薛丁格方程的 Strichartz 估計與長波短波交互作用方程式的半古典極限

國 立 交 通 大 學

應 用 數 學 系

碩士論文

薛丁格方程的 Strichartz 估計與

長波短波交互作用方程式的半古典極限

Strichartz Estimates for Schrödinger Equation and

Semiclassical Limit of the Long Wave-Short Wave

Interaction Equations

研 究 生：陳家豪

薛丁格方程的 Strichartz 估計與

長波短波交互作用方程式的半古典極限

Strichartz Estimates for Schrödinger Equation and

Semiclassical Limit of the Long Wave-Short Wave

Interaction Equations

研 究 生：陳家豪 Student: Jia-Hao Chen

指導教授：林琦焜 Advisor: Chi-Kun Lin

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

in partial Fulfillment of the Requirements

for the Degree of

Master

in

Applied Mathematics

June 2010

薛丁格方程的 Strichartz 估計與

長波短波交互作用方程式的半古典極限

研究生：陳家豪 指導教授：林琦焜 教授

國立交通大學應用數學系碩士班

摘 要

此篇文章分為兩個部分。第一部分主要討論薛丁格方程上的

Strichartz 估計，我們先從量綱分析的角度觀察不等式中指數對

(p,q) 所需滿足的關係式，再給予嚴格的証明。從而結論在推導中可

允許的 (p,q) 符合量綱分析的結果。

第二部分討論長波短波交互作用方程式的半古典極限。首先利用

Madelung 轉換，討論方程式的流體結構與守恆律。再透過修正的

Madelung 轉換與能量估計，證明局部古典解的存在性與唯一性。最

後證明半古典極限解的存在性。

Strichartz Estimates for Schrödinger Equation and

Semiclassical Limit of the Long Wave-Short Wave

Interaction Equations

Student: Jia-Hao Chen Advisor: Chi-Kun Lin

Department of Applied Mathematics

National Chiao Tung University

ABSTRACT

There are two parts in this paper. In part I, we discuss the Strichartz estimates on Schrödinger equation. First, we observe the restrictions on exponent pair (p,q) from the viewpoint of dimension. Then we also provide a rigid proof, and conclude that the so-called admissible pair coincides with the arguments of dimensional analysis.

In part II, we study the semiclassical limit of the three coupled long wave-short wave interaction equations. First, we employ the Madelung transformation to discuss the hydrodynamical structures and the conservation laws. Then, we apply the

modified Madelung transformation and energy estimates to justify the existence and uniqueness of the local classical solution. Finally, we prove the existence of the semiclassical limit of the solution.

誌

誌

誌

誌

謝

謝

謝

謝

首先最要感謝的是我的指導教授林琦焜老師。老師總教我們如何培養直 觀，從最自然的角度看問題，以及老師有一套數學上的哲學思想，我想這對我們 在自然的探索上是一生受用的。老師在交通大學的開放式課程中還分享了很多學 習資源，包含影音課程與課程講義，在傑出研究之餘仍不忘在教學上努力，且其 無私奉獻的精神自然也是令人敬佩的。 還要感謝在碩士班教我實變的王夏聲老師。老師的上課方式是吸引人的，而 其具體表現在我在教室內座位與黑板的距離，學期初我坐在最後一排，到學期末 我坐在第二排(第一排往往是沒坐人的)。另外還要感謝江鑑聲老師，江老師是同 門師兄，待人和善親切，也多次來交大演講，其中讓我獲益良多。吳恭檢是同門 的博士班學長，不論在研討會上或是在私底下與學長的交談中都獲得相當多的寶 貴知識，其數學能力自然是不用多說的，往後出去也一定是位傑出的數學家。再 者要感謝蔡佳穎同學，佳穎與我ㄧ起在林琦焜老師底下學習，她的學習態度積 極、堅毅，是我最佳的學習夥伴。 此篇文章中所提及參考文獻的作者個個都是在該領域中偉大的人物，這些作 者提供了富饒的研究成果，指引著我學習方向，除了敬佩，特此也表達感謝之意。 楊雅如小姐也在我寫作期間幫我檢查英文語法上的問題，沒有她的幫忙，此 篇文章就不算完整。最後要感謝我的家人，從小家裡爸媽就很注重教育，不僅僅 在學業上，更是在待人處事上對我都有所期許，是家人成就了現在的我。 在學習的路上總覺得受之於人太多，在此也期許自己，當自己也有機會教育 別人的時候，你們都是我最好的榜樣，從你們身上所得到的再回報給其他人、下 一代。由衷感謝大家。

Contents

PartI Strichartz Estimates for Schrödinger Equation

1 Introduction ………... 1

2 Preliminaries ………. 2

2.1 Dimensional Analysis ………. 2

2.2 Decay Estimates, Other Inequalities ……….. 4

3 Proof of Theorem 1.2 ……… 5

4. Remarks ……… 6

Part II Semiclassical Limit of the Long Wave-Short Wave

Interaction Equations

5 Introduction……… 8

6 Hydrodynamical Structures and Conservation Laws ……… 9

7 Semiclassical Limit ………. 13

Part I

Strichartz Estimates for

Schr¨

odinger Equation

1

Introduction

In the part I of this paper, we consider the solution of the initial value problem for the nonhomogenerous Schr¨_{odinger equation in R}n

∂tu(t, x) = i∆u(t, x) + f (t, x) (t, x) ∈ [0, T ] × Rn, (1.1)

u(0, x) = u0, (1.2)

where T > 0, ∆ = ∂2

x1 + · · · + ∂

2

xn and f (t, x) is a real-valued function.

By Duhamel principle, the solution u to (1.1),(1.2) can be described as the following integral equation

u(t, x) = eit∆u0(x) +

Z t 0

ei(t−s)∆f (s, x)ds (1.3)

where the operator eit∆ is defined as eit∆u0(x) =  e−4π2it|ξ|2u_b0(ξ) ∨ = e −|x|2_4it (4πit)n2 ∗ u0(x). (1.4)

The main subject here is to earn more inequalities, known as Strichartz estimates, from some existing decay estimates. We have the following results [3, 17] to answer the above question. Before that, we introduce the notion of admissible pair.

Definition 1.1. (1) We say that the exponent pair (p, q) is admissible if n p + 2 q = n 2 (1.5) and    2 6 p 6 ∞ f or n = 1, 2 6 p < ∞ f or n = 2, 2 6 p 6 _n−22n f or n > 3. (1.6) (2) We say that the exponent pair (p, q) is an endpoint if



(p, q) = (∞, 2) f or n = 2,

Theorem 1.2 (Strichartz estimates). For admissible pair (p, q), we have (1) eit∆_u 0 Lq_tLpx 6 c1ku0kL 2 x. (1.8) (2) Z ∞ −∞ eit∆f (t, x)dt L2 x 6 c2kf (t, x)kLq0_t Lp0x . (1.9) (3) Z ∞ −∞ ei(t−s)∆f (s, x)ds Lq_tLpx 6 c3kf (t, x)kLq0tL p0 x . (1.10)

This paper is organized as follows. In section 2, we collect some important preliminaries, including dimensional analysis which provides us an intuitional point of view to treat the equations and inequalities. Furthermore, it gives us a glance why we need the assumption, like the admissible pair. We also provide a rigid proof in section 3. In section 4, there are some remarks on Strichartz estimates.

