SCHRODINGER GROUP ON ZHIDKOV SPACES

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SCHRODINGER GROUP ON ZHIDKOV SPACES

Clément Gallo

To cite this version:

Clément Gallo. SCHRODINGER GROUP ON ZHIDKOV SPACES. Advances in Differential Equa- tions, Khayyam Publishing, 2004, 9, pp.509 - 538. �hal-01960986�

Schr¨ odinger group on Zhidkov spaces

Cl´ement Gallo

UMR de Math´ematiques, Bat. 425 Universit´e Paris-Sud

91405 Orsay, France.

R´esum´e On consid`ere le probl`eme de Cauchy pour l’´equation de Schr¨odinger non lin´eaire surRⁿ, dans des espaces de fonctions non (n´ecessairement) nulles `a l’infini. Ce probl`eme se pose par exemple dans l’´etude de la stabilit´e des “dark solitons”. On montre que l’op´erateur de Schrdinger g´en`ere un groupe sur les espaces de Zhidkov X<sup>k</sup>(R<sup>n</sup>) pour k > n/2, puis que le probl`eme de Cauchy pour SNL est localement bien pos´e dans ces mˆemes espaces de Zhidkov. On montre aussi la conservation des invariants classiques, et, dans certains cas, on en d´eduit que le probl`eme de Cauchy est globalement bien pos´e.

Abstract. We consider the Cauchy problem for nonlinear Schr¨odinger equa- tions on Rⁿ with non zero boundary condition at infinity, a situation which occurs in stability studies of dark solitons. We prove that the Schr¨odinger op- erator generates a group on Zhidkov spacesX^k(Rⁿ) fork > n/2, and that the Cauchy problem for NLS is locally well-posed on the same Zhidkov spaces. We justify the conservation of classical invariants which implies in some cases the global well-posedness of the Cauchy problem.

Key words. Nonlinear Schr¨odinger equations, Dark solitons, Cauchy problem AMS subject classifications. 35Q55, 35A07

1 Introduction

This paper is devoted to the Cauchy problem for the Nonlinear Schr¨odinger equation (NLS)

iut+ ∆u+f(|u|²)u= 0, (t, x)∈R×Rⁿ

u(0) =u0 (1)

with non zero boundary condition at infinity. Such boundary conditions occur in the “defocusing” case (e.g. f(|u|²) = 1− |u|²), and are pertinent to many physical contexts. In nonlinear optics, the so-called dark soliton (see [KLD]) is a solution of (1) of the formu(x, t) = uv(x−vt). For instance, for n= 1 and

f(r) = 1−r, we can compute that, for v ∈(−√ 2,√

2), uv(x−vt) solves (1), whereuvis given by:

uv(x) = r

1−v² 2 tanh

r 1−v²

2

√x 2

! +i v

√2 (2)

The Gross-Pitaevskii equation (see [BeS1], [BeS2] and references therein) iut+ ∆u+ (1− |u|²)u= 0

with the boundary condition u → 1 as |x| → ∞ is a model for Superfluid Helium II at a temperature near zero and for Bose-Einstein condensation. More generally, NLS with the boundary condition |u| → ρ0 as |x| → ∞ where ρ0

is a positive constant such that f(ρ²₀) = 0 occurs in several physical contexts (see [BGMP]), an especially interesting particular case being the cubic-quintic

“ψ³−ψ⁵” NLS.

These nonlinear Schr¨odinger equations possess solitons or solitary waves (see [BeS1], [dB], [M], [KLD]) and it is natural to study the Cauchy problem in spaces the solitary waves belongs to. Of course they can not be the usual Sobolev spaces H^s(Rⁿ) because of the boundary condition at infinity. A possibility would be to work in the affine space 1 +H^s(Rⁿ), when u → 1 as |x| → ∞, and this actually was done in [BeS1]. This approach obviously fails when only|u| (but notu) tends to a constant at infinity, as it is the case for the dark soliton (2).

