Analytic smoothing effect for system of nonlinear Schrodinger equations with general mass resonance

50 (2020), 73–84

Analytic smoothing e¤ect for system of nonlinear Schro¨dinger

equations with general mass resonance

Takayoshi Ogawa and Takuya Sato

(Received January 19, 2019) (Revised October 21, 2019)

Abstract. We prove the analytic smoothing e¤ect for solutions to the system of nonlinear Schro¨dinger equations under the gauge invariant nonlinearities. This re-sult extends the known rere-sult due to Hoshino [Nonlinear Di¤erential Equations Appl. 24 (2017), Art. 62]. Under rapidly decaying condition on the initial data, the solution shows a smoothing e¤ect and is real analytic with respect to the space variable. Our theorem covers not only the case for the gauge invariant set-ting but also multiple component case with higher power nonlinearity up to the fifth order.

1. Introduction

We consider the Cauchy problem of the following nonlinear Schro¨dinger equations: iqtujþ 1 2mj Duj¼ fjðu; uÞ; t A R; x A Rn; j¼ 1; 2; . . . ; k; k A N; ujð0Þ ¼ fj; x A Rn; 8 > < > : ð1:1Þ

where mj>0, fj: Ck
Ck ! C, u ¼ ðu1; . . . ; ukÞ : R
Rn! Ck is the un-known function and f¼ ðf₁; . . . ;f_kÞ : Rn_{! C}k

is a given initial data. The nonlinear Schro¨dinger equation is classified as a dispersive type of partial di¤erential equations and there is no smoothing e¤ect that often appears in solutions to parabolic type equations. However if we restrict the initial data satisfying spacial weight condition, then one can find that solutions to the nonlinear Schro¨dinger equations exhibits a local smoothing e¤ect (cf. Kato [14]). Hayashi-Saitoh [4] proved that if the initial data f decays exponentially as jxj ! y, then there exists an analytic solution to the single nonlinear Schro¨dinger equation:

2010 Mathematics Subject Classification. 35Q55, 35B30.

Key words and phrases. Nonlinear Schro¨dinger equation, analytic smoothing e¤ect, gauge invariance.

iqtuþ 1 2mDu¼ f ðuÞ; t A R; x A R n_; uð0Þ ¼ f; x A Rn; 8 < :

where fðuÞ ¼ juj2u. For the cases of general polynomial nonlinearities fðuÞ, it was shown by several authors that the solution has the analytic smoothing e¤ect ([2], [4], [5], [8]–[13], [15], [16], [18]).

Hoshino-Ozawa [13] showed the analytic smoothing e¤ect for the solution to the following nonlinear Schro¨dinger equations:

iqtu1þ 1 2m1 Du1¼ f1ðu1; u2Þ; t A R; x A Rn; iqtu2þ 1 2m2 Du2¼ f2ðu1; u2Þ; t A R; x A Rn; ðu1ð0Þ; u2ð0ÞÞ ¼ ðf1;f2Þ; x A Rn; 8 > > > > > < > > > > > : ð1:2Þ

where the nonlinear coupling ð f1; f2Þ is given by either of the followings f1ðu1; u2Þ ¼ u1u2; f2ðu1; u2Þ ¼ u21;
for p¼ 2; ð1:3Þ f1ðu1; u2Þ ¼ u12u2þ ju1j2u1þ ju2j2u1; f2ðu1; u2Þ ¼ u31þ ju2j2u2þ ju1j2u2; ( for p¼ 3; ð1:4Þ f1ðu1; u2Þ ¼ u14u2þ ju1j4u1; f2ðu1; u2Þ ¼ u5₁þ ju2j4u2; ( for p¼ 5: ð1:5Þ

The nonlinearities given by (1.3), (1.4), and (1.5) satisfy eiy_f

1ðz1; z2Þ ¼ f1ðeiyz1; eipyz2Þ; eipy_f

2ðz1; z2Þ ¼ f2ðeiyz1; eipyz2Þ

for any y A R and z1; z2 AC.

We prove the analytic smoothing e¤ect of the solution to (1.1) with the p-th powered nonlinearities f fjðu; uÞgj¼1; 2;...; k satisfying a gauge invariance.