Notations. Lp_(Rn_{), 1 6 p < ∞, represents the Lebesgue space with}

norm kf kLp = R Rn|f | p_dx
1_p . L∞_(Rn_{) is with norm kf k} L∞ = ess sup Rn|f |.

The mixed Lebesgue space Lq_tLp_{x(I × R}n) = Lq(I; Lp_(Rn_{)), 1 6 q < ∞,} con-sists of f : I → Lp x with kf kLq_tLpx =  R Ikf (t)k q Lpxdt 1_q < ∞. L∞_t Lp x(I × Rn) = L∞(I; Lp_(Rn_{)) consists of f : I → L}p x with kf kL∞t L p

x = ess supt∈Ikf kLpx < ∞.

2

Preliminaries

2.1

Dimensional Analysis

Dimensional analysis is employed extensively in many fields in science espe-cially physics and mathematics [7]. Here we establish some knowledge about applications on mathematical analysis.

Proposition 2.1 (Operation). We star from two basic operations, differ-entiation and integration. The notation [ · ] stands for the dimension of a function. (1)  d k_f dxk  = 4f (4x)k (1.11) (2) Z Rn f dx  = (4f )(4x)n _(1.12)

Proposition 2.2 (Function space). We use the notation ≈ to describe the dimension of a function space.

(1) (Lp). If f ∈ Lp_(Rn), then kf kLp = Z Rn |f |p_dx 1_p < ∞. Hence [(4f )p(4x)n]1p _{= (4f )(4x)} n

p_{, and formally we say}

Lp ≈ n

p. (1.13) (2) Wk,p
_{. If f ∈ W}k,p_(Rn_{), then roughly we say that}

dk_f dxk _Lp = Z Rn dk_f dxk p dx 1p < ∞. Hence (4f )(4x)−kp(4x)n 1_p

= (4f )(4x)np−k_{, and formally we say}

Wk,p ≈ n

p − k. (1.14) Proposition 2.3 (Differential equation). A differential equation basically is an equality. If it makes sense, the dimension must be balanced. There, we can acquire some properties of this equation before applying any mathematical techniques. For example, the Schr¨odinger equation

∂tu = i∆u.

Matching the dimension on both sides, we have 4u 4t = 4u (4x)2 or 4t = (4x)2 (1.15) This characterizes the relation between time variable and space variable in some sense.

Proposition 2.4 (Inequality). In mathematical analysis, we usually need various inequalities to estimate our solutions of equations. These inequalities usually have annoying restrictions on its exponents. For example, the H¨older inequality: if 1 6 p, q 6 ∞,1p + 1 q = 1 and f ∈ L p_{(Ω), g ∈ L}q_{(Ω) then} Z Ω |f g|dx 6 kfkLpkgk_Lq.

Checking the dimensions, we have

(4f )(4g)(4x)n = (4f )(4x)np_(4g)(4x) n q_.

2.2

Decay Estimates, Other Inequalities

In the following we present useful estimates in studying of Schr¨odinger equa-tions as well as Strichartz estimates.

Proposition 2.5. Let the operator eit∆ _{be defined as (1.4) and t 6= 0, then}

(1) (L1_{− L}∞_). eit∆_f L∞ 6 c4|t| −n 2kf k_L1. (1.16) (2) (L2_{− L}2_). eit∆_f L2 = kf kL2. (1.17) (3) (Lp0− Lp_). eit∆f Lp 6 c5|t| −n 2  1 p0− 1 p  kf k_Lp0, (1.18) if 1 p+ 1 p0 = 1 and p 0 _{∈ [1, 2].}

Proof. (1) By Young’s inequality. (2) By the nature of Fourier transform.

(3) Together with (1),(2) and Riesz-Thorin theorem.

Proposition 2.6 (Hardy-Littlewood-Sobolev inequality). Let 0 < α < n, 1 < p < q < ∞ with n_q + α = n_p, then kIαf kLq = cα Z Rn f (y) |x − y|n−αdy Lq 6 cα,n,pkf kLp, (1.19) where cα = Γ n−α₂
πn22αΓ α 2
.

We ignore the proof. However, from the viewpoint of dimension, we have (4f )(4x)−(n−α)_(4x)nq

(4x)n 1_q

= [(4f )p_(4x)n_]1p_{. Thus, the exponent}

(p, q) satisfies n_q + α = n_p.

Proposition 2.7 (Minkowski integral inequality). For 1 6 p < ∞, Z Rn f (x, y)dx _Lp y 6 Z Rn kf (x, y)kLpydx (1.20)

Proposition 2.8 (Riesz Representation theorem). Let 1 6 p < ∞ with

1

p +

1

q = 1. Then

(Lp(Ω))∗ = Lq(Ω). (1.21) To be more precise, every L ∈ (Lp(Ω))∗ is of the form

L(f ) = Z

Ω

f gdx ∀f ∈ Lp_{(Ω) (1.22)}

for a unique g ∈ Lq_{(Ω). Moreover, we have}

3

Proof of Theorem 1.2

Before setting to prove the theorem, we check the dimension of Theorem 1.2(a). We obtain that the exponent pair (p, q) satisfies n_p + 2_q = n₂.

Proof of Theorem 1.2.

We only give the proof of (p, q) which is non-endpoint, i.e. (p, q) 6= 

2n n − 2, 2

 for n > 3. As for endpoint estimates of admissible pair, we refer to [6]. (3) Employing Minkowski integral inequality, Lp0 _{− L}p _{estimate applying to}

space and Hardy-Littlewood-Sobolev inequality applying to time respectively, we have Z R ei(t−s)∆f (s, x)ds Lq_tLpx 6 Z R ei(t−s)∆f (s, x) Lpxds Lq_t 6 cn,p0 Z R 1 |t − s|n2( 1 p0− 1 p) kf (s, x)k_Lp0 xds Lq_t 6 cn,p0_,q0kf (s, x)k Lq0_t Lp0x. At Lp0 − Lp _{estimate, we need} 1 p + 1 p0 = 1, 1 6 p 0 < 2 < p 6 ∞ (for p = p0 = 2, we have (1.17)), and at Hardy-Littlewood-Sobolev inequality, we need

n 2  1 p0 − 1 p  > 0, 1 < q0 < q < ∞ and 1 q0 = 1 q+ α = 1 q +  1 −n 2  1 p0 − 1 p  . (2) By H¨older inequality and (3), we have

Z R eit∆f (t, x)dt 2 L2 x = Z Rn Z R eit∆f (t, x)dt  Z R eis∆_{f (s, x)ds}  dx = Z Rn Z R f (t, x) Z R ei(t−s)∆f (s, x)ds  dtdx 6 kf (t, x)k_Lq0 t L p0 x Z R ei(t−s)∆f (s, x)ds LqtL p x 6 cn,p0_,q0kf (t, x)k2 Lq0_t Lp0x . At H¨older inequality, we need 1

q+ 1

q0 = 1, and hence (p, q) satisfies

n p+ 2 q = n 2.

