ‘zero-curvature’ representation

∂U

∂t − ∂Υ

∂x + [U,Υ] = 0.

The first equation of (6) can be written in the form

(8) L_u(λ)ψ := (iσ₃∂_x−λ²−iλU)ψ = 0, which defines the scattering transform associated to the DNLS. Let us denote

Ω₊:={λ∈C: Imλ² >0}.

Then givenu∈ S(R)andλ∈Ω₊, there are unique solutions to (8) (the “J¨ost solutions”) exhibiting the following behavior at±∞:

ψ⁻₁(x, λ) =e^−iλ2x 1

0

+o(1)

, asx→ −∞, ψ₂⁺(x, λ) =e^iλ2x

0 1

+o(1)

, asx→+∞.

(9)

Finally, we denote bya_u(λ)the Wronskian of the J¨ost solutions defined above:¹ (10) a_u(λ) = det(ψ₁⁻(x, λ), ψ⁺₂(x, λ)).

Using the second equation in (6), it can be shown that a_u(λ) is time-independent if u is a solution of (1). Furthermore, a_u is a holomorphic function of λ in Ω₊, and one may determine the behavior ofa_u at infinity by transforming (8) into a Zakharov-Shabat spectral problem, linear with respect to the spectral parameter, c.f. [13], [23]. The equivalence between the two problems allows us to write

(11) lim

|λ|→∞, λ∈Ω₊

a_u(λ) =e⁻

i 2kuk²

L2(R). For fixedu, we can thus define the logarithm so that

(12) lim

|λ|→∞,λ∈Ω+

lnau(λ) =−i

2kuk²_L2(R). Moreover,lna_u(λ)admits an asymptotic expansion of the following form:

(13) lna_u(λ) =

∞

X

j=0

E_j(u)

λ^2j as|λ| → ∞, λ∈Ω₊.

Since a_u(λ) is time-independent, the quantities E_j(u) are conservation laws. They are all polynomial in u, u, and their derivatives. Furthermore, the E_j(u)’s inherit scaling properties froma_u(λ). That is, for µ > 0, the fact thata_uµ(λ) = a_u(^√λ_µ)implies that E_j(u_µ) = µ^jE_j(u), for each j ∈ N. The first

1Thetransmission coefficientmentioned earlier is the inverse ofa_u(λ).

several of the E_j(u)’s are (up to multiplicative constants) the conserved quantities (3)–(5) mentioned earlier:

E₀(u) = −i

2kuk²_L2(R)=−i

2M(u), E₁(u) = i

4P(u), E₂(u) = −i 8E(u).

For each ` ∈ N<sup>∗</sup>, the quantity E<sub>2`</sub>(u) can be used to control kuk²_H˙`(

R). Let us define, for ρ positive sufficiently large andL∈N,

(14) ϕ_L(u, ρ) = Im

"

lna_u(p iρ)−

2L+1

X

j=0

E_j(u) (iρ)^j

# .

If u is a solution of the DNLS equation, then ϕ_L(u, ρ) is time-independent, being a sum of time- independent quantities.

In order to establish bounds on the H^snorm ofu, fors ≥ ¹₂, we will show thatR∞

R ρ^2s−1ϕ_[s](u, ρ)dρ withR > 0large enough controls theH˙^sseminorm ofu, in a sense to be made precise later. Here and below, we use[s]to denote the integer part of a real numbers.

Our proof of Theorem1.1relies on a good understanding of the structure of the remainder associated to the expansion (13). Note that whenλ² = iρ, the imaginary part of this remainder (which is what we really use) is simplyϕL(u, ρ). In Section2, we will introduce a determinant characterization ofau(λ);

we use this characterization to formulate a technical statement (Lemma 2.1 below) on the size of the remainder. Assuming the result of Lemma2.1, we will prove Theorem1.1at the end of Section2. Then, in Section 3, we will prove our technical Lemma, completing the circle of ideas. Most of the work is contained in this last section.

Before moving on, let us establish a few notational conventions that we wish to add to the ones intro- duced above. First of all, we use the following normalization for the Fourier transform:

f(ζ) =b 1

√2π Z

R

e^ixζf(x) dx.

The symbolNwill denote the nonnegative integers, andN^∗ = N\{0}. We will use k · k₂ to denote the Hilbert-Schmidt norm, andk · kwill denote the operator norm onL²(R). And we will use the following shorthand for derivatives:

D=−i∂_x, L₀ =iσ₃∂_x.

