Construction of blow-up solutions for Zakharov system on <span class="mathjax-formula">$ {\mathbb{T}}^{2}$</span>

Ann. I. H. Poincaré – AN 30 (2013) 791–824

www.elsevier.com/locate/anihpc

Construction of blow-up solutions for Zakharov system on T ²

Nobu Kishimoto

, Masaya Maeda

aDepartment of Mathematics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan bMathematical Institute, Tohoku University, Sendai 980-8578, Japan

Received 19 September 2011; received in revised form 3 September 2012; accepted 13 September 2012 Available online 17 September 2012

Abstract

We consider the Zakharov system in two space dimension with periodic boundary condition:

i∂tu= −u+nu,

∂t tn=n+|u|², (t, x)∈ [0, T )×T². (Z)

We prove the existence of finite time blow-up solutions of(Z). Further, we show there exists no minimal mass blow-up solution.

©2012 Elsevier Masson SAS. All rights reserved.

Résumé

Nous considérons le système de Zakharov dans l’espace à deux dimensions avec la condition périodique au bord : i∂_tu= −u+nu,

∂t tn=n+|u|², (t, x)∈ [0, T )×T². (Z)

Nous prouvons l’existence de solutions de(Z)explosant au temps fini. En outre, nous prouvons qu’il n’y a aucune solution explosive de masse minimale.

©2012 Elsevier Masson SAS. All rights reserved.

MSC:35Q55

Keywords:Zakharov system; Blow-up solution; Modified energy; Minimal mass blow-up solution

1. Introduction

In this paper, we consider the Zakharov system onT²=(R/2πZ)²:

⎧⎪

⎪⎨

⎪⎪

⎩

i∂tu= −u+nu, 1

c₀²∂t tn=n+|u|²,

u(0)=u₀, n(0)=n₀, n_t(0)=n₁,

(Z)

* Corresponding author. Tel.: +81 (0)75 753 3722; fax: +81 (0)75 753 3711.

E-mail addresses:n-kishi@math.kyoto-u.ac.jp(N. Kishimoto),m-maeda@math.tohoku.ac.jp(M. Maeda).

0294-1449/$ – see front matter ©2012 Elsevier Masson SAS. All rights reserved.

http://dx.doi.org/10.1016/j.anihpc.2012.09.003

wherec₀>0,u: [0, T )×T²→C,n: [0, T )×T²→Randu₀,n₀,n₁are initial data. Further, in our results, we fix c₀=1.

Zakharov system was introduced in[36]to describe the collapse of Langmuir wave (or electron plasma waves) in a non-magnetized plasma. In the context of the dynamics of Langmuir wave,urepresents the slowly varying envelope of the electric field and ndenotes the deviation of the ion density from its mean value. From the physical point of view, the evolution of (Z) leads to the formation of a cavity of ion density and an amplification of the amplitude of the electric field. Further, the collapse of cavity gives an explanation for the mechanism of the dissipation of long- wavelength plasma waves. Therefore, the wave collapse, which is the finite time blowup of the solution of (Z), plays a central role in the strong turbulence of Langmuir waves. See, for example[31,33].

Note that the subsonic limit of Zakharov system (c0→ ∞) formally gives us the nonlinear Schrödinger equation with critical exponent:

i∂_tu= −u− |u|²u,

u(0)=u0. (NLS)

We say critical because it is the smallest power that admits blowup in finite time for initial data in the energy class (u0∈H¹).

The local-in-time well-posedness for these equations have been extensively studied. For a moment we useX to denoteR²orT². For (NLS), it is known for initial data inH^s(X), withs0 for the caseR²[8]and withs >0 for the caseT²[4]. See[34]for a result onR×T. Moreover, for the strong solutionu∈C([0, T];H^s(X))obtained, we have the conservation of mass

u(t )

L²= u₀L², ∀t∈ [0, T], and, ifs1, the conservation of energy

1

2∇u(t )²

L²−1 4u(t )⁴

L⁴=1

2∇u₀²_L2−1

4u₀⁴_L4, ∀t∈ [0, T].

