國
立
交
通
大
學
應用數學系
碩 士 論 文
有旋轉項的 Gross-Pitaevskii 方程之
半古典極限
Semiclassical Limit of the Gross-Pitaevskii
Equation with Rotation
研 究 生：蔡佳穎
指導教授：林琦焜 教授
有旋轉項的 Gross-Pitaevskii 方程之
半古典極限
Semiclassical Limit of the Gross-Pitaevskii
Equation
with Rotation
研 究 生：蔡佳穎 Student：Jia-Ying Tsai
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 ThesisSubmitted 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 2011 Hsinchu, Taiwan
中華民國一百年六月
i
有旋轉項的 Gross-Pitaevskii 方程之
半古典極限
學生：蔡佳穎
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
在本論文中，我們用兩種不同的做法研究有旋轉項的 Gross-Pitaevskii 方
程之半古典極限。首先，我們使用修改過的 Madelung 變換以著重在與量子
流體動力學方程（quantum hydrodynamical equations）等價的擬線性雙
曲對稱系統（quasilinear symmetric hyperbolic system）。我們建立在
極限系統奇點形成之前，當普朗克常數趨近於零時，量子密度與量子動量
收斂到可壓縮的旋轉歐拉方程（compressible rotational Euler equation）
之唯一解。此外，我們證明在維度 2 之可壓縮的旋轉歐拉方程之局部解的
存在性與唯一性。其次，我們考慮量子密度與量子動量在恆定狀態(1,0)附
近的情形。我們建立有旋轉項的 Gross-Pitaevskii 方程弱收斂到等價於線
性波動方程（linear wave equation）的波映射方程（wave map equation）。
這方法的結果引領聲波（acoustic wave）的討論。
Semiclassical Limit of the Gross-Pitaevskii
Equation with Rotation
Student：Jia-Ying Tsai
Advisors：Dr. Chi-Kun Lin
Department of Applied Mathematics
National Chiao Tung University
ABSTRACT
In this paper, we perform the semiclassical limit of the Gross-Pitaevskii equation
with rotation by two different approaches. First, we use the modified Madelung
transformation to focus on the quasilinear symmetric hyperbolic system, which
is equivalent to the quantum hydrodynamical equations. We establish that before
the formation of singularities in the limiting system, the quantum density and
quantum momentum converge to the unique solution of the compressible
rotational Euler equation as the Planck constant
tends to zero. In addition,
we prove the existence and uniqueness of local solutions of the compressible
rotational Euler equation in dimension 2. Second, we consider the case when the
quantum density and quantum momentum are near the constant state (1,0). We
establish that the Gross-Pitaevskii equation with rotation converges weakly to
the wave map equation, equivalently the linear wave equation. The result of this
approach leads the discussion of the acoustic wave.
iii
誌
謝
––––––––––––––––––––––––––––––––––––––––––– You are the stars, the moon, and the sunshine in my life.
「養成隨手的計算紙都能寫得整整齊齊的好習慣。」大二升大三的那年暑假，林琦 焜老師深深烙印在我心靈的話，使我開始養成做數學有條不紊的好習慣。我深深謝謝老 師開啟我清晰學習的思緒。 深深謝謝我的指導教授－林琦焜老師啟發我找到雪亮的勇氣。用批判存疑的眼睛看 世界、看問題、看眼前，用欣賞的心態看別人，用希望的心態看未來。深深謝謝老師引 領我成為獨立的人。直觀的見識，精確的態度，自信的靈魂，省思的自我。我一步步隨 著論文的完成，找到更清澈的自己。論文是我的學習結果，對數學的熱情堅持與認真嚴 謹是老師深刻在我心中的榜樣。每當我出奇不意地至老師辦公室問問題時，老師總是無 時不刻正在做數學與閱讀。我感動不停止的前進。老師的教學渲染我體會數學生命的 美，胸懷科學家的氣度。這份論文的完成是具體的，然而心靈氣度的成長是生命這段旅 程最珍貴的豐收。深深謝謝我的指導教授讓我從辨別、反省、改變中更加進步。 深深謝謝口試委員：中央研究院研究員李志豪老師及交通大學王夏聲老師。李志豪 老師是我大二升大三那年參與暑期研討班的指導教授。從我就讀大學至研究所，老師總 是給我細微的教導及滿滿的關懷和協助，真誠的溫暖蔓延在每一地方每一成長階段。口 試後，老師幫我整理了需弄懂及更正的要點。我深深感謝老師的用心與提攜。王夏聲老 師是我碩一時的實變老師。從我學習當一名研究生開始，他總是看見我學習過程中遭遇 的猶疑與問題，叮嚀、提點、建議我學習的方式，分享、提示、鼓勵我精進的方向，真 切的妙囊出現在每一適時每一關卡。口試後，老師再次提醒我改善的問題點。我深深感 謝老師的教導與開導。深深謝謝老師們的導引，讓我在學習的過程及生活的態度豁然開 朗。 深深謝謝交通大學師長們。謝謝教導我常微分方程的石至文老師與量子力學的孟心 飛老師，奠基我學問的知識。謝謝教導我學術英文寫作的吳思葦老師，精進我英文寫作 能力。謝謝吳恭儉學長在我遇見瓶頸、迷惘、遲疑、困頓時，給予我建議，分享我經驗， 提供我可能的突破方向。每次 seminar 結束，學長總會針對我們遭遇的問題，提出精進 探尋的意見。在學長幫助下，我逐步克服疑惑，成長邁進。另外，謝謝陳冠羽學長和梁 育豪學長在我碩士生涯裡盡其可能為我解惑及給予我協助。舉凡課業和生活上的疑問雜 事，他們總是充滿耐心地教導我，盡力地幫助我解決。特別謝謝梁育豪學長在我口試當 天協助我準備前置作業，細心地提供我完善的器材，讓我保持穩定的心情完成口試。 深深謝謝交通大學給我最好的學習資源與生活。此外，深深謝謝我的好朋友、碩士 班摯友—吳姿慧、同學及學弟妹，給我窩心的扶助與關心。深深謝謝曾經幫助關懷過我 的所有人，讓我在學習及生活中獲得無以倫比的扶持與滋養。因有你們，我越趨豐富、 完善且圓融。 畢業是新旅程的開始。我會成為自信、和煦、分享的人，讓生命光耀且溫暖。我期 許自己懷有寧靜、智慧與勇氣，面對人生挑戰自在、浩然、豪邁。
最後，謹此論文獻給呵護我長大，支持我夢想的母親－黃麗英女士。
蔡佳穎
謹誌于 國立交通大學 2011年六月
Contents
Abstract (in Chinese) i
Abstract (in English) ii
Acknowledgements (in Chinese) iii
Contents v
Chapter 1. Introduction 1
Chapter 2. Hydrodynamical Structure 5 2.1. Euler Equation (First Method) 6 2.2. Euler Equation (Second Method) 8
2.3. Energy Equation 11
Chapter 3. Semiclassical Limit of the Local Smooth Solutions 15 3.1. Quasilinear Symmetric Hyperbolic System 16
3.2. Classical Solutions 19
3.3. Semiclassical Limit 32
Chapter 4. WKB Expansion 37
Chapter 5. The Local Existence and Uniqueness 43
Chapter 6. Acoustic Wave 49
6.1. Dispersion Limit 50
Bibliography 57
CHAPTER 1
Introduction
Bose-Einstein condensation (BEC) is a phenomenon that a macroscopic fraction of the atoms occupy the same quantum level and behave as a coherent matter wave at very low temperature. An important issue is the relationship between BEC and superfluidity. The properties of rotating condensates in traps have increasingly been the object of study in recent years. The readers are referred to [1, 2, 3, 7, 9] for more experimental and theoretical work, and these developments have been reviewed in [6, 17, 18].