Also, the solitary waveφcould be only slowly decaying at infinity, implying that φ−16∈L²(Rⁿ) (see [G] for the travelling wave of Gross-Pitaevskii equation in the 2-dimensional case).

In [Z0] Zhidkov introduces in the one-dimensional case the spacesX^k (with k a natural number), which consist of functions defined on R, bounded and uniformly continuous, with derivatives up to orderkinL², and proved that the Cauchy problem for NLS is locally well-posed inX^k.

Our aim here is to complete and generalize Zhidkov results. We introduce the Zhidkov spaces X^k(Rⁿ) in higher dimensions and prove that the linear Schr¨odinger equation defines a strongly continuous group on X^k(Rⁿ) if and only ifk > n/2, and consequently that the Cauchy problem for NLS is locally well-posed inX^k(Rⁿ) if k > n/2. We also justify rigorously the conservation of natural invariants of the NLS yielding some global well-posedness inX^k(Rⁿ) for some defocusing NLS. A byproduct is the complete justification of the result of Zhiwu Lin in [ZL] that gives a criterion of stability for dark solitons of a class of NLS equations.

This paper is organized as follows. In section 2, we define the Zhidkov spaces X^k(Rⁿ) and state some useful properties of these spaces. In section 3, we prove that the linear Schr¨odinger equation is well posed in X^k(Rⁿ) if and only if k > n/2. In section 4, we show that the Cauchy problem for NLS is locally well-posed inX^k(Rⁿ), ifk > n/2. In section 5, we introduce the renormalized energy for (1) and prove its conservation, forn= 1 or 2, under some hypothesis onf andu0, and we show that in dimension 1, it implies the globalness of the solution of (1) in a defocusing case.

Notations Throughout this paper,Cdenotes a constant that can change from line to line.

If j is a positive integer and u is a map of class C^j from Rⁿ into C, and x, v1, ..., vj ∈ Rⁿ, we denote by D^ju(x)(v1...vj) the j^th differential of u at x applied to (v1, ..., vj).

IfEis a Banach space, we denote byCb(R, E) the space of continuous bounded functions fromRintoE.

2 Some properties of Zhidkov spaces

Definition 2.1 Let n, k ∈N∗. We define the space X^k(Rⁿ) as the closure for the norm

||u||X^k(Rⁿ):=||u||L^∞+ X

16|α|6k

||∂^αu||L²

of the space{u∈C^k(Rⁿ),bounded, uniformly continuous, with∇u∈H^k−1}. Note that a function inX^k is uniformly continuous.

Proposition 2.1 Let n, k∈N^∗. ThenX^k+1(Rⁿ)is dense in X^k(Rⁿ).

Proof. Let u ∈ X^k(Rⁿ), (ρl)l>1 a mollifier sequence (i.e. ρl ∈ C_c^∞(Rⁿ), R

Rⁿρl = 1, ρn >0, Supp ρl ⊂ B(0,1/l)). Thenρl∗u is a sequence inX^k+1

that converges touinX^k.

We show now a regularity result for functions inX^k(Rⁿ), fork > n/2.

Proposition 2.2 Let n∈N∗, let k=⌊n/2⌋+ 1, and p∈R such that p > n if nis even

p= 2n if nis odd .

Then ∇u∈L^p(Rⁿ)for u∈X^k(Rⁿ)and there existsC >0 such that

|u(x)−u(y)|6C|x−y|^1−n/p||∇u||^Lp, x, y∈Rⁿ . (3) Proof. By our choice ofpandk, Sobolev’s embedding implies that

||∇u||L^p(Rⁿ)6C||∇u||H^k−1(Rⁿ)6C||u||X^k(Rⁿ), u∈X^k(Rⁿ). (4) Following the proof of Morrey’s theorem given in [B], we can show that there exists a constantC >0 that depends only onnandpsuch that for any compact setK⊂Rⁿ, and any cubeQ= [−r, r]ⁿ containingK,

|u(x)−u(y)|6C|x−y|^1−n/p||∇u||L^p(Q), x, y∈K, u∈H_loc^k (Rⁿ). (5) SinceX^k(Rⁿ)⊂H_loc^k (Rⁿ), (4) and (5) prove the announced result.