Whenever 2 ≤ p < ∞, we will uses^∗(p)to denote the Sobolev exponents^∗(p) = ¹₂ − ¹_p such that the embeddingH^s∗(p)(R),→L^p(R)holds.

Finally, we set notation for the following subset ofΩ₊:

Γ_δ={λ∈Ω₊:δ <arg(λ²)< π−δ}.

This notation will be useful in some of the intermediate steps we use to prove Theorem 1.1, as our estimates will frequently depend on _Im^|λ|_λ²2 (which is≤C(δ)onΓ_δ). However, the value ofδ > 0will be inconsequential for our final steps, where we will takeλ²to be pure imaginary. Therefore, for simplicity of presentation, we willfixδ >0once and for all and suppress dependence onδin all bounds below.

2. PROOF OF THE MAINRESULT

2.1. The Determinant Characterization of a_u(λ). An important property of a_u(λ) is the fact that it can be realized as a perturbation determinant:

(15) a_u(λ)² = det(I −T_u(λ)²),

where

T_u(λ) =iλ(L₀−λ²)⁻¹U, λ ∈Ω₊.

The operatorT_u(λ)is Hilbert-Schmidt, with

(16) kT_u(λ)k²₂ = |λ|²

Im(λ²)kuk²_L2(R). As a consequence of (15), we may write²

(17) lna_u(λ) =−

∞

X

k=1

Tr(T_u(λ)^2k)

2k , ifkT_u(λ)k<1.

This series will converge wheneverλ ∈ Γ_δ has large enough modulus; indeed, using the explicit kernel of(L₀−λ²)⁻¹, it can easily be shown that for anyp > 2, we have

(18) kT_u(λ)k. |λ|kuk_L^p_(R) Im(λ²)^1−1p

, λ∈Ω₊, u∈L^p(R).

In particular, we can findR₀ =R₀(kuk

H¹³(R))such thatkT_u(λ)k ≤ ¹₂ for allλ∈Γ_δsatisfying|λ|² ≥R₀. We will fix the notationR₀ for use below.

As we shall see later, each term of the series (17) can be expanded in powers ofλ⁻²:

(19) − Tr(T_u(λ)^2k)

2k =

∞

X

j=k−1

µ_j,k(u) λ^2j . According to (13) and (17), theEj(u)’s should then satisfy

(20) E_j(u) =

j+1

X

k=1

µ_j,k(u).

We will use the following notation for the remainders after truncation of the expansions (17) and (19):

(21) lna_u(λ) =−

2L+2

X

k=1

Tr(T_u(λ)^2k)

2k +τ_L^∗(u, λ), L∈N;

(22) −Tr(T_u(λ)^2k)

2k =

2L+1

X

j=k−1

µ_j,k(u)

λ^2j +τ_L^k(u, λ), k ∈ {1, . . . ,2L+ 2}, L∈N.

The primary difficulty of the proof of Theorem1.1—and indeed, the subject of Lemma2.1—is the un- derstanding of the size and structure of the remainder termsτ_L^k(u, λ), and to a lesser extent, theµ_j,k(u)’s.

On the other hand, forλ∈Γ_δwith large enough modulus, it is easy to bound theτ_L^∗(u, λ)’s. For example, ifkT_u(λ)k ≤ ¹₂, then

|τ_L^∗(u, λ)|=

lna_u(λ) +

2L+2

X

k=1

TrT_u^2k(λ) 2k

≤

∞

X

k=2L+3

kT_u(λ)k^2k−2kT_u(λ)k²₂

.kTu(λ)k^4L+4kTu(λ)k²₂ .

kuk^4L+4_Hs∗(p)(R)kuk²_L2(R)

|λ|^{(4L+4)(1−2p)} , 2< p <∞, s^∗(p) = 1 2 −1

p. (23)

The following table summarizes the various relationships among the quantities introduced above and will be helpful to keep track of the numerology. More precise information about the µj,k(u)’s andτ_L^k(u, λ)’s will be provided below.

2This series expansion oflna_u(λ)is consistent with the definition (12).

−TrT_u²(λ)

2 −TrT_u⁴(λ)

4 −TrT_u⁶(λ)

6 · · · −TrTu^4L+2(λ)

4L+ 2 −TrTu^4L+4(λ) 4L+ 4

lnau(λ) = µ0,1(u) E0(u)

+ µ1,1(u)

λ² + µ1,2(u) λ²

E1(u) λ² + µ2,1(u)

λ⁴ + µ2,2(u)

λ⁴ + µ2,3(u) λ⁴

E2(u) λ⁴ ..