The primary research questions to be addressed in this paper are as follows. In the case of a dilute Bose gas at temperature much smaller than the critical condensation temperature, the time-dependent Gross-Pitaevskii equation with rotation is the equation of motion in the frame rotating with the trap. It takes the form
i~∂tψ~= −~ 2 2m∆ψ ~+ g|ψ~|2ψ~+ V (x)ψ~+ i~ωx × ∇ψ~ = −~ 2 2m∆ψ ~+ g|ψ~|2ψ~+ V (x)ψ~+ i~ωx⊥· ∇ψ~, (1.1)
where x⊥ = (−x2, x1). The macroscopic wave function ψ~(t, x) is inherently a complex
function. m is the atomic mass. In the non-linear potential term, g characterizes the strength of the short-range interparticle potential. The external potential V (x) does not depend on the time. The rotating term is composed of the angular velocity ω of the rotating trap and the x3 component of the angular momentum operator
Lx3 = −i~x × ∇ = −i~ (x1∂x2 − x2∂x1) = −i~x
⊥· ∇,
(1.2)
where x = (x1, x2) is the coordinate in the rotating frame. The operator Lx3 is only a scalar
in two dimensional case and represents the rotation. The Gross-Pitaevskii equation with rotation (1.1) is also called the rotating nonlinear Schr¨odinger equation. In this paper, we focus on a simple case in which g and ω are constants.
A singular limit is an interesting problem. It makes a connection between different fields and contributes to the understanding of the nature of the problem. The study of the semiclassical limit has a great importance for determining the limiting behaviour of any function of the field ψ~. It may also describe superfluids and provide rich dynamical
phenomena in rotating BEC gases. In this work, we discuss the semiclassical limit by two different approaches : the modified Madelung transformation and the density fluctuation.
First of all, we use the Madelung transformation
ψ~(t, x) = A~(t, x)e~iS~(t,x)
, (1.3)
where both A~(t, x) and S~(t, x) are real-valued functions. As density ρ~ and momentum
µ~ ω are given by ρ~= |A~|2 = ψ~ 2 , µ~ ω = ρ~ 1 m∇S ~− ωx⊥ , (1.4)
the quantum hydrodynamic equations of the rotating nonlinear Schr¨odinger equation (1.1) then are ∂tρ~+ ∇ · µ~ω = 0, ∂tµ~ω+ ∇ · µ~ ω⊗ µ~ω ρ~ + ∇ g 2m(ρ ~)2 +ρ~∇ 1 mV − 1 2ω 2|x|2 = ~ 2 4m2∇ · ρ ~∇2log ρ~ − 2ω(µ~ ω) ⊥ , (1.5)
with initial data
ρ~(0, x) = ρ~
0(x), µ~ω(0, x) = µ~ω,0(x). (1.6)
To overcome the difficulty caused by the nonlinear term, we introduce the modified Madelung transformation suggested by Grenier [8]. There has been a change of emphasis from the real-valued function A~(t, x) to the complex-valued one. The main method to
carry out this study is based on transforming the rotating nonlinear Schr¨odinger equation (1.1) into a dispersive perturbation of a quasilinear symmetric hyperbolic system, to which the Lax-Friedrich-Kato’s theory can be applied, by the modified Madelung transforma-tion. The readers are referred to good tutorials in [5, 8, 11, 12, 13] for an introduction
1. INTRODUCTION 3
to the above approaches and its applications. When ~ tends to zero, we formally have the compressible rotational Euler equation
∂tρ + ∇ · µω = 0, ∂tµω+ ∇ · µω ⊗ µω ρ + ∇ g 2mρ 2+ ρ∇ 1 mV − 1 2ω 2|x|2 = −2ωµ⊥ω, (1.7)
with initial data
ρ(0, x) = ρ0(x), µω(0, x) = µω,0(x). (1.8)
It follows that the system of quantum mechanics converges to the system obeying Newton mechanics.
It is obviously required that the existence and uniqueness of solutions of the compress-ible rotational Euler equation are determined. Much research has been devoted on the existence and uniqueness of solutions of the Euler equation. For the incompressible Euler equation, the existence of a global solution in the two dimensional case and a local solu-tion in the three dimensional case has been established [20]. Roger Temam also has given another new short proof for the three dimensional case by representing the unknown pres-sure as Poisson’s equation and applying the Galerkin method with a special basis in his paper [20]. For the compressible Euler equation, P. L. Lions has discussed global entropy solutions in the one dimensional case [15]. However, little research has been done on the existence of solutions of the Euler equation with rotation. We propose a clear proof of the existence of a local solution of the compressible rotational Euler equation in dimension 2 by the equivalent relation, resulting from the conclusion of the above semiclassical limit.
Let us then discuss the semiclassical limit in the view of |ψ
~|2− 1
~ . It means that we consider the case when (ρ~, µ~) is near the constant state (1, 0). The semiclassical
(dis-persion) limit concludes that the rotating nonlinear Schr¨odinger equation (1.1) converges weakly to the wave map equation (6.38). Moreover, the wave map equation is equivalent to the linear wave equation (6.39). This is just the beginning of studying the acoustic wave.
The remainder of this paper is organized into five chapters. In Chapter 2, we start by deriving the hydrodynamical structure of the rotating nonlinear Schr¨odinger equation
(1.1). The readers are also referred to [10] for the deriving process. To obtain the lo-cal existence of smooth solutions and perform the semiclassilo-cal limit, the procedure is displayed in Chapter 3 and consists of three main parts. First, we transform the rotat-ing nonlinear Schr¨odinger equation into a quasilinear hyperbolic system by the modified Madelung transformation in Section 3.1. Second, a priori estimate, which allows to pass to the limit ~ → 0 and justfy the WKB hierarchy, is employed in Subsection 3.2.1. For a discussion of a priori estimate, also see [5, 8, 11, 12, 13, 16]. Third, some compactness arguments are the tools of attaining our desired results. For a full account of this part, also see [14]. Chapter 4 contains a description of the WKB expansion. The readers are also referred to [5, 8, 11]. The local existence and uniqueness proof of solutions of the compressible rotational Euler equation is outlined in Chapter 5. We see a connection between the rotating nonlinear Schr¨odinger equation and the acoustic wave in Chapter 6. We perform the semiclassical (dispersion) limit in the view of the density fluctuation in Section 6.1. The readers are also referred to [4, 14] for further details of the density fluctuation.
CHAPTER 2
Hydrodynamical Structure
The physical content of the rotating nonlinear Schr¨odinger equation (1.1) may be revealed by reformulating it as a pair of hydrodynamic equations (1.5), which we will use two methods to derive. Initially, we use Noether’s theorem to determine the conservation laws. The Lagrangian density for the rotating nonlinear Schr¨odinger equation (1.1) is
L= i~ 2 h ψ~∗ ∂tψ~− ψ~∂t ψ~ ∗i − ~ 2 2m∇ ψ ~∗ · ∇ψ~− g|ψ~|2 ψ~∗ ψ~ −V (x) ψ~∗ψ~− i~ω 2 h ψ~∗ x⊥· ∇ψ~− ψ~x⊥· ∇ ψ~∗i . (2.1)
The action S =RR Ldxdt is invariant under the following transformations
ψ~
(t, x) = e
iψ~(t, x) with generator δψ~= iψ~, (2.2)
ψ~
(t, x) = ψ~(t− , x) = ψ~(t, x) with generator δψ~ = ∂tψ~. (2.3)
However, the action S is not invariant under the transformation
ψ~
(t, x) = ψ~(t, x− ) = ψ~(t, x) with generator δψ~ = ∇ψ~, (2.4)
and about x3 axis under the transformation
ψ~
(t, x) = ψ~ t, RTx = ψ~(t, x) with generator δψ~= Lx3ψ
~, (2.5)
where for all ∈ R,
R = cos − sin sin cos , RT = cos sin − sin cos . (2.6)
Therefore, by Noether’s theorem, the invariances generate the conservation laws, and the generators iψ~and ∂
tψ~correspond to the conservation of charge and energy, respectively.