Definition 1. The nonlinear coupling functions f_j: C2k! C satisfy the gauge invariance if each fj ð j ¼ 1; . . . ; kÞ satisfies for any y A R and w; z A Ck,

eimjy_f

jðw1; . . . ; wk; z1; . . . ; zkÞ ¼ fjðeim1yw1; . . . ; eimkywk; eim1yz1; . . . ; eimkyzkÞ ð1:6Þ for some m¼ ðm1; . . . ; mkÞ A Zb0k . We call the condition (1.6) the mass resonance.

We restrict the nonlinearity to the following form: for any u¼ ðu1; . . . ; ukÞ, v¼ ðv1; . . . ; vkÞ : R
Rn! Ck, fjðu; vÞ 1 lj Yk l¼1 uaj; l l v bj; l l for j¼ 1; . . . ; k; ð1:7Þ

where ljAC, aj¼ ðaj; 1; . . . ;aj; kÞ and bj ¼ ðbj; 1; . . . ;bj; kÞ A Zb0k satisfy

aj; 1þ    þ aj; kþ bj; 1þ    þ bj; k¼ p ð1:8Þ with p A N. In this paper, we consider the type of nonlinearity fjðu; uÞ.

Proposition 1. Let k; p A N. The nonlinearity f_j given by (1.7) and (1.8) satisfies the condition (1.6) if and only if given a set of constants m¼ ðm1; . . . ; mkÞ A Zb0k , exponents aj and bjAZ_b0k satisfy

mj ¼ m1ðaj; 1 bj; 1Þ þ    þ mkðaj; k bj; kÞ ð1:9Þ for any j¼ 1; 2; . . . ; k.

In the 2-component case, the following nonlinearity ð f1; f2Þ defined by fjðu1; u2Þ 1 ju1j2u1;ju1j2u2; u12u2; u1ju2j2; u1u22;ju2j2u2 for j¼ 1; 2 satisfies the condition (1.6) with m¼ ð1; 1Þ, p ¼ 3 and the nonlinearity

f1ðu1; u2Þ 1 u12u22; f2ðu1; u2Þ 1 u13u2

satisfies (1.6) with m¼ ð2; 3Þ and p ¼ 4.

In order to state our result, we introduce the following function spaces introduced by Hayashi-Ozawa [10] and Hoshino-Ozawa [12]. For any m > 0 and a A N, L2_{exp; m}ðRnÞ 1 f A L2ðRnÞ; k f kL2 exp; mðRnÞ 1 _sup a A B1ð0Þ kemaxfkL2 < y ( ) ; H_{exp; m}a ðRnÞ 1 f A HaðRnÞ; k f kHa exp; mðR n_Þ1 sup a A B1ð0Þ kemaxfkHa < y ( ) ;

where L2_{¼ L}2_ðRn_{Þ and H}a_{¼ H}a_ðRn_{Þ are the Lebesgue and the Sobolev} spaces, respectively and B1ð0Þ ¼ fx A Rn;jxj < 1g. Let LyðI ; LqðRnÞÞ be the Bochner space where I  R is an open interval. We call Ly_{ðI ; L}q_ðRn_{ÞÞ the} Strichartz space if a pair of exponents ðy; qÞ is admissible as

n 2¼ n qþ 2 y

with 2 a q a 2n n 2; n b 3; 2 a q < y; n¼ 2; 2 a q a y; n¼ 1: 8 > > < > > : Let JmðtÞ 1 x þ it m‘

be the generator of the Galilei transform. The following operator emaJm _has

been introduced in [10], [12] and Hoshino [7]: For a A B1ð0Þ,

emaJmðtÞ_{f 1}X y k¼0 mk k! ða  JmðtÞÞ k f with the domain

DðemaJmðtÞ_{Þ ¼ f A SðR}n_{Þ; sup}

a A B1ð0Þ

kemaJmðtÞ_fk

Lq < y

( )

for any t 0 0 and q b 2. For any t 0 0, emaJm _{is represented as}

emaJm_{f ¼ e}iðt=2mÞD_emax_eiðt=2mÞD_{f ;}

and

emaJmðtÞ_{f ¼ e}iðmjxj2=2tÞ_eita‘_eiðmjxj2=2tÞ_{f ; ð1:10Þ}

where

eita‘f ¼ F1_½etax_ff^_{ for any t A R; a A R}n and F1 is the Fourier inverse transform:

F1½ f ðxÞ ¼ 1 ð2pÞn=2

ð Rn

eixxfðxÞdx:

For m > 0 and an open interval I  R, we define an analytic space: AmðI Þ 1 f A LyðI ; LqÞ; k f kAmðI Þ1 sup

a A B1ð0Þ

kemaJm_fk

Ly_{ðI ; L}q_Þ< y

( )

:

The function space AmðI Þ with weight eax has already been introduced in papers by Hayashi-Ozawa [6] (in space dimension n¼ 1 with open interval like

as D¼ ða; aÞ) and [12] (in space dimension n b 1 with symmetric domain D Rn such that 0 A D, D ¼ D).

We denote emaJmðtÞ _{for m¼ ðm}

1; . . . ; mkÞ A Zb0k as follows: emaJmðtÞ_{f 1ðe}m1aJ_m1ðtÞ_f

1; . . . ; emkaJmkðtÞfkÞ ð1:11Þ for any f : Rn _{! C}k_.

Our main results are the following. Theorem 1. Let 1 a n a 4, 1 < p a 1þ4

n and mj>0. Assume that the nonlinearities f fjðu; uÞg1ajak given by (1.7) satisfy the condition (1.8) and (1.9). Then for any f A Q_j¼1k L_{exp; m}2 _jðRn_{Þ, there exists T > 0 and a unique solution u A} Qk

j¼1AmjððT; TÞÞ to (1.1). In particular, the solution ujðt; xÞ is real analytic in

x A Rn for all t AðT; 0Þ [ ð0; TÞ and any j ¼ 1; 2; . . . ; k.

Since we consider the L2 _{solution of the Cauchy problem (1.1), we need to} restrict 1 < p a 1þ4

n (cf. Tsutsumi [20]). In order to obtain the analytic smoothing e¤ect for the solution to (1.1), the power of the nonlinearity is restricted to a natural number. Hence the spatial dimension n is naturally restricted to n¼ 1; 2; 3; 4. Hoshino [8] showed the existence for the analytic solution to (1.1) with second order nonlinear monomial for 2-component case. Theorem 1 implies the existence for the analytic solution to (1.1) with non-linear p-th order monomial in lower space dimensions and an extension for k-system. If we relax the regularity of initial data, then the result can be extended into higher dimensional case.

Theorem2. Let a A N, 1 a n a 2aþ 4=ðp  1Þ, and p A N such that p < y if n a 2a, p a 1þ 4=ðn  2aÞ if n > 2a. Then for any f A Q_j¼1k Ha

exp; mjðR

n_Þ, there exists T > 0 and a unique solution u A Q_j¼1k AmjððT; TÞÞ to (1.1) with

the nonlinearities given by (1.7) satisfying (1.8) and (1.9). In particular, the solution ujðt; xÞ is real analytic in x A Rn for all t AðT; 0Þ [ ð0; TÞ and any

j¼ 1; 2; . . . ; k.

Remark. All the results by Hoshino-Ozawa [13] involving (1.3), (1.4) and (1.5) can be covered by Theorem 1 under the restriction p¼ 3 for the gauge invariant monomials f f1; f2g satisfying (1.6). In [13], they proved the existence of a global analytic solution to the Cauchy problem (1.2) for small data in higher dimension. On the other hand we proved the existence of an analytic solution to (1.2) for a large initial data in lower dimensional cases and our Theorem 1 also extends the known result by Hoshino [8].