(1) Applying Fubini theorem, we have Z R Z Rn Z Rn ei|x−y|24t (4πit)n2 u0(y)dy ! f (t, x)dxdt = Z R Z Rn Z Rn ei|x−y|24t (4πit)n2 f (t, x)dx ! u0(y)dydt.

By Cauchy-Schwarz inequality and (2)     Z R Z Rn eit∆u0
(x)f (t, x)dxdt     =     Z Rn u0(x) Z R eit∆f (t, x)dt  dx     6 ku0kL2 x Z R eit∆f (t, x)dt _L2 x 6 cn,p0_,q0ku₀k_L2 xkf (t, x)kLq0t L p0 x .

Using Riesz Representation theorem, we conclude that eit∆u0 Lq_tLpx = sup kf k Lq0_t Lp0x =1     Z R Z Rn eit∆u0
(x)f (t, x)dxdt     6 cn,p0_,q0ku₀k_L2 x.

This completes the proof.

From the process of the proof that we establish, we learn that the inequal-ities must be dimensional balanced as well as the results of the theorem. The admissible pair inherits from all the restriction on the exponents of these inequalities. On the other hand, if we conjecture on a phenomenon ahead, then apply dimensional analysis on it. Observing the relations between the dimensions of the units, it also help us to learn more knowledge about the nature of the phenomenon. It even points the way to the proof.

4

Remarks

Here are some observations. First, 1_p and 1_q are linear with slope mn = −n₂,

for fixed n. The increase of p costs the decrease of q. Second, they all pass through (2, ∞) which also means that (2, ∞) is always admissible for all n. We portray as in Figure 1.

Finally, we end Part I by going back to the Theorem 1.2. If the initial datum u0 is given in L2x, the the solution u is in Lpx with p > 2. We gain

more integrability, that is the so-called smooth effect. This also reflects the dispersive nature of Schr¨odinger equation partially.

n p + 2 q = n 2 w, : admissible g: endpoint, non-adimissible ⊗ : endpoint, admissible slope mn= − n 2 -6 1 ∞ 1 6 1 4 1 2 1 p 1 ∞ 1 4 1 2 1 q w w g ⊗ ⊗ ⊗ H H H H H H H H H H H H H H H H H H H @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ J J J J J J J J J J J J J J J J J J J A A A A A A A A A A A A A A A A A A A n = 1 n = 2 n = 3 n = 4 (0, 0) (2, ∞) (∞, 4) (∞, 2) (6, 2) (4, 2) ( 2n n − 2, 2)

Figure 1: exponent pair Dimension n p + 2 q = n 2 Range of p Range of q n = 1 1 p+ 2 q = 1 2 2 6 p 6 ∞ 4 6 q 6 ∞ n = 2 1 p+ 1 q = 1 2 2 6 p < ∞ 2 < q 6 ∞ n > 3 n p + 2 q = n 2 2 6 p 6 2n n − 2 2 6 q 6 ∞ Tabular 1: admissible pair

Part II

Semiclassical Limit of the Long

Wave-Short Wave Interaction

Equations

5

Introduction

In the Part II, we consider the existence and uniqueness of solutions of the initial value problem for the three coupled long wave-short wave interaction (LSI) equations i~∂tψ~+~ 2 2 ∂xxψ ~ _{= β(|ψ}~_|2_{+ w}~_)ψ~ _(5.1) i~∂tφ~ +~ 2 2 ∂xxφ ~ _{= β(|φ}~_|2_{+ w}~_)φ~ _(5.2) ∂tw~ = β∂x |ψ~|2 + |φ~|2
(5.3) with initial values

ψ~_{(0, x) = ψ}~ 0(x) (5.4) φ~_{(0, x) = φ}~ 0(x) (5.5) w~_{(0, x) = w}~ 0(x) (5.6)

where β > 0, w~ _{is real-valued and ψ}~_{, φ}~ _{are complex-valued. w}~

char-acterizes the long wave and ψ~_{, φ}~ _{represent the short waves.This system}

describes the resonance when the group velocity of the short waves and the phase velocity of the long wave coincide.

In section 2, we employ the Madelung transformation to LSI equations (5.1)–(5.3) and rewrite them as a perturbation of the Euler equations. The conservation laws are also derived.

In section 3, we apply the modified Madelung transformation to LSI equa-tions (5.1)–(5.3) and rewrite them as a perturbation of a quasilinear hyper-bolic system. For suitable assumptions on initial data, there exists local classical solution to the quasilinear hyperbolic system as well as the LSI equations. Furthermore, the solution that we establish is uniformly bounded in ~. This allows us to pass to the limit ~ → 0.

Notations. Hs _{= W}s,2 _{represents the Sobolev space with norm kf k} Hs =

kf kWs,2 = P

α6sR |Dαf |2dx

1₂

where Dα_{f , the αth derivatives of f, exists}

in the weak sense. C([0, T ]; X) consists of f : [0, T ] → X with kf kC([0,T ];X)=

6

Hydrodynamical Structures and

Conserva-tion Laws

In this section, we will derive some conservation laws of the LSI equations (5.1)–(5.3) first. For further references (6.1)–(6.26),(6.46)–(6.51), we ignore the superscript ~.