We have not the conservation of momentum due to the rotational term −ωLx3ψ
~
appear-ing in (1.1) and the conservation of angular momentum due to the linear potential term V (x)ψ~. We have the main result :
Theorem 2.1. If ψ~ is a smooth function, then the hydrodynamical formulation of
the rotating nonlinear Schr¨odinger equation (1.1) is (1) the conservation of charge :
∂tρ~+ ∇ · (ρ~(u~− ωx⊥)) = 0,
(2) the equation of momentum :
∂tµ~ω+ ∇ · µ~ ω⊗ µ~ω ρ~ + ∇ g 2m(ρ ~)2 +ρ~∇ 1 mV − 1 2ω 2|x|2 = ~ 2 4m2∇ · ρ ~∇2log ρ~ − 2ω(µ~ ω) ⊥ ,
(3) the conservation of energy :
∂tθ~+ ∇ · µ~ ω ρ~ θ~+ g 2m(ρ ~)2 = ~ 2 4m2∇ · µ~ ω∆ρ~ ρ~ − ∇ · µ~ ω∇ρ~ ρ~ ,
(4) the equation of angular momentum :
∂t x⊥· µ~ + ∇ · µ~ x⊥· µ ~ ρ~ + x⊥ g 2m(ρ ~)2− ωx⊥· µ~ +x⊥· ρ ~ m∇V = ~2 4m2∇ · x⊥∆ρ~− ∇ρ ~ ρ~ x ⊥· ∇ρ~ , where u~ ω = u~− ωx ⊥, µ~ ω = ρ~u~ω = µ~− ρ~ωx⊥, and θ~ = µ~ ω 2 2ρ~ + ~2 8m2 ∇ρ~ 2 ρ~ + g 2m(ρ ~)2+ ρ~ 1 mV − 1 2ω 2|x|2 .
2.1. Euler Equation (First Method)
We introduce the Madelung transformation ψ~(t, x) = A~(t, x)ei
~S ~(t,x)
, where both the amplitude A~(t, x) and the phase S~(t, x) are real-valued functions, and insert it into
the rotating nonlinear Schr¨odinger equation (1.1). Separating the real and imaginary parts leads to ∂tA~+ 1 m∇S ~· ∇A~+ 1 2mA ~∆S~− ωx⊥· ∇A~ = 0, (2.7) ∂tS~+ 1 2m ∇S~ 2 + g A~ 2 + V − ωx⊥· ∇S~ = ~ 2 2m ∆A~ A~ . (2.8)
2.1. EULER EQUATION (FIRST METHOD) 7
To understand the nature of the velocity of the fluid, we multiply (2.7) by 2A~ and define
the density, velocity, and momentum as
ρ~ = |A~|2 = ψ~ 2 , u~ = 1 m∇S ~ = (u~ 1, u~2), µ~ = ρ~u~, (2.9)
respectively. It follows that
∂tρ~+ ∇ · µ~− ωx⊥· ∇ρ~= 0, (2.10)
where ρ~ is a probability density. We also write (2.10) as the total differential form
∂tρ~+ ∇ · µ~− ∇ · ωx⊥ρ~ = 0. (2.11)
Hence, we have the conservation of charge
∂tρ~+ ∇ · (ρ~u~ω) = 0, u~ω = u~− ωx ⊥
, (2.12)
where u~
ω is the modified velocity. Taking the gradient of (2.8) and then multiplying it by
1
m, we obtain the equation of motion for the veloctiy
∂tu~+ u~· ∇ u~+ g m∇ρ ~+ 1 m∇V = ~ 2 2m2∇ ∆pρ~ p ρ~ ! − ω(u~)⊥+ ω(x⊥· ∇)u~ (2.13) or ∂tu~ω+ u~ω· ∇ u~ω+ g m∇ρ ~ +∇ 1 mV − 1 2ω 2|x|2 = ~ 2 2m2∇ ∆pρ~ p ρ~ ! − 2ω(u~ ω) ⊥ , (2.14) where (u~)⊥ = (−u~ 2, u~1) and (u~ω)
⊥ = (u~)⊥+ ωx. To obtain the momentum, we multiply
(2.13) by ρ~ and (2.10) by u~ and then add up to have
∂tµ~+ ∇ · µ~⊗ µ~ ρ~ + ∇ g 2m(ρ ~)2− ωx⊥· µ~+ρ ~ m∇V = ~ 2 4m2∇ · ρ ~∇2log ρ~ + ωx (µ ~)⊥ ρ~ · ∇ρ ~ , (2.15)
where (µ~)⊥ = ρ~(u~)⊥. Therefore, we can derive the equation of the modified momentum ∂tµ~ω+ ∇ · µ~ ω⊗ µ~ω ρ~ + ∇ g 2m(ρ ~)2 +ρ~ 1 m∇V − 1 2ω 2|x|2 = ~ 2 4m2∇ · ρ ~∇2log ρ~ − 2ω(µ~)⊥ ω. (2.16)
Equations (2.12) and (2.16) are analogous to the continuity equation and the momentum equation in fluid mechanics, respectively. The above equations show that the rotation affects both the continuity equation and the momentum equation. Provided that S~ is
not singular, we can conclude that the velocity field is irrotational; that is, the potential flow ∇ × u~ = 0. However, ∇ × u~ ω = −2ω. Moreover, by (2.13), we have ∂t x⊥· u~ + ∇ · x⊥ |u ~|2 2 + g mρ ~+ 1 mV − ωx ⊥· u~ = ~ 2 2m2∇ · ∆pρ~ p ρ~ ! , (2.17)
and by (2.15), we also have the equation of angular momentum
∂t x⊥· µ~ + ∇ · µ~ x⊥· µ ~ ρ + x⊥ g 2m(ρ ~)2− ωx⊥· µ~ +x⊥· ρ ~ m∇V = ~2 4m2∇ · x⊥∆ρ~−∇ρ ~ ρ~ x ⊥· ∇ρ~ . (2.18)
Since angular momentum about the axis of rotation is not conserved, the trap V (x) has no axis of symmetry.