.

.. .

..

. . .. ...

+ µ2L,1(u)

λ^4L + µ2L,2(u)

λ^4L + µ2L,3(u)

λ^4L + · · · + µ2L,2L+1(u) λ^4L

E_2L(u) λ^4L + µ2L+1,1(u)

λ^4L+2 + µ2L+1,2(u)

λ^4L+2 + µ2L+1,3(u)

λ^4L+2 + · · · + µ2L+1,2L+1(u)

λ^4L+2 + µ2L+1,2L+2(u) λ^4L+2

E2L+1(u) λ^4L+2

+ τ_L¹(u, λ) + τ_L²(u, λ) + τ_L³(u, λ) + · · · + τ_L^2L+1(u, λ) + τ_L^2L+2(u, λ) + τ_L^∗(u, λ)

2.2. Structure of the Traces. In this section, we record all the information about the traces that we need in order to prove our main result. We deal first with the easy case of TrT_u(λ)², about which we need more explicit information. A straightforward computation gives us

(24) TrT_u²(λ) = 2iλ²

Z

R

|bu(ζ)|² ζ+ 2λ²dζ.

We determine the expansion ofTrT_u²(λ)by simply substituting into (24) the identity 2λ²

ζ+ 2λ² =

2L+1

X

j=0

− ζ 2λ²

j

+ ζ

ζ+ 2λ² ζ

2λ² 2L+1

, L∈N,

to obtain

(25) − TrT_u²(λ)

2 =

2L+1

X

j=0

1

λ^2j · i (−2)^j+1

Z

R

ζ^j|u(ζ)|b ²dζ

| {z }

=:µj,1(u)

− i

4^L+1λ^4L+2 Z

R

ζ^2L+2|u(ζ)|b ² ζ+ 2λ² dζ

| {z }

=:τ_L¹(u,λ)

, L∈N.

Now we state our main Lemma, which describes the structure of the otherµ_j,k(u)’s andτ_L^k(u, λ)’s.

Lemma 2.1. For anyk ∈N^∗, L∈N, the traces TrT_u^2k(λ)admit the decomposition(22). Theµ_j,k(u)’s andτ_L^k(u, λ)’s satisfy the properties below, where for anyn∈Nwe denoteσ(n) = max{n,¹₃}.

• Each µ_j,k(u) is a homogeneous polynomial of degree 2k in u, u, and their derivatives; it is homogeneous with respect to the natural scaling. We have

|µ<sub>2`,2</sub>(u)|.kuk<sup>3</sup><sub>H</sub>σ(`−1)(R)kuk_H<sup>`</sup><sub>(R)</sub>, `∈N^∗,

|µ<sub>2`,k</sub>(u)|.kuk<sup>2k</sup><sub>H</sub>σ(`−1)(R), `∈N<sup>∗</sup>, k ∈ {3, . . . ,2`+ 1},

|µ<sub>2`+1,k</sub>(u)|.kuk<sup>2k</sup><sub>H</sub>`(R), `∈N<sup>∗</sup>, k ∈ {2, . . . ,2`+ 2}.

(26)

• For|λ|² > R₀,λ∈Γ_δ, we have the following bounds:

|τ_L²(u, λ)|.α

kuk³_Hσ(L)(

R)kuk_HL+α(R)

|λ|^4L+2+2α , L∈N, 0≤α <1;

(27)

|τ_L^k(u, λ)|. kuk^2k_HL(R)

|λ|^4L+4 , L∈N^∗, k∈ {3, . . . ,2L+ 2}.

(28)

We postpone the proof of the Lemma until Section3.

2.3. Proof of Theorem1.1. In this section, we will prove Theorem1.1, assuming the result of Lemma2.1.

Fors ∈ N^∗, the conclusion follows easily from Lemma 2.1, together with (20), (25), and an induction argument; we provide the details presently. Actually, the case s = 1 was already proved in the Intro- duction. Therefore, let us turn to our inductive hypothesis. For k = 1, . . . , `−1, we assume that the following bound holds.

(29) sup

t∈R

ku(t)k_H^k_(R) ≤C(k,ku₀k_H^k_(R)).

We will prove that the same bound holds withk=`≥2.