Remark . In our proof, we did not use the fact that the elements ofX^k(Rⁿ) are uniformly continuous. Therefore, fork > n/2, an equivalent definition for X^k could be

X^k(Rⁿ) ={u∈L^∞(Rⁿ), ∇u∈H^k−1(Rⁿ)} . In particular, fork > n/2,H^k(Rⁿ)⊂X^k(Rⁿ).

3 The Schr¨odinger group on X^k(Rⁿ)

In this section, we prove that if k > n/2, the Schr¨odinger operator defines a group onX^k(Rⁿ). More precisely,

Theorem 3.1 Let n∈N∗,k > n/2 andu0∈X^k(Rⁿ). For t∈Rand x∈Rⁿ, the quantity

S(t)u0(x) =







e^−inπ/4π^−n/2lim

ε→0

R

Rⁿe^(i−ε)|z|2u0(x+ 2√

tz)dz ift>0 e^inπ/4π^−n/2lim

ε→0

R

Rⁿe^{(−i−ε)|z|2}u0(x+ 2√

−tz)dz ift60 (6) makes sense, and the family of operators(S(t))t∈Rdefines a strongly continuous group onX^k(Rⁿ).

Moreover, there exists a constant C >0 that depends only on n and k, such that for every u0∈X^k(Rⁿ)andt∈R,

||S(t)u0||X^k 6 C(1 +|t|^ρ)||u0||X^k (7) where

ρ=

1/2 if nis even 1/4 if nis odd . For convenience, we also denoteS(t)u0(x) byu(t, x).

All the computations below will be performed witht>0. The caset60 is similar. Before starting the proof of the theorem itself, we need to prove some technical lemmas.

Lemma 3.1 Let n, k>1,u0∈X^k(Rⁿ),β >0,x∈Rⁿ andt >0.

We defineg: (β,∞)→Cby g(r) =

Z

Sⁿ⁻¹

u0(x+ 2√

trv)dv (8)

then g ∈ X^k(β,∞) (the definition of which is clear), and for all j ∈ {1...k}, g^(j)∈L²(β,∞, rⁿ⁻¹dr)with

||g^(j)||L²(β,∞,rⁿ⁻¹dr) 6 (2√

t)^j−n/2|Sⁿ⁻¹|^1/2||u0||X^k(Rⁿ) . (9)

Proof. For anyr∈(β,∞),|g(r)|6|Sⁿ⁻¹|||u0||^L∞, henceg∈L^∞(β,∞).

Letε >0. Sinceu0is uniformly continuous, there exists someδ >0 such that

|y−z|6δimplies|u0(y)−u0(z)|6ε. Letr1, r2be such that|r1−r2|6δ/(2√ t).

Then we get: |g(r1)−g(r2)|6|Sⁿ−1|εandg is uniformly continuous.

For j ∈ {1...k}, the Cauchy-Schwarz inequality and a change of variables yield:

Z _∞

β |g^(j)(r)|²rⁿ⁻¹dr 6 (2√

t)^2j|Sⁿ⁻¹| Z _∞

β

Z

Sⁿ⁻¹|D^ju0(x+ 2√

trv)(v...v)|²rⁿ⁻¹dvdr

= (2√

t)^2j|Sⁿ⁻¹| Z ∞

2√ tβ

Z

Sⁿ⁻¹|D^ju0(x+rv)(v...v)|²rⁿ⁻¹dvdr (2√

t)ⁿ 6 (2√

t)^2j−n|Sⁿ⁻¹|||u0||²X^k(Rⁿ)

which is the announced inequality.