2.2. Euler Equation (Second Method)
We consider ψ~ and ψ~∗ to be solutions of the rotating nonlinear Schr¨odinger
equa-tion i~∂tψ~ = −~ 2 2m∆ + g ψ~ 2 + V + i~ωx⊥· ∇ ψ~, (2.19) −i~∂t ψ~ ∗ = −~ 2 2m∆ + g ψ~ 2 + V − i~ωx⊥· ∇ ψ~∗ , (2.20)
respectively. Multiplying (2.19) by ψ~∗ and (2.20) by ψ~, we can write
i~ ψ~∗∂ tψ~ = −~ 2 2m ψ ~∗ ∆ψ~+ g ψ~ 4 + V ψ~ 2 + i~ω ψ~∗ x⊥· ∇ψ~, (2.21) −i~ψ~∂ t ψ~ ∗ = −~ 2 2mψ ~∆ ψ~∗ + g ψ~ 4 + V ψ~ 2 − i~ωψ~x⊥· ∇ ψ~∗ . (2.22)
2.2. EULER EQUATION (SECOND METHOD) 9
Subtracting (2.22) from (2.21) and using the equality
∇ ·h ψ~∗ ∇ψ~i= ψ~∗ ∆ψ~+ ∇ ψ~∗ · ∇ψ~, (2.23) there results ∂t|ψ~|2 = i ~ 2m∇ · h ψ~∗ ∇ψ~− ψ~∇ ψ~∗i + ωx⊥· ∇ ψ~ 2 . (2.24)
Hence, we obtain equation (2.10) by setting
ρ~ = ψ~ 2 , µ~= −i ~ 2m h ψ~∗ ∇ψ~− ψ~∇ ψ~∗i . (2.25)
Next, we do the similar steps to seek for (2.15). We multiply (2.19) by ∇ ψ~∗ and (2.20)
by ∇ψ~ and then add up to yield
−i~h∂t ψ~ ∗ ∇ψ~− ∂ tψ~∇ ψ~ ∗i = − ~ 2 2m h ∆ψ~∇ ψ~∗ + ∆ ψ~∗ ∇ψ~i+ g ψ~ 2 ∇ ψ~ 2 + V ∇ ψ~ 2 +i~ωn x⊥· ∇ψ~ ∇ ψ~∗ −hx⊥· ∇ ψ~∗i ∇ψ~o. (2.26)
On the other hand, we take the gradient of (2.19) and (2.20), multiply them by ψ~∗ and ψ~, respectively, and then add up to yield
−i~nψ~∇h∂ t ψ~ ∗i − ψ~∗ ∇ ∂tψ~ o = − ~ 2 2m n ψ~∗ ∇ ∆ψ~ + ψ~∇h∆ ψ~∗io +3g ψ~ 2 ∇ ψ~ 2 + 2 ψ~ 2 ∇V + V ∇ ψ~ 2 +i~ω ψ~∗∇ x⊥· ∇ψ~ − ψ~∇x⊥· ∇ ψ~∗ . (2.27)
Now subtracting (2.27) from (2.26), we derive −i~∂t ψ~∗∇ψ~− ψ~∇ ψ~∗ = − ~ 2 2m n ∆ψ~∇ ψ~∗ + ∆ ψ~∗ ∇ψ~− ψ~∗ ∇ ∆ψ~ − ψ~∇h∆ ψ~∗io −2g ψ~ 2 ∇ ψ~ 2 − 2 ψ~ 2 ∇V − i~ω∇x⊥· ψ~∗∇ψ~− ψ~∇ ψ~∗ +2i~ωx h ∇ ψ~∗i⊥ · ∇ψ~ , (2.28) where ∇ ψ~∗⊥= −∂ x2 ψ ~∗, ∂ x1 ψ ~∗. After multiplying (2.28) by 1 2m, we have ∂tµ~ = − ~ 2 4m2 ∆ψ~∇ ψ~∗ + ∆ ψ~∗ ∇ψ~ − ψ~∗∇ ∆ψ~ − ψ~∇∆ ψ~∗ − g mρ ~∇ρ~ −ρ ~ m∇V + ω∇ x ⊥· µ~ + ωx (µ ~)⊥ ρ~ · ∇ρ ~ . (2.29) Since ∇∆ρ~ = ψ~∗∇∆ψ~+ ψ~∇∆ ψ~∗ +2∇∇ψ~· ∇ ψ~∗ + ∆ψ~∇ ψ~∗+ ∆ ψ~∗∇ψ~, (2.30) we can rewrite (2.29) as ∂tµ~+ ~ 2 2m2∇ ∇ψ~ 2 + g mρ ~∇ρ~+ρ ~ m∇V + ~2 2m2 ∆ψ~∇ ψ~∗ +∆ ψ~∗∇ψ~ = ~ 2 4m2∇∆ρ ~+ ω∇ x⊥· µ~ + ωx (µ ~)⊥ ρ~ · ∇ρ ~ . (2.31)
Using the equality
~2 2m2 ∇ψ~ 2 = µ~ 2 2ρ~ + ~2 8m2 ∇ρ~ 2 ρ~ , (2.32)
2.3. ENERGY EQUATION 11 we derive ∂tµ~+ ∇ µ~ 2 2ρ~ ! + ∇ g 2m(ρ ~)2+ρ ~ m∇V + ~ 2 2m2 h ∆ψ~∇ ψ~∗ + ∆ ψ~∗ ∇ψ~i = ~ 2 4m2∇ ∆ρ ~− ∇ρ~ 2 2ρ~ ! + ω∇ x⊥· µ~ + ωx (µ ~)⊥ ρ~ · ∇ρ ~ . (2.33) Since ~2 2m2 h ∆ψ~∇ ψ~∗ + ∆ ψ~∗ ∇ψ~i= ∇ · µ ~⊗ µ~ ρ~ − ∇ µ~ 2 2ρ~ ! + ~ 2 4m2 " 1 ρ~∇ · ∇ρ ~⊗ ∇ρ~ − ∇ρ~ 2 2(ρ~)2∇ρ ~−∇ ∇ρ~ 2 2ρ~ # (2.34) and ∇∆ρ~+ ∇ρ~ 2 (ρ~)2 ∇ρ ~− 1 ρ~∇ · ∇ρ ~⊗ ∇ρ~ = ∇ · ρ~∇2log ρ~ , (2.35)
we obtain equation (2.15). In addition, by observing the dimension of (2.32), we can conjecture that the kinetic energy is of the form ~
2 2m2 ∇ψ~ 2 . 2.3. Energy Equation
We can derive the conservation of energy from (2.19) and (2.20) as follows. Multiplying (2.19) by ∂t ψ~
∗
and (2.20) by ∂tψ~ and then adding up, we can write
−~ 2 2m h ∆ψ~∂ t ψ~ ∗ + ∆ ψ~∗ ∂tψ~ i + g ψ~ 2 ∂t ψ~ 2 + V ∂t ψ~ 2 +i~ωx⊥· ∇ψ~∂ t ψ~ ∗ − x⊥· ∇ ψ~∗∂ tψ~ = 0. (2.36)
We use the equalities
∆ψ~∂ t ψ~ ∗ + ∆ ψ~∗∂ tψ~ = ∇ ·∂t ψ~ ∗ ∇ψ~+ ∂ tψ~∇ ψ~ ∗ − ∂t ∇ψ~ 2 (2.37) and x⊥· ∇ψ~∂ t ψ~ ∗ − x⊥· ∇ ψ~∗∂ tψ~ = ∂t ψ~∗ x⊥· ∇ψ~ − x⊥· ∇h ψ~∗ ∂tψ~ i (2.38)
and then multiply (2.36) by 1 m to have ∂t ~2 2m2 ∇ψ~ 2 + g 2m ψ~ 4 + 1 mV ψ~ 2 +i~ω m h ψ~∗ x⊥· ∇ψ~i −∇ · ~2 2m2 h ∂t ψ~ ∗ ∇ψ~+ ∂ tψ~∇ ψ~ ∗i −x⊥· ∇ i~ω m h ψ~∗ ∂tψ~ i = 0. (2.39)
Similarly, we use the equality
x⊥· ∇ψ~∂ t ψ~ ∗ − x⊥· ∇ ψ~∗ ∂tψ~ = x⊥· ∇h ψ~∂ t ψ~ ∗i − ∂tψ~x⊥· ∇ ψ~ ∗ (2.40) to have ∂t ~2 2m2 ∇ψ~ 2 + g 2m ψ~ 4 + 1 mV ψ~ 2 − i~ω m h ψ~x⊥· ∇ ψ~∗i −∇ · ~2 2m2 h ∂t ψ~ ∗ ∇ψ~+ ∂ tψ~∇ ψ~ ∗i + x⊥· ∇ i~ω m h ψ~∂ t ψ~ ∗i = 0. (2.41) Now adding up (2.39) and (2.41) and multiplying it by 1
2, we obtain the energy equation expressed in terms of ψ~ as ∂t ~2 2m2 ∇ψ~ 2 + g 2m ψ~ 4 + 1 mV ψ~ 2 +i~ω 2m h ψ~∗ x⊥· ∇ψ~− ψ~x⊥· ∇ ψ~∗i −∇ · ~2 2m2 h ∂t ψ~ ∗ ∇ψ~+ ∂ tψ~∇ ψ~ ∗i −∇ · x⊥i~ω 2m h ψ~∗ ∂tψ~− ψ~∂t ψ~ ∗i = 0. (2.42) Therefore, we derive the conservation of energy