First of all, for any integer`≥2, and any timet, we have ku(t)k<sup>2</sup><sub>H˙`(</sub>

R) =C(`)µ<sub>2`,1</sub>(u(t)) by (25)

=C(`)

"

E_2`(u(t))−

2`+1

X

k=2

µ_2`,k(u(t))

#

by (20)

≤C(`)E<sub>2`</sub>(u₀) + 1

2ku(t)k²_H˙`(

R)+C(`,ku<sub>0</sub>k<sub>H</sub><sup>`−1</sup>_(R)).

To pass to the last line, we used time-independence of E<sub>2`</sub>(u(t)), the bounds (26), and our inductive hypothesis (29) (withk =`−1). Finally, using that

E_2`(u₀) =

2`+1

X

k=1

µ<sub>2`,k</sub>(u<sub>0</sub>)≤C(`,ku₀k_H^`_(R)), we get

sup

t∈R

ku(t)kH˙<sup>`</sup>(R)≤C(`,ku₀k_H^`_(R)),

which finishes the induction argument, and thus the proof of Theorem1.1fors ∈N^∗.

It remains to consider the situation wheres /∈N^∗. We start by recording the characterization ofϕL(u, ρ) in terms of the remainders τ_L^k(u,√

iρ), and we also set notation for the quadratic part ofϕ_L(u, ρ). We also note that the caseL= 0is included in the definition.

ϕ_L(u, ρ) = Im

"

lna_u(p iρ)−

2L+1

X

j=0

Ej(u) (iρ)^j

#

= Im

"_2L+2 X

k=1

τ_L^k(u,p

iρ) +τ_L^∗(u,p iρ)

#

, L∈N, (30)

ϕ_L,0(u, ρ) = Imτ_L¹(u,p

iρ) = (−1)^L 2^2L+1ρ^2L

Z

R

ζ^2L+2|bu(ζ)|²

ζ²+ 4ρ² dζ, L∈N.

(31)

The conclusion of Theorem1.1for nonintegers≥ ¹₂ will be deduced from the following two Lemmas.

Lemma 2.2. Supposeu∈ S(R),s >0,s /∈N^∗, andR >0. Then the following comparison holds.

(32)

Z

R+

ρ^2s−1|ϕ_[s],0(u, ρ)|dρ.skuk²_H˙s(R) .s

Z ∞ R

ρ^2s−1|ϕ_[s],0(u, ρ)|dρ+R^2(s−[s])kuk²_H˙[s](R).

Proof. Let us define the functionfν :R →R, for0< ν < 1, byfν(z) = ^|z|_1+z^2ν−12 . Note thatfν ∈L¹(R) for this range ofν.

We make a direct substitution of the formula (31) for ϕ_[s],0(u, ρ) into the left side of (32), then we switch the order of integration. Continuing the computation yields

Z ∞ R

ρ^2s−1|ϕ_[s],0(u, ρ)|dρ= 1 2^2[s]+1

Z

R

ζ^2[s]+2|u(ζ)|b ² Z ∞

R

ρ^{2(s−[s])−1} ζ²+ 4ρ² dρdζ

= 1

2^2s+1 Z

R

|ζ|^2s|bu(ζ)|² Z ∞

2R

|ζ|

f_s−[s](z) dzdζ

= 1

4^s+1kfs−[s]k_L¹_(R)kuk²_H˙s(R)− 1 2

Z

R

ζ 2

2s

|bu(ζ)|² Z ^2R_|ζ|

0

fs−[s](z) dzdζ.

We estimate the second term on the right by means of the trivial replacement _1+z¹ 2 ≤1:

1 2

Z

R

ζ 2

2s

|bu(ζ)|² Z ^2R_|ζ|

0

fs−[s](z) dzdζ ≤ 1 2

Z

R

ζ 2

2s

|bu(ζ)|² Z ^2R_|ζ|

0

z^{2(s−[s])−1}dzdζ = R^2(s−[s])

s−[s] · kuk²_H˙[s](

R)

4^[s]+1 .

The comparison (32) follows.

Lemma 2.3. Supposeu∈ S(R),s >0,s /∈N^∗. Denotingβ = max{[s], ^s+[s]+1_4([s]+1), ¹₃}, we have

(33) |ϕ_[s](u, ρ)−ϕ_[s],0(u, ρ)| ≤ C(s,kuk_H^β_(R))

ρ^s+[s]+1 (kuk_H^s_(R)+ 1), ∀ρ≥R₀.