In order to show the analytic smoothing e¤ect for the solutions of (1.2), we introduce the operator emaJm_{. For each fixed a A B}

1ð0Þ, the integrability of emaJm_{f implies real analyticity to a-direction of a function f A Dðe}maJm_Þ.

t AðT; 0Þ [ ð0; TÞ. We employ the norm; sup_{a A B1}ð0ÞkemaJmukLy_{ð0; T; L}q_Þ and

it plays an substituted role of the norm P y k¼1 1 k!kðJmÞ k_uk Ly_{ð0; T; L}q_Þ, where the

norm k  k_Ly_{ð0; T; L}q_Þ is the Bochner norm with the L2-Strichartz admissible

n=qþ 2=y ¼ n=2 fullfilled. Besides the operator emaJm _{commutes with the}

free Schro¨dinger operator Lm¼ iqtþ_2m1 D and satisfies the Leibniz rule for the gauge invariant nonlinearities in (1.1). These properties of the operator emaJm _{simplify the proof of Theorem 1. Because of a commutative relation}

½Lm; Jm ¼ 0, operating Jma with multi-index a A Zb0n to the linear Schro¨dinger equation:

iqtvþ 1

2mDv¼ 0 ð1:12Þ

and taking summation in a, we have the analyticity for the solution to (1.12). We need to show an estimate for the nonlinear term in order to obtain the analyticity for the solution to the Cauchy problem (1.1). We estimate the gauge invariant nonlinearity in (1.1) by using the Galilei transform JmðtÞ ¼ xþit

m‘ that satisfies chain rule for the nonlinear term juj p1 u: JmðtÞðjujp1uÞ ¼ pþ 1 2 juj p1 ðJmðtÞuÞ  p 1 2 juj p3 u2ðJmðtÞuÞ;

where p A N and m > 0. In the case of k-component, the condition (1.6) enables us to compute for the nonlinear term f fjg1ajak. Then we introduce a new unknown function uj; a1emjaJmjuj (see Section 3 below) and it naturally satisfies the similar system:

iqtuj; aþ 1 2mj Duj; a¼ fjðuj; a; uj;aÞ; j¼ 1; 2; . . . ; k; k A N; uj; að0Þ ¼ emjaxfj; 8 > < > : ð1:13Þ

where fjðuj; a; uj;aÞ is a monomial for a unknown function uj; a and uj;a. Solving the system (1.13) in a proper function space, we obtain that the cor-responding solution u also maintains the desired regularity on a direction. Then choosing a A B1ð0Þ for all direction, we can show that the original solu-tion to (1.1) has real analytic regularity in space variable. Therefore, we have the analytic smoothing e¤ect for the solution to the Cauchy problem (1.1).

2. Preliminary

In this section, we state some properties of operator emaJm _{and some basic}

Rn! Ck, we define a function f : C2k! C as follows: fðu; vÞ 1 lY k j¼1 uaj j v bj j ; ð2:14Þ where l A C.

Hoshino-Hyakuna ([9], Lemma 4) proved the following Lemmas 1 and 2.

Lemma 1 ([9]). Let ða₁; . . . ;a_kÞ, ðb₁; . . . ;b_kÞ A Zk

b0, and f be the p-th order monomial given by (2.14). Then for any y A Rn, it holds that

eiy‘fðu; uÞ ¼ f ðeiy‘u; eiy‘_{uÞ for any u A Dðe}iy‘_Þ; where u denotes as complex conjugate of u.

Lemma 2 ([9]). Let p A N, m > 0, a A B₁ð0Þ, ða_{j; l}Þ

1alak, ðbj; lÞ1alakA Z_b0k and f ¼ ð f1; . . . ; fkÞ be a pair of the p-th order monomial with the condition (1.9) given by (1.7). Then for any mass coe‰cient m¼ ðm1; . . . ; mkÞ A Nk satisfying the condition (1.9), it holds that for j¼ 1; 2; . . . ; k,

emjaJmjðtÞ_f

jðu; uÞ ¼ fjðemaJmðtÞu; emaJmðtÞuÞ ð2:15Þ for any t A R and u A DðemaJm_{Þ, where e}maJmðtÞ_{u and e}maJmðtÞ_{u are defined}

by (1.11).

Stein-Weiss [19] proved the following Lemma 3. The essence of Lemma 3 is similar to that of Lemma A.1 in [12].