∂tθ~+ ∇ · µ~ ω ρ~ θ~+ g 2m(ρ ~)2 = ~ 2 4m2∇ · µ~ ω∆ρ~ ρ~ − ∇ · µ~ ω∇ρ~ ρ~ , (2.43)
2.3. ENERGY EQUATION 13 where energy θ~= ~ 2 2m2 ∇ψ~ 2 + g 2m ψ~ 4 + 1 mV ψ~ 2 +i~ω 2m h ψ~∗ x⊥· ∇ψ~− ψ~x⊥· ∇ ψ~∗i = µ~ 2 2ρ~ + ~2 8m2 ∇ρ~ 2 ρ~ + g 2m(ρ ~)2+ 1 mV ρ ~− ω x⊥· µ~ = µ~ ω 2 2ρ~ + ~2 8m2 ∇ρ~ 2 ρ~ + g 2m(ρ ~)2+ ρ~ 1 mV − 1 2ω 2|x|2 = 1 2 µ~ ω p ρ~ !2 + ~ 2 2m2 ∇pρ~ 2 + g 2m ρ~+ m g 1 mV − 1 2ω 2|x|2 2 −m 2g 1 mV − 1 2ω 2|x|2 2 . (2.44) If we confine V (x) to satisfying 1 mV (x) − 1 2ω
CHAPTER 3
Semiclassical Limit of the Local Smooth Solutions
Let us consider the family, parameterized by ~, of solutions
ψ~(t, x) = A~(t, x) exp i
~
S~(t, x)
, t ∈ R+, x ∈ R2, (3.1)
of the rotating nonlinear Schr¨odinger equation (1.1) with rapidly oscillating initial condi-tion ψ~(0, x) = ψ~ 0(x) = A~0(x) exp i ~ S~ 0(x) , (3.2)
where the complex-valued function A~(t, x) denotes the amplitude, and the real-valued
function S~(t, x) denotes the phase. Unlike the usual WKB method to look for the solution
of the form
ψ~(t, x) = A~(t, x) exp i
~S(t, x)
, (3.3)
where S is independent of ~, we allow S~ to depend on ~. The initial density and
momentum satisfying the Euler equation (2.12) and (2.16) are then
ρ~(0, x) = A~ 0(x) 2 , µ~ ω(0, x) = A~ 0(x) 2 1 m∇S ~ 0(x) − ωx ⊥ . (3.4)
We will use the hydrodynamical structure derived in the preceding chapter to study the asymptotic behaviour of solutions ψ~(t, x) of the rotating nonlinear Schr¨odinger equation
(1.1) with initial condition (3.2) as ~ tends to zero. If we argue formally, it is natural to conjecture that the O(~2) dispersive term appearing in (2.16) is negligible as ~ tends
to zero, and the limiting density ρ and momentum µω satisfy the compressible rotational
Euler equation (1.7) with initial condition inferred from (3.4) given by
ρ(0, x) = |A0(x)|2, µω(0, x) = |A0(x)|2 1 m∇S0(x) − ωx ⊥ . (3.5) 15
Because the O(~2) dispersive term is nonlinear, we still have difficulty treating the problem directly from the hydrodynamical structure. According to Grenier [8], the mod-ified Madelung transformation can be employed in the study of the semiclassical limit. The procedure for expounding and proving are divided into three sections.
3.1. Quasilinear Symmetric Hyperbolic System
The first step in studying the semiclassical limit is to show the existence of a smooth solution ψ~ of the rotating nonlinear Schr¨odinger equation (1.1) on a finite time interval
[0, T ], independent of ~, for initial data A~
0(x) and S0~(x) with Sobolev regularity. We
will transform the rotating nonlinear Schr¨odinger equation into a dispersive perturbation of a quasilinear symmetric hyperbolic system. We will look for solutions (3.1) where A~ = a~+ ib~. After inserting (3.1) into (1.1), we obtain
i~∂tA~− A~∂tS~= −i~ m∇S ~· ∇A~− ~ 2 2m∆A ~+ 1 2mA ~ ∇S~ 2 − i~ 2mA ~∆S~+ g A~ 2 A~+ V A~+ i~ω x⊥· ∇A~ − ω x⊥· ∇S~ A~. (3.6) We split (3.6) into ∂tS~+ 1 2m ∇S~ 2 + g A~ 2 + V − ω x⊥· ∇S~ = 0, (3.7a) ∂tA~+ 1 m∇S ~· ∇A~+ 1 2mA ~∆S~− ω x⊥· ∇A~ = i~ 2m∆A ~, (3.7b)
based on whether the term is of order O(1) or of order O(~) and O(~2). The expression
(3.7a)–(3.7b) differs from (2.7)–(2.8), which are split into the real and imaginary parts, by the criteria of separating (3.6) into two parts. Notice that the second derivative term in (2.8) is highly nonlinear, whereas that in (3.7b) is linear. Therefore, the classical quasi-linear hyperbolic theory provides an approach to the semiclassical limit of the rotating nonlinear Schr¨odinger equation (1.1). The change of variable u~ = (u~
1, u~2) = 1 m∇S ~ leads to ∂tu~1 + 1 2∂x1(u ~ 1) 2+ (u~ 2) 2 + g m∂x1 A~ 2 + 1 m∂x1V − ω∂x1 x ⊥· u~ = 0, (3.8a) ∂tu~2 + 1 2∂x2(u ~ 1) 2+ (u~ 2) 2 + g m∂x2 A~ 2 + 1 m∂x2V − ω∂x2 x ⊥· u~ = 0, (3.8b) ∂tA~+ u~· ∇A~+ 1 2A ~∇ · u~− ω x⊥· ∇A~ = i~ 2m∆A ~. (3.8c)