Proof. Choosep > 2to solve2([s] + 1)(1− ²_p) =s+ [s] + 1. (Note thats^∗(p) = ^s+[s]+1_4([s]+1) for this choice ofp.) Then forρ > R₀, we have

|ϕ_[s](u, ρ)−ϕ_[s],0(u, ρ)| ≤

2[s]+2

X

k=2

|τ_[s]^k(u,p

iρ)|+|τ_[s]^∗ (u,p

iρ)| by (30),(31)

≤C(s)

kuk³_Hβ(R)kuk_H^s_(R) ρ^s+[s]+1 +

2[s]+2

X

k=3

kuk^2k_H[s](R)

ρ^2[s]+2 +

kuk^4[s]+4_Hs∗(p)(R)kuk²_L2(R)

ρ(2[s]+2)(1−_p²)

by (27),(28),(23)

≤ C(s,kuk_H^β_(R))

ρ^s+[s]+1 (kuk_H^s_(R)+ 1).

In the second line, we understand the sum overkto be empty if[s] = 0.

The conclusion of Theorem 1.1 for noninteger s ≥ ¹₂ follows from Lemmas2.2 and 2.3, the time- independence of the quantityϕ_[s](u, ρ)for solutions of the DNLS equation, and the bound

(34) sup

t∈R

ku(t)k_H^β_(R) ≤C(β,ku₀k_H^β_(R)),

whereβ = max{[s],_4([s]+1)^s+[s]+1,¹₃}is as in the statement of Lemma2.3. The bound (34) follows from our induction argument ifs >1and from the result of Harrop-Griffiths, Killip, and Vis¸an [9] if ¹₂ ≤s <1.

Let us give the remaining details of the proof of Theorem1.1presently. For anyt ∈R, we have ku(t)k²_H˙s(R).s

Z ∞ R0

ρ^2s−1|ϕ_[s],0(u(t), ρ)|dρ+R^2(s−[s])₀ ku(t)k²_H[s](R)

≤C(s) Z ∞

R0

ρ^2s−1|ϕ_[s](u(t), ρ)|dρ+C(s, R₀,ku(t)k_Hβ(R))(ku(t)k_H^s_(R)+ 1)

≤C(s) Z ∞

R0

ρ^2s−1|ϕ_[s](u₀, ρ)|dρ+C(s,ku₀k_H^s_(R))(ku(t)k_H^s_(R)+ 1)

≤ 1

2ku(t)k²_Hs(R)+C(s,ku₀k_H^s_(R)),

which establishes the desired conclusion. Note that the first line in the calculation above is simply the upper bound in Lemma2.2. To pass from the first line to the second, we use Lemma2.3, followed by the lower bound of Lemma2.2. We use (34) and the time independence ofϕ_[s](u(t), ρ)to pass to the third line. Finally, we justify the last line by noting that

Z ∞ R0

ρ^2s−1|ϕ[s](u0, ρ)|dρ.^sC(s,ku0kH^s(R)),

which follows from an application of Lemma2.3, followed by the lower bound in Lemma2.2.

3. PROOF OFLEMMA 2.1

3.1. Outline of the Proof. In this section, we expand eachTr(T_u^2k(λ))in powers ofλ⁻², up to a spec- ified order, and we establish bounds on the remainders, in order to prove our key Lemma 2.1. In Sec- tion3.2, we consider the caseL = 0, which is easy to treat explicitly but does not fit naturally into our argument for the other cases. WhenL≥1, we follow the strategy of [5], deducing the expansions of the traces from the expansion of the resolvent L_u(λ)⁻¹. The relationship between T_u(λ) andL_u(λ)is the following:

(35) L_u(λ) = (L₀−λ²)(I−T_u(λ)).

Therefore,

(36) L_u(λ)⁻¹ = (I −T_u(λ))⁻¹(L₀−λ²)⁻¹ =

∞

X

n=0

T_u(λ)ⁿ(L₀ −λ²)⁻¹

| {z }

=:Rn

, kT_u(λ)k<1.

The point is that

(37) T_u^2k(λ) = iλR2k−1U.

Thus, the part of L⁻¹_u (λ)that is of relevance to us is R2k−1, i.e., the term in the expansion (36) that is homogeneous of degree2k−1inu, u. In particular, we seek an expansion ofλR2k−1in powers ofλ⁻², up to orderλ^4L+2for a givenL∈N^∗, and a good understanding of the remainder term.