We state next two elementary lemmas which are straightforward conse- quences of the Leibniz formula

Lemma 3.2 Let k∈N. There exists constants(bl,k)06l6k such that d

dr 1

r. k

= Xk

l=0

bl,k

1 r^2k−l

d^l

dr^l . (10)

Lemma 3.3 There exists constants (ak,j)06j6k such that d

dr 1

r. k

(rⁿ⁻¹.) = Xk

j=0

ak,j

1 r^2k−n+1−j

d^j

dr^j. (11)

We are now ready to prove theorem 3.1. We fixn∈N^∗, and we introduce a functionχ∈C^∞(Rⁿ) such that:

• χis radial

• χincreases along any half-line issued at 0

• χ≡0 on{x,|x|61}

• χ≡1 on{x,|x|>2}

For any β >0, we define χβ =χ(./β). For convenience, we will also use the notationχβ(|.|) =χβ(.).

Proof of theorem 3.1. We assume first thatk =⌊n/2⌋+ 1. The first step of the proof is to show that the limit in formula (6) is well defined. Let us fix t >0, β >0 andε >0. We split the integral in (6) into two parts:

Z

Rⁿ

e^(i−ε)|z|2u0(x+ 2√

tz)dz = Z

Rⁿ

e^(i−ε)|z|2(1−χβ(z))u0(x+ 2√ tz)dz

(I)

+ Z

Rⁿ

e^(i−ε)|z|2χβ(z)u0(x+ 2√ tz)dz

(II)

We first consider the term (I):

e^(i−ε)|z|2(1−χβ(z))u0(x+ 2√ tz)

6||u0||^L∞1B(0,2β)(z) Then by Lebesgue’s theorem, the limit asε→0 of (I) exists and

lim

ε→0

Z

Rⁿ

e^(i−ε)|z|2(1−χβ(z))u0(x+ 2√ tz)dz

6 ||u0||^L∞|B(0,1)|(2β)ⁿ .(12) We now consider the term (II). We compute it by using polar coordinates, and with the notations of lemma 3.1, we integrate by parts (which is justified by the regularity results ongproved in lemma 3.1) using lemma 3.3:

Z

Rⁿ

e^(i−ε)|z|2χβ(z)u0(x+ 2√ tz)dz

= Z _∞

0

e^(i−ε)r2χβ(r)rⁿ⁻¹ Z

Sⁿ⁻¹

u0(x+ 2√ trv)dv

dr

= Z ∞

β

1 2(i−ε)r

d dr.

k

e^(i−ε)r2

χβ(r)rⁿ⁻¹ Z

Sⁿ⁻¹

u0(x+ 2√ trv)dv

dr

=

−1 2(i−ε)

kZ ∞ β

e^(i−ε)r2 Xk

j=0

ak,j 1 r^2k−n+1−j

d^j

dr^j[χβ(r)g(r)]dr

=

−1 2(i−ε)

k k

X

j=0

ak,j

Xj

l=0

j l

Z ∞ β

e^(i−ε)r2 1

r^2k−jg^(l)(r)χ^(j_β^−l)(r)rⁿ⁻¹dr(13) where we have used the Leibniz formula in the last equality. We will now apply Lebesgue’s theorem to each term of this sum.

Forl= 0,j∈ {0...k}, we have:

e^(i−ε)r2 1

r^2k−jg(r)χ^(j)_β (r)rⁿ⁻¹

6 |Sⁿ⁻¹|||u0||L^∞

r^2k−n+1







||χ||L^∞ ifj= 0

r^j

β^j||χ^(j)||^L∞ ifj∈ {1...k}andr62β 0 ifj∈ {1...k}andr >2β 6 |Sⁿ⁻¹|||u0||L^∞

r^2k−n+1 2^j||χ^(j)||L^∞

Sincek > n/2,R_∞

β dr/r^2k−n+1=β^n−2k/(2k−n)<∞and we can pass to the limit asε→0 by Lebesgue’s theorem. We obtain