By Madelung transformation, we introduce the complex-valued wave func-tions ψ = A1exp  iS1 ~  , (6.1) φ = A2exp  iS2 ~  , (6.2) where A1, A2, S1 and S2 are real-valued functions. A1, A2 are called the

amplitudes, and S1, S2 the classical actions. Substituting (6.1) (resp.(6.2))

into (5.1) (resp.(5.2)), (A1, S1, A2, S2) obeys the following equations

∂tA1+ ∂xA1∂xS1+ 1 2A1∂xxS1 = 0, (6.3) ∂tS1+ 1 2(∂xS1) 2 + βA2₁+ βw = ~ 2 2 ∂xxA1 A1 , (6.4) ∂tA2+ ∂xA2∂xS2+ 1 2A2∂xxS2 = 0, (6.5) ∂tS2+ 1 2(∂xS2) 2_{+ βA}2 2+ βw = ~2 2 ∂xxA2 A2 . (6.6) Consider the new variables

ρ1 ≡ A21, u1 ≡ ∂xS1, (6.7)

ρ2 ≡ A22, u2 ≡ ∂xS2, (6.8)

we have the following two conservation laws

∂tρ1+ ∂x(ρ1u1) = 0, (6.9) ∂tu1+ ∂x  1 2u 2 1+ βw  = ~ 2 2 ∂x ∂xx √ ρ1 √ ρ1 , (6.10) ∂tρ2+ ∂x(ρ2u2) = 0, (6.11) ∂tu2+ ∂x  1 2u 2 2+ βw  = ~ 2 2 ∂x ∂xx √ ρ2 √ ρ2 . (6.12)

Equations (6.9)–(6.12) have the form of a perturbation of the Euler equations with w satisfying ∂tw = β∂x(ρ1+ ρ2), (6.13) which is equivalent to w(t, x) = w0(x) + β Z t 0 ∂x(ρ1+ ρ2)dτ. (6.14)

Here (6.9) and (6.11) are conservation laws of mass. From (6.9), (6.10) (resp.(6.11), (6.12)), we can also derive the equation of the canonical mo-mentum ρ1u1 (resp. ρ2u2) ∂t(ρ1u1) + ∂x  ρ1u21+ β 2ρ 2 1  + βρ1∂xw = ~ 2 4 ∂x(ρ1∂xxlog ρ1), (6.15) ∂t(ρ2u2) + ∂x  ρ2u22+ β 2ρ 2 2  + βρ2∂xw = ~ 2 4 ∂x(ρ2∂xxlog ρ2), (6.16) which is not conservative. However, adding (6.15), (6.16) together and em-ploying (6.13), we have the conservation law of momentum as follows

∂t  ρ1u1+ ρ2u2− 1 2w 2  + ∂x  ρ1u21+ β 2ρ 2 1 + βρ1w + ρ2u22+ β 2ρ 2 2+ βρ2w  = ~ 2 4 ∂x(ρ1∂xxlog ρ1+ ρ2∂xxlog ρ2). (6.17) So far, we complete the conservation laws of mass and momentum. Next, we will seek for the conservation laws of energy. Multiply (6.9) by −1₂u2₁ and βw respectively, and (6.15) by u1, we have

−1 2u 2 1∂tρ1− 1 2u 2 1∂x(ρ1u1) = 0, (6.18) βw ∂tρ1+ βw ∂x(ρ1u1) = 0, (6.19) u1∂t(ρ1u1) + u1∂x  ρ1u21+ β 2ρ 2 1  + βρ1u1∂xw = ~ 2 4u1∂x(ρ1∂xxlog ρ1). (6.20) Summing (6.18), (6.19) and (6.20), we obtain

∂t  1 2ρ1u 2 1 + ~2 8 (∂xρ1)2 ρ1  + ∂x  1 2ρ1u 3 1+ ~2 8 u1(∂xρ1)2 ρ1 + βρ1u1w  + βw∂tρ1+ u1∂x  β 2ρ 2 1  = ~ 2 4∂x  ρ1u1∂xxρ1− ∂x(ρ1u1)∂xρ1 ρ1  . (6.21)

Also, from the symmetry point of view, we have ∂t  1 2ρ2u 2 2 + ~2 8 (∂xρ2)2 ρ2  + ∂x  1 2ρ2u 3 2+ ~2 8 u2(∂xρ2)2 ρ2 + βρ2u2w  + βw∂tρ2+ u2∂x  β 2ρ 2 2  = ~ 2 4∂x  ρ2u2∂xxρ2− ∂x(ρ2u2)∂xρ2 ρ2  . (6.22) Equations (6.21) and (6.22) are not in the conservative forms yet. Adding (6.21), (6.22) together and employing (6.13), we then have the conservation law of energy ∂t  1 2ρ1u 2 1+ ~2 8 (∂xρ1)2 ρ1 + β 2ρ 2 1+ βρ1w +1 2ρ2u 2 2+ ~2 8 (∂xρ2)2 ρ2 +β 2ρ 2 2+ βρ2w  + ∂x  1 2ρ1u 3 1+ ~2 8 u1(∂xρ1)2 ρ1 + βρ2₁u1+ βρ1u1w +1 2ρ2u 3 2+ ~2 8 u2(∂xρ2)2 ρ2 + βρ2₂u2+ βρ2u2w − β2 2 (ρ1+ ρ2) 2  = ~ 2 4 ∂x  ρ1u1∂xxρ1− ∂x(ρ1u1)∂xρ1 ρ1 − ρ2u2∂xxρ2− ∂x(ρ2u2)∂xρ2 ρ2  . (6.23) Define energy densities Eψ, Eφ by

Eψ = Eψ,1+ Eψ,2+ Eψ,3+ Eψ,4 ≡ 1 2ρ1u 2 1+ ~2 8 (∂xρ1)2 ρ1 + β 2ρ 2 1+ βρ1w, (6.24) Eφ= Eφ,1+ Eφ,2 + Eφ,3+ Eφ,4 ≡ 1 2ρ2u 2 2+ ~2 8 (∂xρ2)2 ρ2 + β 2ρ 2 2+ βρ2w, (6.25)

then we can rewrite (6.23) as ∂t(Eψ+ Eφ) + ∂x  (Eψ + Eψ,3)u1+ (Eφ+ Eφ,3)u2− β2 2 (ρ1+ ρ2) 2  = ~ 2 4 ∂x  ρ1u1∂xxρ1− ∂x(ρ1u1)∂xρ1 ρ1 − ρ2u2∂xxρ2− ∂x(ρ2u2)∂xρ2 ρ2  . (6.26) The total energy of the LSI equations (5.1)–(5.3) is constituted by the classi-cal part, Eψ,1+ Eφ,1 the kinetic energy, Eψ,3+ Eψ,4+ Eφ,3+ Eφ,4 the potential

The general problem of the semiclassical limit is to determine the limiting behavior of any function of the field ψ~_{, φ}~ _{and w}~ _{as ~ → 0. It is natural to}

conjecture that the dispersive term O(~2_{) which appears in (6.15) and (6.16)}

is negligible as ~ → 0 and the limiting density (ρ1, u1, ρ2, u2) satisfies the

limiting Euler system with initial values

∂tρ1+ ∂x(ρ1u1) = 0, (6.27) ∂t(ρ1u1) + ∂x  ρ1u21+ β 2ρ 2 1  + βρ1∂xw = 0, (6.28) ∂tρ2+ ∂x(ρ2u2) = 0, (6.29) ∂t(ρ2u2) + ∂x  ρ2u22+ β 2ρ 2 2  + βρ2∂xw = 0, (6.30)

with initial values

ρ1,0(x) = ρ1(0, x) = A21,0(x), (6.31) u1,0(x) = u1(0, x) = ∂xS1,0(x), (6.32) ρ2,0(x) = ρ2(0, x) = A22,0(x), (6.33) u2,0(x) = u2(0, x) = ∂xS2,0(x), (6.34) which w satisfies ∂tw = β∂x(ρ1+ ρ2), (6.35) w(0, x) = w0(x). (6.36)