Our strategy will be to examine the symbolR(x, ζ)of the pseudodifferential operatorL_u(λ)⁻¹. In Sec- tion3.3, we will expand the diagonal and antidiagonal partsR^d(x, ζ)andR^a(x, ζ)ofR(x, ζ)in powers of λ⁻², determining recursively the form of each term of the expansion. Homogeneity considerations will then give us the desired expansion of λR_2k−1 (and thus of TrT_u^2k(λ)) in powers of λ⁻². In Sec- tion3.4, we identify theµ_j,k(u)’s from (22) and separate them from the remainder term. In Section3.5 we estimate the remainder term, finishing the proof of the Lemma. The final Section3.6consists of the proof by induction of a technical result stated in Section3.3.1, on the form of the terms of the expansions forR^dandR^a.

3.2. CaseL= 0. Let us note first of all that the desired decomposition in the caseL= 0reads lna_u(λ) = [µ_0,1(u) +λ⁻²µ_1,1(u) +τ₀¹(u, λ)

| {z }

=−¹₂TrT_u²(λ)

] + [λ⁻²µ_1,2(u) +τ₀²(u, λ)

| {z }

=−¹₄TrT_u⁴(λ)

] +τ₀^∗(u, λ).

(See the table in Section2.1.) The only term which we have not already understood isτ₀²(u, λ); in order to treat it, we decomposeT_u⁴(λ)explicitly as follows. A computation (the details of which are contained, for instance, in [2]) tells us that

TrT_u⁴(λ) =i(2λ²)² Z

R

u(x) (D+ 2λ²)⁻¹u(x)2

(D−2λ²)⁻¹u(x) dx.

Then, making a few simple manipulations, we can bring the right side of the equation above into the following form.

TrT_u⁴(λ) = i

−2λ² Z

R

u(x)

u(x)−(D+ 2λ²)⁻¹Du(x) 2

u(x)−(D−2λ²)⁻¹Du(x) dx

=− i 2λ²

Z

R

|u(x)|⁴dx− Z

R

|u|²u(x)(D−2λ²)⁻¹Du(x) dx−2 Z

R

|u|²u(x)(D+ 2λ²)⁻¹Du(x) dx

+ 2 Z

R

|u(x)|²(D+ 2λ²)⁻¹Du(x)(D−2λ²)⁻¹Du(x) dx+ Z

R

((D+ 2λ²)⁻¹Du(x))²u(x)²dx

− Z

R

u(x)((D+ 2λ²)⁻¹Du(x))²((D−2λ²)⁻¹Du(x)) dx

=− 4 λ²

i

8kuk⁴_L4(R)

| {z }

=µ1,2(u)

−4τ₀²(u, λ).

To estimateτ₀²(u, λ), we use the following simple Lemma, the proof of which we omit.

Lemma 3.1. The following estimates hold, forλ∈Γ_δ.

• If0≤α₁ ≤α₂ ≤1, then

(38)

(D±2λ²)⁻¹Du _˙

H^α1(R) .^α2−α1

kukH˙^α2(R)

(2 Im(λ²))^α2−α1, ∀u∈H^α2(R).

• If2≤p <∞, then

(39)

(D±2λ²)⁻¹Du

L^p(R) .^p kuk_H^s∗(p)_(R), ∀u∈H^s∗(p)(R).

We estimate one of the terms defining τ₀²(u, λ)explicitly; the others can be dealt with in an entirely similar way.

1 λ²

Z

R

u(x)((D+ 2λ²)⁻¹Du(x))²((D−2λ²)⁻¹Du(x)) dx

≤ 1

|λ|²kuk_L⁶_(R)k(D+ 2λ²)⁻¹Duk²_L6(R)k(D−2λ²)⁻¹Duk_L²_(R) .α

kuk³

H¹³(R)kuk_H^α_(R)

|λ|^2+2α . We conclude thatτ₀²(u, λ)satisfies the required bound, finishing the caseL= 0.

3.3. Expanding the Resolvent.

3.3.1. Formal Expansion ofR^aandR^d. As stated above, forL≥1we seek an expansion of the symbol ofL⁻¹_u (λ), in powers ofλ⁻². That is, we seek to understandR(x, ζ)in the expression

(40) L⁻¹_u (λ)f = 1

√2π Z

dζe^ixζR(x, ζ)fb(ζ).

The identityL_u(λ)R(x, D) = Iimplies

(41) iσ₃∂_xR(x, ζ)−(ζσ₃+λ²)R(x, ζ)−iλU(x)R(x, ζ) =I.

Introducing the new variablep= _λ^ζ2, this reads

(42) iσ3∂xR(x, ζ)−λ²(pσ3+ 1)R(x, ζ)−iλU(x)R(x, ζ) =I.