Lemma 3 ([12], [19]). Let M > 0 and f satisfy

keiy‘_fk L2a M; for any y A B1ð0Þ: Then eiy‘f ¼ ð2pÞn=2 ð Rn eiðxþiyÞxff^ðxÞdx is analytic in Rn_{þ iB} 1ð0Þ ¼ fx þ iy; x A Rn; y A B1ð0Þg.

Lemma 4. Let n b 1, m > 0, and ðy; qÞ be an admissible pair. Then there exists C0¼ C0ðn; qÞ such that for any f A L2ðRnÞ,

keði=2mÞtD_fk

Moreover, for any T > 0, ðT 0 eði=2mÞðtsÞDfðsÞds        _L2 a C0k f kLy 0_{ðT; T; L}q 0_Þ; ð2:17Þ

where p0 is the dual exponent to p defined by 1=pþ 1=p0_{¼ 1.}

See for the proof, Ginibre-Velo [3], Yajima [21], Cazenave-Weissler [1] and Keel-Tao [17].

3. Proof of Theorem 1

Proof. Let u 1ðu₁; . . . ; u_kÞ, u 1ðu₁; . . . ; u_kÞ : R
Rn! Ck, u_{j; a}1 emjaJmj_u

j, and an initial data f¼ ðf1; . . . ;fkÞ A Qk

j¼1L2exp; mj. We define a

p-th order function fðu; vÞ : Ck
Ck ! Ck by fðu; vÞ ¼ ð f1ðu; vÞ; . . . ; fkðu; vÞÞ and fjðu; vÞ 1 lj Yk l¼1 uaj; l l v bj; l l for j¼ 1; . . . ; k;

where ljAC, aj¼ ðaj; lÞ1alak, bj¼ ðbj; lÞ1alak and m¼ ðmjÞ1ajakAZ_b0k sat-isfy the condition (1.8) and (1.9). We operate emjaJmj _{to the j-th equation in}

(1.1). Then, by using the commutative relation JmjðtÞ; iqtþ

1 2mjD

h i

¼ 0 and the identity (2.15) in Lemma 2, we see that

iqtuj; aþ 1 2mj Duj; a¼ fjðuj; a; uj;aÞ; j¼ 1; 2; . . . ; k; k A N; uj; að0Þ ¼ emjaxfj: 8 > < > :

In order to show the analytic smoothing e¤ect for the solution, we solve the following system of integral equations on Ly_{ðT; T; L}q_Þ:

uj¼ eði=2mjÞtDfj ilj ðt 0 eði=2mjÞðtsÞDY k l¼1 uaj; l l ulbj; l ds

for j¼ 1; . . . ; k. Let I 1ðT; TÞ and M 1 4kC0kfkL2

exp; m, where C0 >0

depends only on the dimension n and exponent q b 2 in the Strichartz estimate (2.16) and (2.17). We define the metric space

XMðI Þ 1 ( u¼ ðu1; . . . ; ukÞ A Yk j¼1 AmjðI Þ; Xk j¼1

ðkujkLy_{ðI ; L}q_Þþ kujkAmjðI ÞÞ a M

for any admissible pair ðy; qÞ )

with the metric

dðu; vÞ 1X k

j¼1

ðkuj vjkLy_{ðI ; L}q_Þþ kuj vjkA_mjðIÞÞ

for some admissible pair ðy; qÞ. We see that ðXMðI Þ; dÞ is a complete metric space. Therefore we define the operator F½u ¼ ðF1½u1; . . . ; Fk½ukÞ on XMðI Þ by Fj½uj 1 eði=2mjÞtDfj i ðt 0 eði=2mjÞðtsÞD_f jðu; uÞds ð3:19Þ

for j¼ 1; . . . ; k and apply Banach’s fixed point theorem to the operator F defined by (3.19). For 1 a n a 4, by the Strichartz estimates (2.16), (2.17) and the representation for the operator defined by (1.10), we can estimate for Fj

kemjaJmj_F

j½ujkLy_{ðI ; L}q_Þakðeði=2mjÞtDemjaxeði=2mjÞtDÞeði=2mjÞtDf_jk_Ly_{ðI ; L}q_Þ

þ ðt 0 eði=2mjÞðtsÞD_emjaJmj_f jðu; uÞds         Ly_{ðI ; L}q_Þ 1Lj; 1þ Lj; 2; ð3:20Þ

where Lj; 1 and Lj; 2 are the linear part and the Duhamel part of the in-tegral equation (3.19), respectively. By the Strichartz estimate (2.16), we have