This argument is self-consistent only if the limiting Euler system (6.27)– (6.36) remains classical. Furthermore, the limiting energy densities will be given by Eψ = Eψ,1+ Eψ,3+ Eψ,4 = 1 2ρ1u 2 1+ β 2ρ 2 1+ βρ1w, (6.37) Eφ= Eφ,1+ Eφ,3 + Eφ,4 = 1 2ρ2u 2 2+ β 2ρ 2 2+ βρ2w, (6.38)

and will satisfy

∂t(Eψ + Eφ) + ∂x  (Eψ+ Eψ,3)u1+ (Eφ+ Eφ,3)u2− β2 2 (ρ1+ ρ2) 2  = 0. (6.39)

Moreover we introduce the modified Madelung transformation as follows ψ = A1exp  iS1 ~  , (6.40) A1 = √ ρ1exp(iθ1), u1 = ∂xS1, (6.41) φ = A2exp  iS2 ~  , (6.42) A2 = √ ρ2exp(iθ2), u2 = ∂xS2, (6.43)

which A1and A2 are complex-valued. Plugging (6.40)–(6.43) into (5.1),(5.2),

(ρ1, θ1, u1, ρ2, θ2, u2) satisfies ∂tρ1+ ∂x(ρ1u1 + ~ρ1∂xθ1) = 0, (6.44) ∂tθ1+ u1∂xθ1+~ 2(∂xθ1) 2 = ~ 2 ∂xx √ ρ1 √ ρ1 , (6.45) ∂tu1+ u1∂xu1+ β∂x(ρ1+ w) = 0, (6.46) ∂tρ2+ ∂x(ρ2u2 + ~ρ2∂xθ2) = 0, (6.47) ∂tθ2+ u2∂xθ2+~ 2(∂xθ2) 2 ₌ ~ 2 ∂xx √ ρ2 √ ρ2 , (6.48) ∂tu2+ u2∂xu2+ β∂x(ρ2+ w) = 0, (6.49) which w is given by ∂tw = β∂x(ρ1+ ρ2), (6.50) or is equivalent to w(t, x) = w0(x) + β Z t 0 ∂x(ρ1+ ρ2)dτ. (6.51)

It is remarkable that the quantum effect in this system is of order O(~) different from the perturbation of the Euler equations (6.9)–(6.14) of order O(~2_).

7

Semiclassical Limit

In this section, we will derive the existence and uniqueness of local clas-sical solutions for LSI equations (5.1)–(5.3) with initial values (5.4)–(5.6). Then we will study their semiclassical limit by utilizing the hydrodynamical structures presented in the previous section.

First, we employ the modified Madelung transformation [4] to rewrite (5.1)–(5.3) into a perturbation of a quasilinear hyperbolic system [5, 14]. Let

ψ~ _{= A}~ 1exp  iS ~ 1 ~  , (7.1) A~ 1 = a~1+ ib~1, u~1 = ∂xS1~, (7.2) φ~ _{= A}~ 2exp  iS ~ 2 ~  , (7.3) A~ 2 = a~2+ ib~2, u~2 = ∂xS2~, (7.4)

then substituting (7.1) (resp.(7.3)) into (5.1) (resp.(5.2)), we have ∂tA~1+ ∂xS1~∂xA~1+ 1 2A ~ 1∂xxS1~ = i ~ 2∂xxA ~ 1, (7.5) ∂tS1~+ 1 2(∂xS ~ 1) 2_{+ β|A}~ 1| 2_{+ βw}~ _{= 0, (7.6)} ∂tA~2+ ∂xS2~∂xA~2+ 1 2A ~ 2∂xxS2~ = i ~ 2∂xxA ~ 2, (7.7) ∂tS2~+ 1 2(∂xS ~ 2) 2_{+ β|A}~ 2| 2_{+ βw}~ _{= 0. (7.8)}

Differentiating (7.6) (resp.(7.8)) w.r.t. x and replacing (A~

1, S1~) (resp.(A~2, S2~)) by (7.2) (resp.(7.4)), we have ∂ta~1+ u~1∂xa~1+ 1 2a ~ 1∂xu~1 = − ~ 2∂xxb ~ 1, (7.9) ∂tb~1+ u~1∂xb~1+ 1 2b ~ 1∂xu~1 = ~ 2∂xxa ~ 1, (7.10) ∂tu~1+ u~1∂xu~1+ 2βa~1∂xa~1+ 2βb~1∂xb~1+ β∂xw~ = 0, (7.11) ∂ta~2+ u~2∂xa~2+ 1 2a ~ 2∂xu~2 = − ~ 2∂xxb ~ 2, (7.12) ∂tb~2+ u~2∂xb~2+ 1 2b ~ 2∂xu~2 = ~ 2∂xxa ~ 2, (7.13) ∂tu~2+ u~2∂xu~2+ 2βa~2∂xa~2+ 2βb~2∂xb~2+ β∂xw~ = 0, (7.14)

with initial values a~

1(0, x) = a~1,0(x), b~1(0, x) = b~1,0(x), u~1(0, x) = u~1,0x = ∂xS1~(0, x), (7.15)

a~

2(0, x) = a~2,0(x), b~2(0, x) = b~2,0(x), u~2(0, x) = u~2,0x = ∂xS2~(0, x). (7.16)

According to (5.3), w~ _{is given explicitly by}

w~_{(x, t) = w}~ 0(x) + β Z t 0 ∂x(a~1) 2_{+ (b}~ 1) 2_{+ (a}~ 2) 2_{+ (b}~ 2) 2_{ dτ. (7.17)}

Hence, (7.9)–(7.17) form a quasilinear hyperbolic system which is equivalent to the LSI equations (5.1)–(5.3) with initial values (5.4)–(5.6). The system can be rewritten in the vector form