The Zakharov system (Z) has similar conservation laws. First, the mass ofu(t ) is also conserved. In addition, assumen₁∈ ˆH⁻¹(X;R), where

Hˆ⁻¹(X;R):=

φ∈H⁻¹(X;R) ∃ψ∈L² X;R²

s.t.φ= −∇ψ

(see also Remark1below). Then,∂tn(t )∈ ˆH⁻¹for allt and the wave part of (Z) may be written in the form

∂_tn+ ∇v=0; 1

c₀²∂_tv+ ∇n= −∇

|u|² ,

for somev(t )∈L²(X;R²). In this case, we have the conservation of energyE(t)=E(0), whereEis defined by E=E(u, n, v):= ∇u²_L2+1

2

n²_L2+ v²_L2

+

X

n|u|²dx. (1)

The local well-posedness of (Z) on R² in the energy spaceH¹×L²× ˆH⁻¹ was first obtained by Bourgain and Colliander[5], which was improved toH¹×L²×H⁻¹and wider spaces by Ginibre, Tsutsumi, and Velo[11]. The lowest regularity in which the local well-posedness is known so far isL²×H^−1/2×H^−3/2[3]. The caseT²is more involved, but the local well-posedness in the energy space (actually inH¹×L²×H⁻¹and some wider spaces) was recently proved, see[17].

Remark 1.The definition of the Hˆ⁻¹ norm ofφ= −∇ ψ is a bit tricky. It would not be well-defined if we used simply theL²norm ofψ. For example, consider ψˆ =(χB,−^ξ_ξ²₁χB)forX=R², whereB:= [1,2]²∪ [−2,−1]²and χ_B is the characteristic function ofB, and ψ =(cos(x1+x₂),−cos(x1+x₂))for X=T². Then, ψL² >0 but φ= −∇ ψ=0.

Before the definition we recall the Helmholtz decomposition L²(X;R)²=L²_σ ⊕^⊥G into the solenoidal space L²_σ = { ψ: ∇ ψ=0}and the gradient spaceG= {∇η: η∈ ˙H¹(X;R)}. LetPG:L²(X;R)²→Gbe the orthogonal projection ontoG. We now defineφ_Hˆ−1:= PGψL² forφ= −∇ ψ, which is a well-defined norm onHˆ⁻¹.

The time global existence and blow-up problem of (NLS) onR²andT²have also been studied by many authors[14, 19–29,35]. It is well known that ifu₀∈H¹(R²)andu_0L²_(R²₎<Q_L²_(R²₎, then the solution of (NLS) onR²exists globally in time. (In fact, it was recently shown by Dodson[10]that the assumptionu0∈H¹can be replaced with u0∈L².) Here,Qis the unique positive radial solution of

−Q+Q−Q³=0, x∈R². (2)

This also holds for the caseT². That is, ifu₀∈H¹(T²)andu₀L²(T²)<QL²(R²), then the solution of (NLS) onT²exists globally in time. On the other hand, ifMQ²_L2(R²), it is known that there existsu₀∈H¹(X)such that u0²_L2(X) =M and the solution of (NLS) on X blows up in finite time (for the case T² see [2,6]). In this sense,QL²(R²) is the sharp threshold for the global existence and blowup of (NLS) both onR² and onT². For the blow-up problem of (Z) onR², Glangetas and Merle[12]constructed a blow-up solution withu₀L²(R²) arbi- trarily nearQL². Further, in[13]they showed that ifu0L²(R²)QL²(R²), then the solution of (Z) onR²with (u(0), n(0), nt(0))=(u₀, n₀, n₁)∈H¹(R²)×L²(R²)×H⁻¹(R²)exists globally in time. However, it seems there is no counterpart of the results of Glangetas and Merle forT²as far as the authors know.

In this paper, we construct a blow-up solution of (Z) by using the fixed point argument. Further, as in theR²case, we show that ifu₀L²(T²)QL²(R²), then the solution of (Z) exists globally in time. For (NLS), Burq, Gérard, and Tzvetkov[6]constructed a blow-up solution onT²by adapting an idea of Ogawa and Tsutsumi[27], who treated a similar problem onT. In[6]they cut off the explicit blow-up solution onR² and solved the perturbed equation.