lim

ε→0

Z _∞

β

e^(i−ε)r2 1

r^2k−jg(r)χ^(j)_β (r)rⁿ⁻¹dr

6 |Sⁿ⁻¹|||u0||^L∞2^j||χ^(j)||^L∞ (2k−n)β^2k−n (14). Forl>1,j∈ {l...k}, we have:

e^(i−ε)r2 1

r^2k−jg^(l)(r)χ^(j_β^−l)(r)rⁿ⁻¹

6 ||χ^(j−l)||L^∞

1 β^j−l

1

r^2k−j|g^(l)(r)|rⁿ⁻¹ . Notice that 2(2k−j)−(n−1) >1, so that r→1/r^2k−j ∈L²(β,∞, rⁿ⁻¹dr).

The Cauchy-Schwarz inequality together with (9) lead to Z _∞

β

1

r^2k−j|g^(l)(r)|rⁿ⁻¹dr

6 1

(4k−2j−n)^1/2β^2k−j−n/2(2√

t)^l−n/2|Sⁿ⁻¹|^1/2||u0||X^k .

So Lebesgue’s theorem can be applied, and

lim

ε→0

Z _∞

β

e^(i−ε)r2 1

r^2k−jg^(l)(r)χ^(j_β^−l)(r)rⁿ⁻¹dr

6 |Sⁿ−1|^1/2||χ^(j−l)||^L∞

(4k−2j−n)^1/2 ||u0||X^k

(2√ t)^l−n/2

β^2k−l−n/2 (15) We fix nowβ = 1. For j∈ {1...k}, the quantities (2√

t)^j−n/2 are majorized byC(t^(1−n/2)/2+t^(k−n/2)/2) whereC is a positive constant. Therefore, using (12),(14) and (15), there exists a positive constantCsuch that for allu0∈X^k, for allt >0,

||u(t)||L^∞(Rⁿ) 6 C(1 +t^(1−n/2)/2+t^(k−n/2)/2)||u0||X^k(Rⁿ) (16) The second step of the proof consists in proving that u(t)→ u0 in L^∞ as t→0. We first introduce some definitions.

Definition 3.1 Forl∈ {0...k} andh∈L²(β,∞, rⁿ⁻¹dr), we define T_h^l:=

Z _∞

β

e^(i−ε)r2h(r)g^(l)(r)rⁿ⁻¹dr and we will say that such a quantity is “of typeT^(l)”.

If h can be written h(r) = χ^(p)_β (r)/r^q with 2q−n > 0, we will say that the

“order” ofT_h^l inβ isp+q−n/2.

Ifhis given as a linear combination of such terms, we define the “order” ofT_h^l as the lowest order of non-zero monomials in the expression of h(remark that our “definition” of the order ofT_h^l depends on the decomposition ofh).

Letα∈(0,1/(2n+1)), andm >0 such thatm>(n/2−1)/α. The following technical lemma gives a new expression of the term (II). It consists in doing as much integration by parts as necessary, in order to express (II) as a sum of terms of orders>m, which we are able to estimate in an appropriate way (see estimate (19) below.

Lemma 3.4 (II) can be written as a linear combination of terms of type T^(l) withl∈ {0...k}, such that forl∈ {1...k−1}, the order of the terms of type T^(l) in this linear combination is>m.