∂tU~+ A(U~)∂xU~+ G(w~) = ~ 2LU ~_{, (7.18)} U~_{(0, x) = U}~ 0(x) = (a~1,0(x), b~1,0(x), u~1,0(x), a~2,0(x), b~2,0(x), u~2,0(x))t, (7.19) w~_{(0, x) = w} 0(x), (7.20) where U~ _{= (a}~ 1, b~1, u~1, a~2, b~2, u2~)t, G(w~) = (0, 0, β∂xw~, 0, 0, β∂xw~)t, A(U~_{) =}               u~ 1 0 a~ 1 2 0 0 0 0 u~ 1 b~ 1 2 0 0 0 2βa~ 1 2βb~1 u~1 0 0 0 0 0 0 u~ 2 0 a~ 2 2 0 0 0 0 u~ 2 b~ 2 2 0 0 0 2βa~ 2 2βb~2 u~2               , and L =         0 −∂xx 0 0 0 0 ∂xx 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −∂xx 0 0 0 0 ∂xx 0 0 0 0 0 0 0 0         . Now, we introduce S, S =         4β 0 0 0 0 0 0 4β 0 0 0 0 0 0 1 0 0 0 0 0 0 4β 0 0 0 0 0 0 4β 0 0 0 0 0 0 1         , (7.21) which is symmetry and positive define for β > 0. Multiplying (7.18) by S, we have the quasilinear symmetry hyperbolic system

S∂tU~+ eA(U~)∂xU~+ eG(w~) = ~

2LUe

where eG(w~_{) = SG(w}~_{), eL = SL and eA}~ _{= SA}~ _{is symmetry. The local}

existence in time for the initial values (7.19) of the quasilinear symmetry hyperbolic system (7.22) follows the iteration scheme as below. For con-venience, we ignore the superscript ~ in (7.23)–(7.30) and some calculating process. Define U0_{(t, x) = U}

0(x), w0(t, x) = w0(x) where U0(x), w0(x) are

the given initial values and define Uk+1_{(t, x), w}k+1_{(t, x) inductively as the}

solution of the linear initial value problem

S∂tUk+1+ eA(Uk)∂xUk+1+ eG(wk+1) = ~ 2LUe k+1_{, (7.23)} wk+1(t, x) = w0(x) + β Z t 0 ∂x(ak1) 2 + (bk₁)2+ (ak₂)2+ (bk₂)2 dτ, (7.24) Uk+1(0, x) = U₀k+1(x) = U0(x), (7.25)

for k = 0, 1, 2, . . .. Assume U0 ∈ Hs and w0 ∈ Hs+1 where s is to be

determined. Let U be a solution of (7.18) and belongs to C1([0, T ]; C2(Ω)) which is of compact support for each t. The canonical energy associated with the quasilinear symmetry hyperbolic system (7.18) is defined by

(SU, U ) = Z

UtSU dx. (7.26) The classical energy estimate follows immediately by the symmetry of S, eA and antisymmetry of eL. Indeed,

( eLU, U ) = Z UtLU dx =e Z (UtLU )e tdx = Z UtLe t U dx = − Z UtLU dxe = −( eLU, U )

and this implies ( eLU, U ) = 0. So, if eA together with its derivatives of any de-sire order are continuous and bounded uniformly in [0, T ] × Ω, by integration by parts, then d dt(SU, U ) = (S∂tU, U ) + (SU, ∂tU ) = 2(S∂tU, U ) = ~( eLU, U ) − 2( eA∂xU, U ) − 2( eG, U ) = 0 + ((∂xA)U, U ) − 2( ee G, U ) 6 c1(t)(SU, U ).

By applying Gronwall inequality, we deduce the energy inequality (SU, U ) ≤ (SU0, U0)e Rt 0c1(τ )dτ, (7.27) and hence max 06t6TkU ~_(t)k L2 6 c₂kU₀~k_L2. (7.28)

The higher energy estimate can be obtained in the similar way. We differen-tiate (7.18) w.r.t. x, then multiply on both sides by S, we have

S∂x∂tU + eA∂x2U + ∂xA∂e _xU + ∂_xG =e ~

2L∂e xU, (7.29) ∂xU (0, x) = ∂xU0(x). (7.30)

With similar calculation, d dt(S∂xU, ∂xU ) = (S∂t∂xU, ∂xU ) + (S∂xU, ∂t∂xU ) = 2(S∂t∂xU, ∂xU ) = ~( eL∂xU, ∂xU ) − 2(∂xA∂e _xU, ∂_xU ) − 2( eA∂_x∂_xU, ∂_xU ) − 2(∂_xG, ∂e _xU ) = 0 − 2(∂xA∂e _xU, ∂_xU ) + (∂_xA∂e _xU, ∂_xU ) − 2(∂_xG, ∂e _xU ) = −(∂xA∂e _xU, ∂_xU ) − 2(∂_xG, ∂e _xU ) 6 c3(t)(S∂xU, ∂xU ).

By Gronwall inequality again, we have max

06t6Tk∂xU ~_(t)k

L2 6 c₄k∂_xU₀~k_L2. (7.31)

Moreover, the estimate of the time derivative ∂tU is directly derived from

the equation (7.18) itself. max 06t6Tk∂tU ~_k Hs−2 = max 06t6T ~ 2LU ~_{− A∂} xU~− G(w~) Hs−2 6 c5 max 06t6TkU ~_k Hs+ c₆ max 06t6TkG(w ~_)k Hs. (7.32)

∂tU~ only belongs to Hs−2 because of the twice derivative appearing in L.

So far, we have shown that for fixed ~,

for all k. Hence U~,k

k∈N is uniformly bounded in k. Moreover, by mean

value theorem, max 06t6TkU ~,k_{(t + h) − U}~,k_(t)k Hs−2 = max 06t6Tk∂tU ~,k_{(ξ) · hk} Hs−2, ξ ∈ (t, t + h) ⊂ [0, T ] = h · max 06t6Tk∂tU ~,k_(t)k Hs−2

tends to 0 as h goes to 0, for all k. Thus the sequence U~,k

k∈N is

equicon-tinuous. Following the Arzela-Ascoli theorem, there exists U~ _{∈ L}∞_{([0, T ]; H}s_{) ∩ Lip([0, T ]; H}s−2_),

such that as k → ∞

U~,k _{→ U}~ _{in C([0, T ]; H}s−2_).

Thus, by interpolation inequality, max 06t6TkU ~,k1 _{− U}~,k2_k Hs−θ 6 c₇ max 06t6TkU ~,k1_{− U}~,k2_k Hs−2 max 06t6TkU ~,k1 _{− U}~,k2_k Hs 6 c8 max 06t6TkU ~,k1_{− U}~,k2_k Hs−2

for 0 < θ < 2, we have the convergence

U~,k _{→ U}~ _{in C([0, T ]; H}s−θ_).