Thus we use the blow-up solution of (Z) onR²which was constructed by Glangetas and Merle. However, in contrast to (NLS), (Z) has a derivative in the nonlinearity. Therefore, we cannot directly apply the argument of[6]because of the so-called “loss of derivative.” To overcome this difficulty, we introduce a modified energy and derive an a priori estimate for the approximate solutions. Our main result is as follows.

Theorem 1. For arbitrary M >Q²_L2(R²), there exist T =T (M) >0 and a solution (u, n) of (Z) in the class (u, n, nt)∈C([0, T );H¹(T²)×L²(T²)× ˆH⁻¹(T²))with the following properties.

(i) u(t )²_L2(T²)< M.

(ii) C1(T−t )⁻¹u(t )H¹(T²)+ n(t )L²(T²)+ n_t(t)_Hˆ−1(T²)C2(T−t )⁻¹for someC1, C2>0.

(iii) limt→T∇u(t )L²(T²\B(0,r)) =0, limt→Tn(t )L²(T²\B(0,r)) =0 for any r > 0 sufficiently small, where B(a, r):= {x∈T²| |x−a|< r}.

Our approach is easily generalized to the case of exactlypblow-up points. A similar generalization was mentioned by Godet[15]for (NLS).

Corollary 1.Let{x₁, . . . , x_p}be distinct points inT². For arbitraryM > pQ²_L2(R²), there existT =T (M) >0and a solution(u, n)of(Z)in the class(u, n, n_t)∈C([0, T );H¹(T²)×L²(T²)× ˆH⁻¹(T²))with the following properties.

(i) u(t )²_L2(T²)< M.

(ii) C₁(T−t )⁻¹u(t )H¹(T²)+ n(t )L²(T²)+ n_t(t)_Hˆ−1(T²)C₂(T−t )⁻¹for someC₁, C₂>0.

(iii) lim

t→T(T−t )∇u(t )

L²(B(xj,r))>0, lim

t→T(T−t )n(t )

L²(B(xj,r))>0, and

tlim→T

∇u(t )

L²(T²\p

j=1B(x_j,r))=0, lim

t→T

n(t )

L²(T²\p

j=1B(x_j,r))=0 for anyr >0sufficiently small and any1jp.

The global existence of the solution for the case u₀L²(T²)QL²(R²) is a corollary of the following mass concentration result.

Theorem 2.Suppose(u, n)is the solution of(Z)which blows up att=T ∈(0,∞). Then, there existt_n→T and y_n∈T²such that

lim inf

n→∞

|x−yn|<R

u(t_n, x) ²dxQ²_L2(R²)

for anyR >0.

Corollary 2.If(u₀, n₀, n₁)∈H¹(T²)×L²(T²)×H⁻¹(T²)satisfiesu₀L²(T²)QL²(R²), then the corresponding solution of(Z)exists globally in time.

Corollary2can be derived by using the same argument as Glangetas and Merle with a sharp Gagliardo–Nirenberg inequality onT²by Ceccon and Montenegro[9]. However, for the proof of Theorem2, it is not sufficient by itself to replace the sharp Gagliardo–Nirenberg inequality onR²with that onT². This is because the terms∇u²_L2andu²_L2

appearing in the Gagliardo–Nirenberg inequality on T² have different scalings. Therefore, we use a concentration compactness argument and splituin many pieces so that we can use the Gagliardo–Nirenberg inequality onR².

This paper is organized as follows. In Section 2, we formulate the perturbed equation which we have to solve to construct a finite time blow-up solution. In Section3, we construct an approximate solution by regularizing the perturbed equation. In Section4, we introduce a modified energy and derive an a priori estimate for the approximate solution. This estimate will allow us to construct a solution to the perturbed equation. Also, the idea for multi-point blowup is given in Section 4. In Section5, we prove Theorem2and Corollary2. In Appendix A of this paper, for readers’ convenience, we give a brief sketch of the proof of the modified concentration compactness lemma which we will use for the proof of Theorem2.

We define some notations which we use in the following. We denote the Fourier series ofu(t, x) in the spatial variable as

u(t, x)=

m∈Z²

e^imxu(t, m).ˆ

We define the Sobolev spacesH^k(T²)fork∈Ras H^k

T² :=

u∈D

T² uH^k<∞ , u²_Hk:=

m∈Z²

m^2k u(m) ˆ ²,

wherex :=(1+ |x|²)^1/2. We writeABto denote the estimateACBwith a constantC >0, which may depend on some parameters in a harmless way, and writeA∼BifABA. We use the notations likeε,λwhen we need to emphasize the dependence of constants on some parameters.