Proof. Letl∈ {1...k−1},p, q∈Nsuch that 2q−n >0 andh(r) =χ^(p)_β (r)r^q. We transformT_h^lby integrations by parts, as in (13). After some computations, we get (this is actually (13) where we have replacedkbyk−l,gbyg^(l)andχβ

byχ^(p)_β (r)/r^q):

T_h^l= −1

2(i−ε)

k−l k−l

X

j=0

ak−l,j

Xj

c=0

j c

^j−c

X

b=0

j−c b

(−q)...(−q−(j−c−b−1))

× Z _∞

β

e^(i−ε)r2χ^(p+b)_β (r)g^(l+c)(r)

r^{2(k−l)+q−c−b} rⁿ⁻¹dr . (17) Since 2(2(k−l) +q−c−b)−n > 2(k−l) + 2q−n, k > l and 2q−n > 0, we have writtenT_h^l as a linear combination of terms of typesT^(l), T^(l+1)...T^(k). Moreover, the order of the terms of typeT^(l)(that correspond toc= 0) in this sum is:

p+b+ 2(k−l) +q−b−n/2 =p+q−n/2

| {z }

order ofT_h^l

+2 (k−l)

| {z }

>1

.

The conclusion of this computation is that passing from the canonical expression ofT_h^l to its new expression, the order of typeT^(l)terms increases at least by 2.

We now use the above calculation to show the result by induction on l ∈ {0...k−1}.

Let us consider, for l ∈ {0...k−1}, the induction hypothesis Hl: “(II) can be written as a sum of terms of typeT^(γ), 06 γ 6k, so that if 16 γ 6l, the term of typeT^(γ)in this sum has order>m.”

Formula (13) implies that (II) =

Xk

l=0

T_h^l(l), whithh^(l)(r) = Xk

j=l

akj

j l

χ^(j_β^−l)(r) r^2k−j . We have 2(2k−j)−n>2k−n >0, so thatH0is true thanks to (13).

Let us now takel∈ {1...k−1}and supposeHl−1. The induction hypothesis im- plies that (II) can be written under the form: (II) =Pk

γλγT_h^γ_γ whereλγ ∈C, hγ is a linear combination of χ^(p)_β (r)/r^q, and for γ ∈ {1...l−1}, T_h^γ_γ’s order

is at leastm. Applying the former calculation to the term T_h^l_l, we get a new expression of (II) where the terms of type T^(γ) with γ 6l−1 are unchanged (in particular, the new expression still verifiesHl−1), the order of the term of typeT^(l)has increased by 2(k−l)>2. We can start this process again, as long as it is necessary (but with a finite number of steps) to ensure that the term of typeT^(l)is of order>m, and henceHl is true.

So we have proved thatHk−1 is true, which is the result of the lemma.

We give now an upper bound on|T_h^l|, forl∈ {1...k}, withh(r) =χ^(p)_β (r)/r^q and 2q−n >0:

|T_h^l| 6 ||χ^(p)||∞

β^p

Z ∞ β

1

r^q|g^(l)(r)|rⁿ⁻¹dr 6 ||√χ^(p)||∞

2q−n||u0||X^k|Sⁿ⁻¹|^1/2

| {z }

=:C

(2√ t)^l−n/2

β^p+q−n/2 (18)

where we have used the Cauchy-Schwarz inequality and lemma 3.1. If we assume thatβ>1 andT_h^l’s order is at least m, then

|T_h^l| 6 C(2√ t)^l−n/2

β^m . (19)

We writeβ on the formβ = ˜β/(2√

t)^α, fort61 and ˜β >2^α. Let us fix δ >0.

The choice ofmand the fact thatl>1 ensure thatl−n/2 +αm>0.

We choose ˜β >2^α large enough such that for allt61, (2√

t)^l−n/2+αm

β˜^m 6δ , l∈ {1...k} and |Sⁿ⁻¹|||u0||∞2^j||χ^(j)||L^∞

(2k−n) ˜β^2k−n (2√

t)^α(2k−n)6δ , j∈ {0...k} .

The propertyHk−1proven in Lemma 3.4, (19) and (14) imply that there exists a constantC >0 (which does not depend onδ) such that

∀ε >0, Z

Rⁿ

e^(i−ε)|z|2χβ/(2˜ √

t)^α(z)u0(x+ 2√ tz)dz

6 Cδ . (20)