In addition, we discuss the convergence A(Uk_)∂

xUk+1 to A(U )∂xU . Indeed,

it can be done with the fact that

∂xU~,k → ∂xU~,

as k → ∞, since kA(Uk_)∂

xUk+1− A(U )∂xU kHs−1

= kA(Uk)∂xUk+1− A(Uk)∂xU + A(Uk)∂xU − A(U )∂xU kHs−1

6 kA(Uk)kHs−1k∂_xUk+1− ∂_xU k_Hs−1+ kA(Uk) − A(U )k_Hs−1k∂_xU k_Hs−1

Consequently, we have

Then the original equation (7.18) implies U~ _{∈ C}1_{([0, T ]; H}s−2_{); hence we}

have the solution

U~ _{∈ C([0, T ]; H}s_{) ∩ C}1_{([0, T ]; H}s−2_{). (7.34)}

Also, from the relation between U~ _{and w}~ _{in (7.17), we have}

w~ _{∈ C([0, T ]; H}s−1_{) ∩ C}1_{([0, T ]; H}s−3_{). (7.35)}

Furthermore, by Sobolev type inequality, if s > 1₂ + 4 then Hs−2 ,→ C2.

This can be easily checked by the dimensions of two function spaces Hs−2 and C2_, 1 2 − (s − 2) < 1 ∞ − 2. Then we have U~ _{∈ C([0, T ]; H}s_{) ∩ C}1_{([0, T ]; H}s−2_{) ,→ C}1_{([0, T ]; C}2_{), (7.36)} w~ _{∈ C([0, T ]; H}s−1_{) ∩ C}1_{([0, T ]; H}s−3_{) ,→ C}1_{([0, T ]; C}1_{), (7.37)}

and hence the solution (U~_{, w}~_{) of the quasilinear hyperbolic system (7.18)–}

(7.20) is classical.

The uniqueness of the classical solution of (7.18) follows from the energy estimate for the difference of two given solutions. Make U and V two so-lutions with the same initial data. Define U∗ = U − V , and we have the equation

S∂tU∗+ eA(U )∂xU∗+ [ eA(U ) − eA(V )]∂xV = ~

2LUe

∗

. (7.38) With previously similar arguments and U , V are of compact support, we have d dt(SU ∗ , U∗) = (S∂tU∗, U∗) + (SU∗, ∂tU∗) = 2(S∂tU∗, U∗)

= ~( eLU∗, U∗) − 2( eA(U )∂xU∗, U∗) − 2([ eA(U ) − eA(V )]∂xV, U∗)

= 0 + ((∂xA(U ))Ue ∗, U∗) − 2([ eA(U ) − eA(V )]∂xV, U∗)

6 c9(t)(SU∗, U∗).

By Gronwall inequality, we have

(SU∗, U∗_{) 6 (SU0}∗, U₀∗)eR0tc9(τ )dτ = 0. (7.39)

This implies U∗ = 0 and hence U = V . Therefore the classical solution (U~_{, w}~_{) is unique.}

Theorem 7.1. Let s > 1₂ + 4. Assume the initial values U~ 0 = (a~1,0, b~1,0, u~1,0, a~2,0, b~2,0, u~2,0) ∈ H s_{× H}s_{× H}s_{× H}s_{× H}s_{× H}s_, (7.40) w~ 0 ∈ Hs+1, (7.41)

then there exists T > 0 such that the quasilinear hyperbolic system (7.18) with initial values (7.19),(7.20) has a unique classical solution

U~ _{∈ C([0, T ]; H}s_{) ∩ C}1_{([0, T ]; H}s−2_{) ,→ C}1_{([0, T ]; C}2_{), (7.42)}

w~ _{∈ C([0, T ]; H}s−1_{) ∩ C}1_{([0, T ]; H}s−3_{) ,→ C}1_{([0, T ]; C}1_{), (7.43)}

for all t ∈ [0, T ].

As an immediate consequence, we have the similar result for the LSI equations (5.1)–(5.6).

Theorem 7.2. Let s > 1₂ + 4. Assume the initial values (A~

1,0, S1,0~ , A~2,0, S2,0~ , w0~) ∈ H

s_{× H}s+1_{× H}s_{× H}s+1_{× H}s+1_{, (7.44)}

then there exists T > 0 such that the LSI equations (5.1)–(5.3) with initial values (5.4)–(5.6) have a unique classical solution (ψ~_{, φ}~_{, w}~_{) of the form}

ψ~ _{= A}~ 1exp  iS ~ 1 ~  , φ~ _{= A}~ 2exp  iS ~ 2 ~  , w~_{(t, x) = w}~ 0(x) + β Z t 0 ∂x(A~1) 2_{+ (A}~ 2) 2_{ dτ,} which A~

1, S1~, A~2, S2~ (resp. w~) are bounded in L

∞_{([0, T ]; H}s_{) (resp. L}∞_{([0, T ];} Hs−1_{)) uniformly in ~.} Proof. Since ψ~ _{= A}~ 1exp  iS ~ 1 ~  and φ~ _{= A}~ 2exp  iS ~ 2 ~  where A~ 1 = a~1+ ib~1, u~1 = ∂xS1~, A~2 = a~2 + ib~2 and u~2 = ∂xS2~, by Theorem 7.1, we have A~ 1 ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2), ∂xS1~ ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2),

and hence S~ 1 ∈ C([0, T ]; H s_{) ∩ C}1_{([0, T ]; H}s−2_). Similarly, A~ 2 ∈ C([0, T ]; H s ) ∩ C1([0, T ]; Hs−2), ∂xS2~ ∈ C([0, T ]; H s ) ∩ C1([0, T ]; Hs−2), and hence S~ 2 ∈ C([0, T ]; H s_{) ∩ C}1_{([0, T ]; H}s−2_).

By Moser type calculus inequality, we conclude that

ψ~ _{∈ C([0, T ]; H}s_{) ∩ C}1_{([0, T ]; H}s−2_{) ,→ C}1_{([0, T ]; C}2_), φ~ _{∈ C([0, T ]; H}s_{) ∩ C}1_{([0, T ]; H}s−2_{) ,→ C}1_{([0, T ]; C}2_). Moreover, w~_{(t, x) = w}~ 0(x) + β Z t 0 ∂x(A~1) 2_{+ (A}~ 2) 2_{ dτ} ∈ C1_{([0, T ]; C}1_),

and thus the theorem follows.

Because of the nature of the antisymmetry of e_{L, the term ~(LU, U)} van-ishs in our estimates. The time interval [0, T ] and the boundary for U~ _in

Hs _{are independent of ~. These will allow us to pass to the limit ~ → 0 in}

(7.18).