2. Formulation

First of all, we recall the result by Burq, Gérard, and Tzvetkov[6], which constructed blow-up solutions to (NLS) onT². (NLS) onR²has a family of explicit blow-up solutions{ ˜R_λ}λ>0which blow up att=T, where

R˜λ(t, x)= 1 λ(T −t )eⁱ⁽

1

λ2(T−t)−_4(T^|x|₋²_t))

Q x

λ(T−t )

,

andQis given in (2). Letψ∈C_0,r^∞(R²)be such that 0ψ1, suppψ⊂ {|x|<2}andψ (x)=1 for|x|<1, then the functionψR˜_λ(t), which is restricted on a ballB(0,2)⊂ [−π, π]², can be regarded as a function onT². Consider the function

u(t, x)=ψ (x)R˜_λ(t, x)+v(t, x)

withv: [0, T] ×T²→C. Then,uis a blow-up solution to (NLS) onT²ifvsolves the equation

(i∂_t+)v=Q_2,3(v)−ψ²R˜²_λv¯+ | ˜R_λ|²v +

1−ψ²

ψ| ˜R_λ|²R˜_λ−2∇ψ∇ ˜R_λ−ψR˜_λ,

v(t )→0 ast→T , (3)

where quadratic and cubic terms with respect tovhave been written asQ2,3(v).

Applying the fixed point argument to the associated integral equation, one can solve (3), for example, inH²(T²).

First, notice that the external force in (3) decays exponentially ast→T, namely 1−ψ²

ψ R˜λ(t) ²R˜λ(t)−2∇ψ∇ ˜Rλ(t)−ψR˜λ(t)

H²e^{−λ(Tδ−t)}

for someδ >0. Thus, we can expect that the solutionvalso decays exponentially ast→T.

This decay ofexponentialorder is essential for the treatment of the linear terms ψ²(R˜²_λv¯+ | ˜R_λ|²v)in the fixed point argument. To see this, we assumev(t )L²∼e⁻

μ

λ(T−t) for someμ >0 and consider the estimate for theL²norm of the Duhamel integral term, then

T

t

e^i(t−s)

(ψR˜_λ)²v (s) ds

L²

T

t

ψR˜_λ(s)²

L^∞v(s)

L²ds

∼ T

t

1 λ²(T−s)²e⁻

μ λ(T−s)ds

= 1 μλe⁻

μ λ(T−t)∼ 1

μλv(t )

L².

From the above estimate, the linear terms seem to be harmless (at least inL²) wheneverμλis sufficiently large. In fact, under the assumption thatλis sufficiently large, Burq, Gérard, and Tzvetkov obtained an exponentially decaying solution by performing the fixed point argument in the spaceC([0, T );H²)with an appropriate weight int which grows exponentially ast→T. Note that anypolynomialdecay inT −t of solution will not be sufficient for us to close the fixed point argument.

Let us return to the Zakharov system (Z) and take the same approach as (NLS). Let(Pλ, Nλ):R²→R² be a radially symmetric solution of

−P_λ+P_λ=N_λP_λ, λ²

r²∂_rrN_λ+6r∂rN_λ+6Nλ

−N_λ=|P_λ|², (4) wherer= |x|. Then it is easy to check that(u, n)defined as

u(t, x)= 1 λ(T−t )eⁱ⁽

1

λ2(T−t)−4(T^|x|−²t))

P_λ x

λ(T −t )

,

n(t, x)= 1

λ(T−t ) 2

N_λ x

λ(T −t )

is a solution of (Z) in R² which blows up as t →T. It was shown by Glangetas and Merle [12]that for λ >0 sufficiently small Eq. (4) actually has a solution with the following properties.

Proposition 1. (See[12].) There exists a family of radially symmetric solutions{(P_λ, N_λ)}0<λ1 to(4) such that (P_λ, N_λ)→(Q,−Q²)inH¹(R²)×L²(R²)asλ→0. Further,(P_λ, N_λ)∈H^k×H^k for allk∈N∪ {0}and

P_λ^(k)(x) ke^−δ|x|, N_λ^(k)(x) kx^−(3+k) for someδ >0.