3.1. QUASILINEAR SYMMETRIC HYPERBOLIC SYSTEM 17 Let A~= a~+ ib~; we have ∂t u~1 + ωx2 + 2g ma ~∂ x1a ~+2g mb ~∂ x1b ~+ u~ 1 + ωx2 ∂x1 u ~ 1+ ωx2 + u~ 2 − ωx1 ∂x2 u ~ 1 + ωx2 − ω2x1+ 1 m∂x1V = 2ω u ~ 2 − ωx1 , (3.9a) ∂t u~2 − ωx1 + 2g ma ~∂ x2a ~+2g mb ~∂ x2b ~+ u~ 1 + ωx2 ∂x1 u ~ 2 − ωx1 + u~ 2− ωx1 ∂x2 u ~ 2− ωx1 − ω2x2+ 1 m∂x2V = −2ω u ~ 1 + ωx2 , (3.9b) ∂ta~+ u~1 + ωx2 ∂x1a ~+ u~ 2 − ωx1 ∂x2a ~ +1 2a ~∂ x1 u ~ 1 + ωx2 + 1 2a ~∂ x2 u ~ 2 − ωx1 = − ~ 2m∆b ~, (3.9c) ∂tb~+ u~1 + ωx2 ∂x1b~+ u~2− ωx1 ∂x2b~ +1 2b ~∂ x1 u ~ 1+ ωx2 + 1 2b ~∂ x2 u ~ 2 − ωx1 = ~ 2m∆a ~, (3.9d)
with initial data
a~(0, x) = a~ 0(x), b~(0, x) = b~0(x), u~ ω(0, x) = u~1(0, x) + ωx2, u~2(0, x) − ωx1 = u~ 1,0(x) + ωx2, u~2,0(x) − ωx1 = u~0(x) − ωx⊥= u~ω,0(x), (3.10) satisfying a~ 0(x) 2 +b~ 0(x) 2 = A~ 0(x) 2 , u~ ω,0(x) = 1 m∇S ~ 0(x) − ωx ⊥ . (3.11) Let U~ ω = a~, b~, u~1 + ωx2, u~2 − ωx1 T
; the system can be written in the form
∂tUω~+ M1(Uω~)∂x1U ~ ω+ M2(Uω~)∂x2U ~ ω + G = L(Uω~), U~ ω(0, x) = Uω,0~ (x) = a~0(x), b~0(x), u~1,0(x) + ωx2, u~2,0(x) − ωx1 T , (3.12)
where M1(Uω~) = u~ 1 + ωx2 0 1 2a ~ 0 0 u~ 1 + ωx2 1 2b ~ 0 2g ma ~ 2g mb ~ u~ 1 + ωx2 0 0 0 0 u~ 1 + ωx2 , (3.13a) M2(Uω~) = u~ 2− ωx1 0 0 1 2a ~ 0 u~ 2 − ωx1 0 1 2b ~ 0 0 u~ 2 − ωx1 0 2g ma ~ 2g mb ~ 0 u~ 2 − ωx1 , (3.13b) G = 0 0 −ω2x 1+ 1 m∂x1V −ω2x 2+ 1 m∂x2V , (3.13c) and L(U~ ω) = 0 − ~ 2m∆ 0 0 ~ 2m∆ 0 0 0 0 0 0 2ω 0 0 −2ω 0 a~ b~ u~ 1 + ωx2 u~ 2 − ωx1 . (3.13d)
3.2. CLASSICAL SOLUTIONS 19
The matrix L is antisymmetric and reflects the dispersive nature of (3.12). For all (ξ, η)T ∈ R2, ξM1(Uω~) + ηM2(Uω~) = λ 0 ξ 2ma ~ η 2ma ~ 0 λ ξ 2mb ~ η 2mb ~ 2ξga~ 2ξgb~ λ 0 2ηga~ 2ηgb~ 0 λ , (3.14) where λ = ξ(u~
1 + ωx2) + η(u~2 − ωx1). The matrix (3.14) can be symmetrized by
M0(Uω~) = 1 0 0 0 0 1 0 0 0 0 1/4mg 0 0 0 0 1/4mg , (3.15)
which is symmetric and positive definite if g > 0 for all U~
ω, and has only real eigenvalues λ,
λ, λ ±r g m ξ
2
+ η2 (a~)2+ (b~)2. Thus, we write (1.1) as the dispersive perturbation
of the quasilinear symmetric hyperbolic system
M0(Uω~)∂tUω~+ fM1(Uω~)∂x1U ~ ω+ fM2(Uω~)∂x2U ~ ω + eG = eL(Uω~), U~ ω(0, x) = Uω,0~ (x), (3.16)
where fM1 = M0M1, fM2 = M0M2, eG = M0G, and eL = M0L. Here fM1 and fM2 are
symmetric, and eL is antisymmetric.
3.2. Classical Solutions
In order to carry out the existence of classical solutions, we proceed along the lines of the existence proof concerning the initial value problem for the quasilinear symmetric hyperbolic system with modifications. We utilize the iteration scheme for establishing the local existence in time. As a first approximation to the solution of (3.16), we consider U0
define successively Uωp+1(t, x, ~) as the solution of the linear equation M0(Uωp)∂tUωp+1+ fM1(Uωp)∂x1U p+1 ω + fM2(Uωp)∂x2U p+1 ω + eG = eL(Uωp+1), Up+1 ω (0, x, ~) = Uω,0~ (x). (3.17)
for p = 1, 2, 3, · · · . We call Uωp+1(t, x, ~), p = 0, 1, 2, 3, · · · successive approximations to a solution of (3.16). We might expect that Up
ω tends to Uω~ as p tends to ∞. For
further reference, we ignore the superscripts p and consider both Uω ∈ C∞ and Wω ∈ C∞
satisfying
M0(Wω)∂tUω+ fM1(Wω)∂x1Uω+ fM2(Wω)∂x2Uω + eG = eL(Uω),
Uω(0, x, ~) = Uω,0~ (x).
(3.18)
3.2.1. A Priori Estimate. The energy estimate is used to prove the existence of approximate solutions Uωp. Assume that the matrices fM1 and fM2 together with their
derivatives of any desired order are continuous and bounded uniformly in [0, T ] × R2. We
perform the process of the energy estimate in three stages.
Stage 1. L2-norm. The canonical energy associated with (3.18) is defined by the scalar product E(t) = (M0Uω, Uω) = Z Z M0Uω· Uωdx1dx2 = Z Z UωTM0Uωdx1dx2. (3.19)
For a certain T , let the function Uω(t, x, ~) be a solution of (3.18) of class C2([0, T ] × R2).