Proposition 7.3. Let (ρ~

1, θ~1, u~1, ρ~2, θ~2, u~2, w~) be in C1([0, T ]; C2) and be

the solution of equations (6.44)–(6.51). For i = 1, 2, if ρ~

i,0(x) > 0 then

ρ~

i(t, x) > 0, ∀t > 0. Furthermore, when the ~ varies, ρ~i will not be too

small; that is, too closed to zero. Proof. Since u~ i, θ~i ∈ C1([0, T ]; C2), u~i + ~∂xθ~i ∈ C1([0, T ] × R). From (6.44), we have ∂tρ~i + ∂xρ~i(u~i + ~∂xθ~i) = 0, (7.45) or ∂tρ~i + (u~i + ~∂xθ~i)∂xρ~i = −ρ~i∂x(u~i + ~∂xθi~). (6.46)

In addition, the ordinary differential equations dx

dt = u

~

i + ~∂xθ~i, (7.47)

has a unique solution x = Γ(t) which belongs to C1_{([0, T ] × R). Equation} (7.46) implies d dtρ ~ i(t, Γ(t)) = −ρ~i(t, Γ(t))∂x(u~i + ~∂xθi~). (7.49)

Integrating over [0, τ ], we have ρ~ i(τ, ξ) = ρ~i(0, Γ(0)) exp  − Z τ 0 ∂x(u~i + ~∂xθi~)dt  . (7.50) Hence ρ~

i(t, x) > 0 if ρ~i,0(x) > 0. Moreover, the integration in the r.h.s. of

(7.50) will not tend to the infinity when the ~ varies, hence ρ~

i will not be

too closed to zero.

The limiting system of the quasilinear hyperbolic system (7.18) with ini-tial value (7.19) is also a quasilinear hyperbolic system as the following shows: (formally letting ~ → 0) Ut+ A(U )Ux+ G(w) = 0 (7.51) U (0, x) = U0(x) (7.52) w(0, x) = w0(x) (7.53) where w is given by ∂tw = β∂x(a21+ b 2 1+ a 2 2+ b 2 2), (7.54) or is equivalent to w(t, x) = w0(x) + β Z t 0 ∂x(a21+ b 2 1+ a 2 2+ b 2 2)dτ. (7.55)

This is equivalent to the limiting Euler system (6.27)–(6.36) as long as the solutions are smooth. Next, we will show the existence and uniqueness of the local smooth solution to the system (6.27)–(6.36).

Theorem 7.4. Let s > 1

2+ 4 and [0, T ] be the fixed time interval determined

in Theorem 3.1. Given initial values U~

0, U0 ∈ Hs, and U0~ converges to U0

in Hs _{as ~ → 0. Then, there exists}

U ∈ C([0, T ]; Hs) ∩ C1([0, T ]; Hs−2) ,→ C1([0, T ]; C2), w ∈ C([0, T ]; Hs−1) ∩ C1([0, T ]; Hs−3) ,→ C1([0, T ]; C1),

which is a classical solution to the IVP for the limiting quasilinear hyperbolic system (7.51)–(7.55), and so is to the IVP for the limiting Euler system (6.27)–(6.36).

Proof. SinceU~

~ is bounded uniformly in ~, by Arzela-Ascoli theorem and

interpolation inequality, we have a function U such that, as ~ → 0 U~ _{→ U in C([0, T ]; H}s−θ_),

for 0 < θ < 2. Also, from the equation (7.18) itself, we have U~ _{→ U in C}1_{([0, T ]; H}s−2−θ_),

for 0 < θ < 2. LU~ _{is uniformly bounded in H}s−2_{, so the perturbation term} ~

2LU~ tends to 0 as ~ → 0. Hence the sequence converges to a solution of the

limiting quasilinear hyperbolic system (7.51)–(7.55). The solution w is then given by (7.55) and belongs to C1([0, T ]; C1).

Theorem 7.5. Let (ρ1, u1, ρ2, u2, w) be a solution of the limiting Euler system

(6.27)–(6.36) on [0, T ], which initial value (ρ1,0, u1,0, ρ2,0, u2,0, w0) belongs to

Hs× Hs_{× H}s_{× H}s_{× H}s+1_{. Assume A}~

1,0 (resp. A~2,0, w~0) converges strongly

to A1,0 (resp. A2,0, w0) in Hs (resp. Hs, Hs+1) as ~ → 0. Then, for ~ small

enough, there exists a unique classical solution (ψ~_{, φ}~_{, w}~_{) to the IVP for the}

LSI equations (5.1)–(5.6).

Proof. Consider the difference of (7.18) and (7.51). Define eU~ _{= U}~ _{− U ,}

then we have ∂tUe~ + A( eU~+ U )∂_xUe~ + [A( eU~+ U ) − A(U )]∂_xU +G(w~) − G(w)  = ~ 2L( eU ~ _{+ U ). (7.56)}

We introduce S = S( eU~ _{+ U ) which is symmetry, positive define and can}

symmetrize A( eU~ _{+ U ). Multiplying (7.56) by S, we have}

S∂tUe~+ SA( eU~+ U )∂_xUe~ + S[A( eU~+ U ) − A(U )]∂_xU + SG(w~) − G(w)  = ~

2SL( eU

~_{+ U ). (7.57)}

The energy associated with (7.56) is defined by (S eU~_{, eU}~_{) =}

Z

We apply the energy estimate again. d dt(S eU ~_{, eU}~_{) = (S∂} tUe~, eU~) + (S eU~, ∂_tUe~) = 2(S∂tUe~, eU~) = ~(SL( eU~_{+ U ), eU}~_{) − 2(SA( eU}~_{+ U )∂} xUe~, eU~) − 2(S[A( eU~ _{+ U ) − A(U )]∂} xU, eU~) − 2(S[G(w~) − G(w)], eU~).

By the antisymmetry of L, we have

~(SL eU~, eU~) = 0. The Cauchy-Schwarz inequality implies

~(SLU, eU~) 6 ~c10kLU kL2k eU~k_L2 6 ~c₁₁kU k_H2k eU~k_L2 6 c₁₂k eU~k2_L2 ; − 2(SA( eU~_{+ U )∂} xUe~, eU~) = (S(∂_xA( eU~+ U )) eU~, eU~) 6 c₁₃k eU~k2_L2 ; (S[A( eU~_{+ U ) − A(U )]∂} xU, eU~) 6 c14k[A( eU~+ U ) − A(U )]∂xU kL2k eU~k_L2 6 c15k∂xU kL2k eU~k_L2 6 c₁₆kU k_H1k eU~k_L2 6 c17k eU~k2L2 ; (S[G(w~_{) − G(w)], eU}~_{) 6 c} 18k eU~k2L2 .

Hence we have the inequality d dt(S eU ~_{, eU}~_{) 6 c} 19(t)(S eU~, eU~). By Gronwall inequality, (S eU~_{, eU}~_{) 6 (S eU}~ 0, eU0~)e Rt 0c19(τ )dτ, (7.59)

which the r.h.s. tends to 0 as ~ → 0 because of eU~

0 = U0~ − U0 tends to 0.

Then the theorem follows.

We conclude that the behavior of the quasilinear hyperbolic system (7.18) resembles the limiting system (7.51). That is to say, the ~ appearing in the Euler equations (6.9)–(6.13) is negligible. Hence the quantum equations can be depicted by the classical hydrodynamics equations.

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