These blow-up solutions are similar to the solutionsR˜_λ of (NLS), but different from them in the following two points:

– A largeλis not allowed.

– The solutions for the wave part decay only polynomially int.

From time reversal symmetry, it suffices to consider solutions which blow up backward in time att=0. For small λ >0, let

U˜λ(t, x):= 1 λte⁻ⁱ⁽

1 λ2t−^|x4t^|2)

Pλ

x λt

, W˜_λ(t, x):= 1

(λt )²N_λ x

λt

be the blow-up solution of (Z) inR²constructed in[12]. With the cut functionψdefined above, set U_λ(t, x):=ψ (x)U˜_λ(t, x),

W_λ(t, x):=ψ (x)W˜_λ(t, x), (t, x)∈R× [−π, π]²R×T².

We construct a blow-up solution of the form(U_λ+u, W_λ+w), where(u, w)does not blow up in the energy space ast→0. Assuming that(U_λ+u, W_λ+w)solves (Z), we obtain

(i∂_t+)u=uw+(W_λu+U_λw)+

(ψ−1)ψU˜_λW˜_λ−2∇ψ∇ ˜U_λ−ψU˜_λ ,

(∂_{t t}−)w=|u|²+(U¯_λu+U_λu)¯ +F_λ, (5)

where

F_λ:=

|U_λ|²

−ψ

| ˜U_λ|²

+2∇ψ∇ ˜W_λ+ψW˜_λ

=(ψ−1)ψ

| ˜Uλ|² +2∇

ψ²

∇

| ˜Uλ|² +

ψ²

| ˜Uλ|²+2∇ψ∇ ˜Wλ+ψW˜λ.

Here, the first difficulty arises due to the fact that the external force termF_λ(t)only decays polynomially. We can thus expect only polynomial decay forw, then the same forU_λwin the Schrödinger part, then the same foru, which would not be enough to remove the singularity in the termW_λu.

The idea to overcome this difficulty is to decomposewinto polynomially decaying part and exponentially decay- ing part. Note that the slowly-decaying external forceF_λ(t)is restricted outside a ballB(0,1). The finite speed of propagation then suggests that the slowly-decaying part ofw(t)also vanishes around the origin for a short time. Since U_λ has an exponential decay outside a neighborhood of the origin, we can still expect the exponential decay for the productU_λw.

To make this argument rigorous, letZ_λ,a(t, x)be the solution of the following inhomogeneous linear wave equation fora∈R:

(∂_{t t}−)Z_λ,a=F_λ,

Z_λ,a(0, x)=0, ∂_tZ_λ,a(0, x)=aψ (x)

1−ψ (x)

. (6)

Note thatZ_λ,ais explicitly written as

Z_λ,a(t, x)= sin(t|∇|)

|∇|

aψ (1−ψ ) (x)−

t 0

sin((t−s)|∇|)

|∇| F_λ(s, x) ds

=:Ψa(t, x)+Zλ(t, x).

A direct calculation using Proposition1shows thatZ_λ,a∈C¹([0,∞);H^k(T²))for anyk0 and sup

0<t <T

t⁻¹Z_λ,a(t)

H^k(T²)+∂_tZ_λ,a(t)

H^k(T²)

k,T ,λ,a1 (7)

forT >0. Moreover, both the external force term and the initial data in (6) vanish in a ballB(0,1), which together with the finite speed of propagation yields thatZ_λ,a(t, x)≡0 on a ballB(0,1/2)for 0< t <1/2. Actually any initial data that vanish around the origin may be sufficient for the fixed point argument, but we have selected the above ones for another reason to be mentioned below.

We shall construct a blow-up solution of the form(U_λ+u, W_λ+Z_λ,a+z), where(u, z)converges to 0 ast→0, solving

(i∂t+)u=uz+(Wλ+Zλ,a)u+Uλz+

UλZλ,a+(ψ−1)ψU˜λW˜λ−2∇ψ∇ ˜Uλ−ψU˜λ

,

(∂_{t t}−)z=|u|²+(U¯_λu+U_λu).¯

(8) We notice that the external force in the Schrödinger part decays exponentially ast→0.