Lj; 1¼ keði=2mjÞtDemjaxfjkLy_{ðI; L}q_Þa C0kemjaxfjkL2a

1

4kM: ð3:21Þ

Similarly applying (2.17) to Lj; 2 with admissible pair ð4q=ð4q þ nq  2npÞ; q=pÞ and using Ho¨lder’s inequality, we have

Lj; 2a C0 Yk l¼1 ðemlaJml_u lÞaj; lðemlaJmlulÞbj; l           L4q=ð4qþnq2npÞ_{ðI ; L}q=p_Þ a C0 Yk l¼1 kemlaJ_mlu lk aj; l LqkemlaJmlulk bj; l Lq          _L4q=ð4qþnq2npÞ_{ðI Þ} a C0T1ðn=4Þð p1Þ Yk l¼1 kemlaJ_mlu lk aj; l Ly_{ðI ; L}q_Þke mlaJ_mlu lk bj; l Ly_{ðI ; L}q_Þ a C0T1ðn=4Þð p1ÞMp: ð3:22Þ

Conbining (3.20), (3.21) and (3.22), we have kFj½ujkA_mjðI Þa

1

4kMþ C0T

1ðn=4Þð p1Þ_Mp_:

In the same way as above, we have that for u; v A XMðI Þ, kemjaJmj_ðF j½uj  Fj½vjÞkLy_{ðI ; L}q_Þ a ðt 0 eði=2mjÞðtsÞD_emjaJmj_{ð f} jðu; uÞ  fjðv; vÞÞds        _Ly_{ðI ; L}q_Þ a C0T1ðn=4Þð p1ÞMp1 Xk j¼1 kuj vjkAmjðI Þ: Hence we obtain dðF½u; F½vÞ a C0T1ðn=4Þð p1ÞMp1dðu; vÞ: Choosing T > 0 such that

C0T1ðn=4Þð p1ÞMpa 1 4kM; C0T1ðn=4Þð p1ÞMp1a 1 2; 8 > > < > > :

we have that the map F is a contraction over XMðI Þ. Then Banach’s fixed point theorem implies that F has a unique fixed point u in XMðI Þ which is a solution of the integral equation (3.19). In other words, the solution u¼ ðu1; . . . ; ukÞ satisfies sup a A B1ð0Þ kemjaJmj_u jkLy_{ðI ; L}q_Þ¼ sup a A B1ð0Þ keita‘eðimjjxj2=2tÞ_u jkLy_{ðI ; L}q_Þ< y

for any a A B1ð0Þ and j ¼ 1; . . . ; k. Therefore Lemma 3 yields that eimjjxj

2

=2t_u j is analytic over Rnþ iBtð0Þ. Since eimjjxj

2

=2t _{is analytic over R}n_{þ iR}n_{, u is} real analytic in x A Rn.

In the critical case for n¼ 4, we define

M¼ sup

a A B1ð0Þ

keðit=2mjÞD_emjax_fk

Ly_{ðI ; L}q_Þ

in the space XMðI Þ defined by (3.18). Taking su‰ciently small T > 0, we can construct the local solution v¼ emaJm_{u in the Stricartz space L}y_{ðI ; L}q_{Þ by}

The same argument for the proof of Theorem 1 can be applied to the proof of Theorem 2.

Acknowledgments

The authors express their thanks to the referees and the editor for their helpful suggestions. The first author is supported by Grant-in-aid for Scientific Research A, No. 19H00638.

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Takayoshi Ogawa Mathematical Institute/

Research Alliance Center of Mathematical Science Tohoku University

Sendai 980-8578, Japan E-mail: [email protected]

Takuya Sato

Mathematical Institute, Tohoku University Sendai 980-8578, Japan