We use the symmetry of M0, fM1, and fM2 and integration by parts to have the basic
energy equality of Friedrich d dtE(t) = (ΓUω, Uω) + 2 e L(Uω), Uω − 2G, Ue ω , (3.20)
where Γ = ∂tM0+ ∂x1Mf1 + ∂x2Mf2, so that the classical energy estimate can be obtained
immediately. Since eL is an antisymmetric matrix, and we derive e L(Uω), Uω = Z Z UωTLUe ωdx1dx2 = Z Z UωTLUe ω T dx1dx2 = Z Z e LUω T Uωdx1dx2 = Z Z UωTLeTUωdx1dx2 = − Z Z UωTLUe ωdx1dx2, (3.21)
3.2. CLASSICAL SOLUTIONS 21
the term L(Ue ω), Uω
= 0 contributes nothing to the energy estimate. This means that the singular perturbation does not create energy. If G ∈ L2(R2), then we apply
Cauchy-Schwarz’s inequality and Young’s inequality to have
e G, Uω
< C + CkUωk2L2. (3.22)
Using the positive definite and symmetric matrix M0, we obtain
d
dtE(t) ≤ (kΓkL∞+ C) kUωk
2
L2 + C ≤ k (kΓkL∞+ C) E(t) + C, (3.23)
where an appropriate constant k > 1 is set to ensure that the last inequality holds. Because of the initial data
E(0) = M0Uω,0~ , Uω,0~ ≤ kM0kL∞kU~
ω,0k 2
L2, (3.24)
it follows from Gronwall’s inequality that for t ∈ [0, T ],
E(t) ≤ exp [k (kΓkL∞ + C) t] kM0kL∞kU~ ω,0k 2 L2 + Ct . (3.25) Furthermore, max 0 ≤ t ≤ T kUω(t, ~)k2L2 ≤ exp [k (kΓkL∞+ C) T ] kU~ ω,0k 2 L2 + CT kM0kL∞ . (3.26)
Stage 2. H1-norm. If we multiply (3.18) by (M
0(Wω))−1, differentiate with respect
to x1, and multiply it by M0(Wω), then we have
M0(Wω)∂t∂x1Uω+ fM1(Wω)∂x1∂x1Uω+ fM2(Wω)∂x2∂x1Uω = eL(∂x1Uω) + F1x1, ∂x1Uω(0, x, ~) = U ~ ω x1,0(x), (3.27a) where F1x1 = −∂x1Mf1∂x1Uω− ∂x1Mf2∂x2Uω− ∂x1G.e (3.27b)
Similarly, we differentiate with respect to x2 to have M0(Wω)∂t∂x2Uω+ fM1(Wω)∂x1∂x2Uω+ fM2(Wω)∂x2∂x2Uω = eL(∂x2Uω) + F1x2, ∂x2Uω(0, x, ~) = Uω~ x2,0(x), (3.28a) where F1x2 = −∂x2Mf1∂x1Uω− ∂x2Mf2∂x2Uω− ∂x2G.e (3.28b)
We expect to bound (M0∂x1Uω, ∂x1Uω) and (M0∂x2Uω, ∂x2Uω), where (·, ·) is the usual L
2
scalar product. Assume Uω ∈ C2([0, T ]; C3(R2)). Since M0, fM1, and fM2 are symmetric,
we have ∂t(M0∂x1Uω, ∂x1Uω) = (Γ∂x1Uω, ∂x1Uω) + 2 e L(∂x1Uω), ∂x1Uω + 2 (F1x1, ∂x1Uω) (3.29) and ∂t(M0∂x2Uω, ∂x2Uω) = (Γ∂x2Uω, ∂x2Uω) + 2 e L(∂x2Uω), ∂x2Uω + 2 (F1x2, ∂x2Uω) , (3.30)
where Γ = ∂tM0+ ∂x1Mf1+ ∂x2Mf2. The antisymmetry of eL yields the result
e L(∂x1Uω), ∂x1Uω =L(∂e x2Uω), ∂x2Uω = 0. (3.31)
From the structure of M0, fM1, and fM2and the application of Cauchy-Schwarz’s inequality
and Young’s inequality, we find the following estimates
2−∂xiMf1∂x1Uω− ∂xiMf2∂x2Uω, ∂xiUω ≤ C k∂x1Uωk 2 L2 + C k∂x2Uωk 2 L2 (3.32) for i = 1, 2. If ∂xiG ∈ L
2(R2) for i = 1, 2, then we apply again Cauchy-Schwarz’s
inequality and Young’s inequality to obtain
2−∂x1G, ∂e x1Uω + 2−∂x2G, ∂e x2Uω ≤ C + C k∂x1Uωk 2 L2 + C k∂x2Uωk 2 L2. (3.33)
Since M0 is a positive definite and symmetric matrix, we have
∂t[(M0∂x1Uω, ∂x1Uω) + (M0∂x2Uω, ∂x2Uω)]
≤ k (k Γ kL∞ +C) [(M0∂x
1Uω, ∂x1Uω) + (M0∂x2Uω, ∂x2Uω)] + C
3.2. CLASSICAL SOLUTIONS 23
for some k > 1. Because of the initial data M0 Uω~ x1,0, U ~ ω x1,0 +M0 Uω~ x2,0, U ~ ω x2,0 ≤ kM0kL∞ U ~ ω x1,0 2 L2 + U ~ ω x2,0 2 L2 , (3.35)
we deduce from Gronwall’s inequality and the strict positivity of M0 that
max 0 ≤ t ≤ T k∂x1Uω(t, ~)k 2 L2 + k∂x2Uω(t, ~)k 2 L2 ≤ exp [k (k Γ kL∞ +C) T ] · U ~ ω x1,0 2 L2 + U ~ ω x2,0 2 L2 + CT kM0kL∞ . (3.36)
Stage 3. H2-norm. From equations (3.27a) and (3.28a), we use the method similar to that in Stage 2 to obtain
M0(Wω)∂t ∂xi∂xjUω +Mf1(Wω)∂x1 ∂xi∂xjUω + fM2(Wω)∂x2 ∂xi∂xjUω =L ∂e xi∂xjUω + F2xixj, (3.37a) where F2xixj = −∂xi∂xjMf1∂x1Uω− ∂xi∂xjMf2∂x2Uω− ∂xiMf1∂x1∂xjUω −∂xiMf2∂x2∂xjUω− ∂xjMf1∂x1∂xiUω− ∂xjMf2∂x2∂xiUω− ∂xi∂xjG,e (3.37b)
for i, j = 1, 2. Assume Uω ∈ C2([0, T ]; C4(R2)). Because of the symmetry of M0, fM1, and
f M2, we have ∂t M0∂xi∂xjUω, ∂xi∂xjUω = Γ∂xi∂xjUω, ∂xi∂xjUω +2L ∂e xi∂xjUω , ∂xi∂xjUω + 2 F2xixj, ∂xi∂xjUω , (3.38)
where Γ = ∂tM0 + ∂x1Mf1 + ∂x2Mf2. The first term on the right side of (3.38) can be
bounded by Γ∂xi∂xjUω, ∂xi∂xjUω ≤ kΓkL∞ ∂xi∂xjUω 2 L2. (3.39)
The antisymmetry of eL leads to e L ∂xi∂xjUω , ∂xi∂xjUω = 0, (3.40)
and the usual estimates on commutators lead to
2 − ∂xi∂xjMf1∂x1Uω− ∂xi∂xjMf2∂x2Uω− ∂xiMf1∂x1∂xjUω −∂xiMf2∂x2∂xjUω− ∂xjMf1∂x1∂xiUω− ∂xjMf2∂x2∂xiUω, ∂xi∂xjUω ≤ C k∂x1Uωk 2 L2 + C k∂x2Uωk 2 L2 + C X |α|=2 k∂α xUωk2L2, (3.41)
where α is a multi-index of length |α|; that is, α = (α1, α2), |α| = α1 + α2. If ∂xi∂xjG ∈
L2(R2), meaning ∂ xi∂xj
1
m∇V (x)
∈ L2(R2), then we apply Cauchy-Schwarz’s
inequal-ity and Young’s inequalinequal-ity to obtain
2−∂xi∂xjG, ∂e xi∂xjUω < C + C ∂xi∂xjUω 2 L2. (3.42) Therefore, ∂t X |α|=2 (M0∂xαUω, ∂xαUω) ≤ (kΓkL∞ + C) X |α|=2 k∂α xUωk 2 L2 + C k∂x1Uωk 2 L2 + C k∂x2Uωk 2 L2 + C ≤ k (kΓkL∞ + C) X |α|=2 (M0∂xαUω, ∂xαUω) + C k∂x1Uωk 2 L2 + C k∂x2Uωk 2 L2 + C (3.43)
for some k > 1. It follows from Gronwall’s inequality and the strict positivity of M0 that
max 0 ≤ t ≤ T X |α|=2 k∂α xUω(t, ~)k2L2 ≤ exp [k (kΓkL∞+ C) T ] · X |α|=2 ∂xαU~ ω,0 2 L2 + C k∂x1Uωk 2 L2 + C k∂x2Uωk 2 L2 + C T kM0kL∞ . (3.44)
3.2. CLASSICAL SOLUTIONS 25 Setting Uω(t, x, ~) = U (t, x, ~) − ω(0, 0, −x2, x1) and Uω,0~ (x) = U0~(x) − ω(0, 0, −x2, x1), we can rewrite (3.44) as max 0 ≤ t ≤ T X |α|=2 k∂α xU (t, ~)k 2 L2 ≤ exp [k (kΓkL∞ + C) T ] · X |α|=2 ∂xαU~ 0 2 L2 + C k∂x1Uωk 2 L2 + C k∂x2Uωk 2 L2 + C T kM0kL∞ . (3.45)
to present a clear perspective.