It is easy to see that a solution(u, z)to the above problem satisfies

T²

z_t(t, x) dx=czˆ_t(t,0)≡0

for allt. Thus,|∇|⁻¹ztcan be defined by

m=(0,0)e^imx|m|⁻¹zˆt(t, m). Set r=z+i|∇|⁻¹zt.

Then, if(u, z)solves (8),(u, r)satisfies

(i∂_t+)u=uRer+(W_λ+Z_λ,a)u+U_λRer+

U_λZ_λ,a+(ψ−1)ψU˜_λW˜_λ−2∇ψ∇ ˜U_λ−ψU˜_λ ,

i∂_t− |∇|

r= |∇|

|u|²+ ¯U_λu+U_λu¯

. (9)

Sincezis real valued, we can recover the solution of (8) from that of (9) byz:=Rer. In this casez_t = |∇|(Imr) holds, and(z, z_t)∈C((0, T];L²(T²;R)× ˆH⁻¹(T²;R))if and only ifr∈C((0, T];L²(T²;C)).

We will construct a local solution to this problem (9) that decays exponentially inH¹(T²)×L²(T²)ast→0.

Theorem 3.For anya∈Rand sufficiently smallλ >0, there existsT =T (λ, a) >0such that Eq.(9)has a solution (u, r)∈C((0, T], H¹(T²)×L²(T²))which decays exponentially ast→0.

Here, a solution of (9) means a distributional solution of the associated integral equation of (9).

Now, we admit Theorem3for a moment and show Theorem1.

Proof of Theorem1. Recall that we have replaced the forward blowup att=T with the backward blowup att=0.

For givenM >Q²_L2(R²), we chooseλ >0 so thatP_λ²_L2(R²)< M, which is possible from Proposition1. Next, in the following way, we choosea∈Rso that(W_λ+Z_λ,a)_t∈ ˆH⁻¹. We first notice that

Ψˆ_a,t(t,0)≡a

ψ (1−ψ ) ˆ(0) and[ψ (1−ψ )]ˆ(0)=c

ψ (1−ψ ) dx >0. We also see from the equation that[(Wλ+Zλ)t]ˆ(t,0)is conserved.

Then, we choosea∈Rso that[(Wλ+Zλ)t]ˆ(t,0)+a[ψ (1−ψ )]ˆ(0)=0. With this choice ofλanda, we set our blow-up solution of (Z) as(u, n)=(U_λ+v, W_λ+Z_λ,a+Rer), where(v, r)is the solution of (9) (withureplaced byv) obtained in Theorem3. Note that this solution belongs to the energy space.

It is easily verified by theL²conservation law for (Z) and the monotone convergence theorem that

T²

u(t, x) ²dx=lim

t→0

R²

ψ (x)U˜_λ(t, x) ²dx=

R²

P_λ(x) ²dx < M. (10)

Hence, we have proved (i).

To prove (ii) we claim the following stronger estimates: for 0< t1, ∇u(t )

L²(T²)∼t⁻¹, (11)

n(t )

L²(T²)∼t⁻¹, (12)

n_t(t)_ˆ

H⁻¹(T²)∼t⁻¹. (13)

For (11) and (12) it is sufficient to consider the main partsU_λ(t)andW_λ(t), respectively. Similarly to (10), we have

tlim→0(λt )²

T²

W_λ(t, x) ²dx=

R²

N_λ(x) ²dx >0,

which implies (12). Also, we have ∇Uλ(t, x) ²=

ψ (x) (λt )²e⁻ⁱ⁽

1 λ2t−^|x4t^|2)

(∇Pλ) x

λt

+iψ (x)x 2λt² e⁻ⁱ⁽

1 λ2t−^|x4t^|2)

Pλ

x λt

+(∇ψ )U˜λ

².

Since

tlim→0(λt )²

T²

ψ (x) (λt )²(∇P_λ)

x λt

²dx=

R²

∇P_λ(x) ²dx

and

T²

ψ (x)x 2λt² P_λ

x λt

²dx λ

2 2

R²

1 λt

x λtP_λ

x λt

²dx= λ

2 2

xP_λ²_L2,

T²

(∇ψ )(x)U˜_λ(t, x) ²dx∇ψ²_L∞U˜_λ(t)²

L²= ∇ψ²_L∞P_λ²_L2, we conclude that

tlim→0(λt )²

T²

∇U_λ(t, x) ²dx=

R²

∇P_λ(x) ²dx >0, which implies (11).