We splitRinto its diagonal and antidiagonal partsR^dandR^a, respectively, R(x, ζ) =R^d(x, ζ) +R^a(x, ζ),

and we also split equation (42) accordingly:

(43) iσ₃∂_xR^d(x, ζ)−λ²(pσ₃+ 1)R^d(x, ζ)−iλU(x)R^a(x, ζ) =I; (44) iσ₃∂_xR^a(x, ζ)−λ²(pσ₃+ 1)R^a(x, ζ)−iλU(x)R^d(x, ζ) = 0.

Setting the notation

R^d(x, ζ) =X

k≥0

1

λ^2+2kR^d_k(x, p), R^a(x, ζ) =X

k≥0

1

λ^3+2kR^a_k(x, p), we rewrite (43) and (44) in expanded form:

I =−(pσ₃+ 1)R^d₀+

∞

X

k=1

iσ₃∂_xR^d_k−1−(pσ₃+ 1)R_k^d−iU R^a_k−1

λ^2k ;

(45)

0 = −(pσ₃+ 1)R^a₀−iU R^d₀+

∞

X

k=1

iσ₃∂_xR^a_k−1−(pσ₃+ 1)R^a_k−iU R^d_k

λ^2k .

(46)

We thus obtain the recursive system (47)–(49) below.

(47) R^d₀(x, p) =−pσ₃−1

p²−1 , R^a₀(x, p) =− iU p²−1, (48) R_k^d(x, p) = 1

p²−1

−iU R^a_k−1(x, p) +i∂_xR^d_k−1(x, p)σ₃

(pσ₃−1), k≥1,

R_k^a(x, p) = 1 p²−1

iU R^d_k(x, p) +i∂_xR^a_k−1(x, p)σ₃

(pσ₃ + 1)

= 1

p²−1

U²R^a_k−1(x, p)−U ∂_xR_k−1^d (x, p)σ₃+i∂_xR^a_k−1(x, p)σ₃(pσ₃+ 1) ,

k ≥1.

(49)

Note that we used the formula for R^d_k(x, p)to pass to the second line in the formula for R^a_k(x, p). We also used several times the fact thatσ₃A=−Aσ₃ for any antidiagonal matrix.

We use the computations above to clarify the form of the R^d_k’s and R^a_k’s; the precise statement is contained in the following Lemma.

Lemma 3.2. TheR^d_k’s andR^a_k’s take the following form:

(50) R_k^d(x, p) =

k

X

r=1

R^d_k,r(x, p), k≥1,

(51) R_k^a(x, p) =

k

X

r=0

R^a_k,r(x, p), k≥0,

where the entries of the R^d_k,r’s and R_k,r^a ’s are homogeneous polynomials of degrees 2r and 2r + 1, respectively, inu, u, and their derivatives. More specifically, settingQ_γ = ∂_x^γ1U· · ·∂_x^γnU, forγ ∈ Nⁿ, we have

R^d_k,r(x, p) = 1 (p²−1)^k+1

X

γ∈N^2r

|γ|=k−r

Q_γ(x)P_|γ|(p)(pσ₃−1), (52)

R^a_k,r(x, p) = 1 (p²−1)^k+1

X

γ∈N^2r+1

|γ|=k−r

Q_γ(x)P|γ|(p).

(53)

Here and below we use the notationP_nto denote any diagonal matrix whose diagonal entries are poly- nomials inphaving degree at mostn.

We postpone the proof of this Lemma until Section3.6, so as not to interrupt the flow of ideas.

3.3.2. The Truncated Expansion, and a Formula forR2m−1. For a fixedN ∈ N^∗, we set the following notation. (Later we will setN = 2L.)

R^(N)(x, p) =

N

X

k=0

R^d_k(x, p) λ^2+2k

| {z }

=:R^(N)_d (x,p)

+

N−1

X

k=0

R^a_k(x, p) λ^3+2k

| {z }

=:R^(N)a (x,p)

= R^d₀(x, p) λ²

| {z }

=:R^(N)_d,0(x,p)

+

N

X

r=1 N

X

k=r

R^d_k,r(x, p) λ^2+2k

| {z }

=:R^(N)_d,r(x,p)

+

N−1

X

r=0 N−1

X

k=r

R^a_k,r(x, p) λ^3+2k

| {z }

=:R^(N)a,r(x,p)

. (54)