For (13), it suffices to consider(W_λ+Z_λ,a)_t(t)instead ofn_t(t). We see that (W_λ)_t(t, x)= −2ψ (x)

λ²t³ N_λ x

λt

−ψ (x)x λ³t⁴ (∇N_λ)

x λt

= −ψ (x)∇ 1

λt² x λtN_λ

x λt

=(∇ψ )(x) 1

λt² x λtNλ

x λt

− ∇ ψ (x)

λt² x λtNλ

x λt

. Note that the second term is inHˆ⁻¹, then

(W_λ+Z_λ,a)_t(t)_ˆ

H⁻¹= ∇

ψ (x) λt²

x λtN_λ

x λt

_ˆ

H⁻¹

+O ∇ψ·

1 λt²

x λtN_λ

x λt

L²

+(Z_λ,a)_t(t)

L²

. Since∇ψ (x)≡0 for|x|<1, we have

t²

T²

(∇ψ )(x) 1

λt² x λtN_λ

x λt

²dxt²

R²

1 λ^1/2t^3/2

|x| λt

³

2

N_λ x

λt ²dx

=λt|x|³²N_λ²

L²→0 (t→0).

Recalling the definition of theHˆ⁻¹norm (Remark1), we also have

tlim→0t ∇

ψ (x) λt²

x λtN_λ

x λt

_ˆ

H⁻¹

=lim

t→0

ψ (x) λt

x λtN_λ

x λt

L²(T²)

= xN_λL²(R²),

where the first equality follows from the fact that ^ψ(x)_{λt2 λt}^xN_λ(_λt^x)∈Gfor allt >0, which is verified by observing that

ψ(x)

λ²t³Nλ(_λt^x)=:f (|x|)is spherically symmetric andxf (|x|)= ∇(_|x|

0 rf (r) dr+C). This concludes (13).

(iii) follows from a similar argument to the proof of (11) and (12). For instance,

T²\B(0,r)

ψ (x) (λt )²(∇P_λ)

x λt

²dx

T²

ψ (x)(1−ψ (2x/r)) λt r

|x| λt (∇P_λ)

x λt

²dx,

which goes to 0 ast→0 by the dominated convergence theorem, giving the claim for∇u. We make a similar argument fornand obtain (iii). 2

All we have to do is to solve (9). However, compared to the case of (NLS), there are two major difficulties left:

(i) the loss of one derivative in the equation, and (ii) how to control the linear termW_λu(∼(λt )⁻²u) without assuming λto be large.

When we construct solutions in the spaceH^k×H^k−1, the loss of one derivative appears in the Schrödinger part and prevents us from applying the usual fixed point argument. We shall employ the method of parabolic regularization to overcome this issue. This method is also helpful in treating another issue (ii), because the viscosity effect will ease the singularity ofWλand give an extra small factorT⁰⁺to the corresponding term in the estimate. The details will be discussed in Section3.

What is the most important is then the a priori estimate for the approximate solutions constructed via the parabolic regularization. We meet the difficulties (i) and (ii) here again.

If we use the standard energy estimate, we will have only the estimate of _dt^du(t )²_Hk in terms ofu(t )H^k and r(t )_H^k, which forces us to assume one more regularity forr(t ). To obtain the a priori estimate in H^k ×H^k−1, we shall introduce a “modified energy.” More precisely, we modify the standard energy (theH^k×H^k−1norm of solutions) with harmless terms so that in the estimate of the time derivative of them the term including∇^kr(t )will be canceled (see Section4for details). This approach was recently taken by Kwon[18]and by Segata[32]for the fifth order KdV equation and the fourth order nonlinear Schrödinger equations with a derivative in the nonlinear term, respectively. Note that this kind of modification on energy has a lot of ideas in common with the concept of “correction terms” in the context of theI-method introduced in a series of papers by Colliander, Keel, Staffilani, Takaoka, Tao.

Concerning (ii), the factW_λ(t, x)∈Rwill be essential. For instance, when we derive the identity for _dt^du(t )²_L2

the term corresponding toW_λu will not appear. Similarly, in the estimate of _dt^d∇^ku(t )²_L2, there will be the terms like

T²

∇^lW_λ(t)∇^k−lu(t )∇^ku(t ) dx¯ (14)

for l=1,2, . . . , k, but the term corresponding to l=0 will vanish. On the other hand, note that ∇^lW_λL^∞ (λt )^−2−l. Then, if∇^k−lu(t )_L² has a decay faster thant^l∇^ku(t )_L² for eachl=1,2, . . . , k, we can obtain the extra small factor again and control (14) by shrinking the time interval. We will actually construct solutions with such a property, by carefully choosing the weight function in the norm for fixed point argument. The precise definition of the norm will be given in Section3.

Remark 2.In[30], Ozawa and Tsutsumi proved the local well-posedness of the initial value problem for the Zakharov system onR^d (d =1,2,3) in the spaceH²×H¹×L². They pointed out that the loss of derivative does not occur when the Zakharov system is considered as the system of equations for∂_tuandn. This technique may be a solution to our difficulty (i), but it seems difficult to settle another issue (ii) by this idea. That is why we employ the method of parabolic regularization.

3. Parabolic regularization

We first look for solutions(u_ε, r_ε)to a regularized equation

⎧⎪

⎨

⎪⎩

i∂_t++iε²

u=uRer+(W_λ+Z_λ,a)u+U_λRer +

U_λZ_λ,a+(ψ−1)ψU˜_λW˜_λ−2∇ψ∇ ˜U_λ−ψU˜_λ ,

i∂_t− |∇| +iε²

r= |∇|

|u|²+ ¯U_λu+U_λu¯ ,

(15)

forε >0 in the space XT_ε=

(u, r)∈C

(0, Tε];H³ T²

×C

(0, Tε];H² T²

: (u, r)

X_Tε<∞ (u, r) ,

X_Tε= sup

t∈(0,Tε]H[u, r](t),

H[u, r](t)²:=

e^2λtμ u(t )_˙

H³(T²)

2

+

t⁻⁴e^2λtμu(t )

L²(T²)

2

+

t⁻²³e^2λtμ r(t )_˙

H²(T²)

2

+

t⁻¹⁰³e^2λtμ r(t )

L²(T²)

2

, whereμ >0 is the constant to be given in Lemma3.

Remark 3.Note that by a simple interpolation, we have sup

t∈(0,Tε]e^2λtμ

t⁻⁸³uH¹+t⁻⁴³uH²+t⁻²rH¹

C(u, r)

X_Tε. We prepare several lemmas.

Lemma 1.LetV_ε(t)u₀be the solution of i∂_t++iε²

u=0, u(0)=u₀. Then, we have

t

0

Vε(t−s)u(s) ds H^k

ε^−4l t 0

(t−s)^−4luH^k−l.

Proof. From the definition ofV_ε, we have t

0

Vε(t−s)u(s) ds= t 0

m∈Z²

e^{imx−i|m|2(t−s)−ε|m|4(t−s)}uˆm(s) ds.

Therefore, we have

t

0

V_ε(t−s)u(s) ds H^k

t 0 m∈Z²

m^2ke^{−ε|m|4(t−s)} uˆ_m(s) ²¹

2

ds

t 0

sup

˜

m∈Z² ˜m^le^{−ε| ˜m|4(t−s)}

m∈Z²

m^2(k−l) uˆm(s) ²¹₂ ds

ε^−4l t 0

(t−s)^−4luH^k−l,

where we have used sup

˜

m∈Z² ˜m^le^{−ε| ˜m|4(t−s)}ε^−4l(t−s)^−l4. 2 We estimate theL^pnorms of∇^kU_λand∇^kW_λ.

Lemma 2.Letk0,λsufficiently small andp∈ [1,∞]. Then we have ∇^kUλ

L^p(λt )^−k−1+p2, ∇^kWλ

L^p(λt )^−k−2+2p.

Proof. By direct calculation using the definition ofU_λ,W_λand Proposition1, we have the conclusion. 2 We next estimate the inhomogeneous term of the Schrödinger part of (15).