The symbol R^(N)(x, p)is a truncated expansion of R(x, p) in inverse powers of λ, having diagonal and antidiagonal partsR^(N)_d ,R^(N)a , respectively. The point of this definition is that, using Lemma3.2, we know thatR^(N)_d,r is homogeneous of degree2rinu,u, and their derivatives, whileR^(Na,r⁾is homogeneous of degree 2r + 1 in these quantities. Expanding R^(N) according to (54) and applying the recursive identities (45)–(46), we see thatR^(N)(x, p)satisfies

[iσ₃∂_x−λ²(pσ₃+ 1)−iλU(x)]R^(N)(x, p) =I+Y^(N)(x, p), whereY^(N)(x, p) = Y_d^(N)(x, p) +Ya^(N)(x, p),

Y_d^(N)(x, p) = 1

λ^2+2Niσ3(∂xR^d_N)(x, p), Y_a^(N)(x, p) =− 1

λ^1+2NR^a_N(x, p)(pσ3−1).

This implies

(55) L⁻¹_u (λ) = R^(N)(x, λ⁻²D)−L⁻¹_u (λ)Y^(N)(x, λ⁻²D).

Recall thatR2m−1is the term in the expansion (36) which is homogeneous of order2m−1inu,u, and their derivatives. On the other hand, the portion ofR^(N)which is of this homogeneity is preciselyR^(N)_a,m−1. Combining these considerations with (55), we see thatR2m−1is the difference betweenR^(N)_a,m−1 and the part ofL⁻¹_u (λ)Y^(N)(x, λ⁻²D)that is homogeneous of degree2m−1inu,u, and their derivatives. Using the expansion (36) to isolate this part, we obtain:

R2m−1 =R_a,m−1^(N) (x, λ⁻²D)− 1 λ^2+2N

X

k+r⁰=m−1 k≥0,1≤r⁰≤N

T_u(λ)^2k+1(L₀−λ²)⁻¹(L₀R^d_N,r0)(x, λ⁻²D)

− 1 λ^1+2N

X

k+r⁰=m−1 k≥0,0≤r⁰≤N

Tu(λ)^2k(L0−λ²)⁻¹R^a_N,r⁰(x, λ⁻²D)(λ⁻²L0+ 1).

(56)

3.4. Extracting theµ_j,m(u)’s. Combining (56) with (37), (54), and pulling out inverse powers ofλ, we easily find the following formula forTrT_u^2m(λ)withm≥2, truncated atN = 2L.

Tr(T_u^2m(λ)) =

2L−1

X

j=m−1

1

λ^2j+2 Tr[iU R^a_j,m−1(x, λ⁻²D)]

+ X

k+r=m−1 k≥0,1≤r≤2L

(−1)^k λ^4L+4+2kTr

(U(λ⁻²L₀−1)⁻¹)^2k+2(L₀R^d_2L,r)(x, λ⁻²D)

+ X

k+r=m−1 k≥0,0≤r≤2L

i(−1)^k+1 λ^4L+2+2kTr

(U(λ⁻²L₀−1)⁻¹)^2k+1R_2L,r^a (x, λ⁻²D)(λ⁻²L₀+ 1) . (57)

We will refer to the three sums above asI,II, andIII, respectively.

We now identify the coefficients µ_j,m(u)’s and verify that they satisfy the properties claimed in Lemma 2.1. The claimed homogeneity properties will be clear from the formulas that we derive be- low; we will just need to verify the bounds (26). The latter are also straightforward to verify but will require us to use the structure of theR^d_k,r’s andR^a_k,r’s from (52)–(53).

The first sum has the form

−2m

2L−1

X

j=m−1

µ_j,m(u) λ^2j with

µj,m(u) = − i

2mλ² Tr

U R_j,m−1^a (x, λ⁻²D)

=− i 4mπ

X

γ∈N^2m−1

|γ|=j−(m−1)

Tr

"

Z

U(x)Q_γ(x) dx

Z P|γ|(_λ^ζ2) ((_λ^ζ2)²−1)^j+1

dζ λ²

# (58) .

Note that ‘Tr’ denotes an operator trace in the first line, whereas it refers to the2×2matrix trace in the second and third lines. We will use the notation ‘Tr’ similarly in what follows without further comment.

Sinceλis presumed to lie inΓ_δ, a comparison of the degrees in the numerator and denominator ensures that the integrals overζare finite and their values are independent ofλ. The total number of derivatives in the x-integrals isj −(m−1), and we distribute these so that the highest order of the derivatives that fall on a single U is as small as possible. We list the bounds onµ_j,m(u) according tomand the parity ofj. In each case below,`is a strictly positive integer.