As described in stage 3, the results of the higher energy estimate are obtained. Sum-marizing the above estimates, we conclude that
max 0 ≤ t ≤ T k Uω(t, ~) k2Hs≤ exp [k (kΓkL∞ + C) T ] kU~ ω,0k 2 Hs + CT kM0kL∞ , (3.46)
and we find that for sufficiently small T , we can estimate all ∂α
xUω for |α| ≤ s, s > 3.
This shows that if 1
m∇V (x) − ω
2
x ∈ Hs(R2) and U~
ω,0 ∈ Hs(R2), then the iteration
scheme defined by (3.17) is well-defined, and we also obtain a priori estimate on the space derivatives of the type
k Uωp(t, ~) kHs≤ C, t ∈ [0, T ], (3.47)
which denotes
Uωp ∈ L∞([0, T ]; Hs(R2)). (3.48)
In addition, it follows that every component of Uωp belongs to L∞([0, T ]; Hs(R2)) and then from (3.16) or (3.17) that for t ∈ [0, T ],
k∂tap(t, ~)kHs−2 ≤ C, k∂tbp(t, ~)kHs−2 ≤ C,
k∂tup1(t, ~)kHs−1 ≤ C, k∂tup2(t, ~)kHs−1 ≤ C.
(3.49)
The inclusion relation Hs−1(R2) ⊂ Hs−2(R2) leads to
k∂tUωp(t, ~)kHs−2 ≤ C, t ∈ [0, T ], (3.50)
which denotes
∂tUωp ∈ L ∞
Remark 3.1. Assume U~ = a~, b~, u~ 1, u~2
T
. It is convenient to rewrite U~
ω = U~−
ω(0, 0, −x2, x1)T and is helpful for the finer analysis. If
1 m∇V (x) − ω 2x ∈ H1 (R2), 1 m∇V (x) ∈ H s (R2), U~
ω,0 ∈ H1(R2), and U0~ ∈ Hs(R2), then we construct approximate
solutions Up ω(t, ~) = Up(t, ~) − ω(0, 0, −x2, x1)T, p = 0, 1, 2, · · · satisfying Up ω ∈ L ∞([0, T ]; H1(R2)), Up ∈ L∞([0, T ]; Hs(R2)), ∂ tUp ∈ L∞([0, T ]; Hs−2(R2)).
3.2.2. The Existence and Uniqueness Results.
Proposition 3.2. Let s > 3 and the potential V (x) satisfy 1
m∇V (x) − ω
2
x ∈ Hs(R2).
Assume that the initial data U~
ω,0 = a~0, b~0, u~1,0+ ωx2, u~2,0− ωx1
T
∈ [Hs
(R2)]4 satisfies the uniform bound
k U~
ω,0 kHs< C1.
Then there is a time interval [0, T ] with T > 0, so that the IVP for (3.12) has a unique classical solution U~ ω = a~, b~, u~1 + ωx2, u~2− ωx1 T ∈ C1([0, T ]; C2 (R2)). Furthermore, U~ ω ∈ C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)),
and T depends on the bound C1, and in particular, not on ~. In addition, the solution Uω~
satisfies the estimate
kU~
ω(t, ·)kHs < C2
for all t ∈ [0, T ]. The constant C2 is also independent of ~.
Proof. Following the results obtained in Subsection 3.2.1, for any fixed ~, we have constructed a sequence {Up
ω} ∞
p=0 belonging to
C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)), (3.52)
and satisfying (3.17) as well as the uniform estimate
max
0 ≤ t ≤ T
3.2. CLASSICAL SOLUTIONS 27
We use the mean value theorem to show that for every p, for 0 < t1, t2 < T and ξ ∈ (t1, t2),
kUp ω(t2, ~) − Uωp(t1, ~)kHs−2 = k∂tUωp(ξ, ~)(t2− t1)kHs−2 = |t2 − t1| k∂tUωp(ξ, ~)kHs−2 ≤ max 0 ≤ t ≤ T k∂tUωp(t, ~)kHs−2|t2− t1|. (3.54) Thus, if Up
ω : [0, T ] → Hs(R2) is continuous and differentiable on [0, T ], and ∂tUωp is
bounded for every t, then Uωp is the Lipschitz continuous function on [0, T ] with values in the norm topology of Hs−2(R2). This also explains that {Up
ω} ∞
p=0 is equicontinuous.
It follows from the Arzela-Ascoli theorem that there exists U~
ω ∈ L∞([0, T ]; Hs(R2)) ∩
Lip([0, T ]; Hs−2(R2)) such that
max 0 ≤ t ≤ T Uωp− U~ ω Hs−2 → 0, (3.55) as p tends to ∞. Hence, Up ω → Uω~ in C([0, T ]; Hs−2(R2)). (3.56)
Furthermore, we use the interpolation inequality to show that for 0 < θ < 1, Up ω− Uω~ Hs−α ≤ Up ω− Uω~ θ Hs Up ω− Uω~ 1−θ Hs−2 ≤ kUp ωkHs+ U~ ω Hs θ Uωp− U~ ω 1−θ Hs−2 ≤ C Uωp− U~ ω 1−θ Hs−2 → 0 (3.57)
as p tends to ∞ so that we have
Up
ω → Uω~ in C([0, T ]; Hs−α(R2)) (3.58)
for an appropriate constant α with 0 < α = 2 − 2θ < 2. When we choose s such that s − α − 2 > 2
2 = 1, the space H
s
(R2) becomes an algebra, in which we can perform multiplication and keep the product stay, to overcome the difficulty of the nonlinearity. Indeed, it can be shown that
U~
ω ∈ C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)), (3.59)
and U~
ω is a solution of (3.12). The Sobolev embedding theorem tells us that
if s ≥ 5. Thus, we have C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)) ,→ C1([0, T ]; C2(R2)) (3.61) and U~ ω ∈ C 1 ([0, T ]; C2(R2)) (3.62) which indicates that the constructed solution is classical.
In order to show that no extraction of subsequence is needed, we still prove the unique-ness of the classical solution of IVP for (3.12) by doing a straightforward energy esti-mate for the difference of two solutions. Let (U~
ω)1 and (Uω~)2 be two solutions satisfying
(U~
ω)1(0, x) = Uω,0~ (x) and (Uω~)2(0, x) = Uω,0~ (x). Let H~ = (Uω~)1− (Uω~)2; we get
M0∂tH~+ fM1(Uω~)1 ∂x1H ~+ fM 2(Uω~)1 ∂x2H ~= eL(H~) + F, (3.63a) where F = n f M1(Uω~)2 −Mf1(Uω~)1 o ∂x1(Uω~)2+ n f M2(Uω~)2 −Mf2(Uω~)1 o ∂x2(Uω~)2. (3.63b) We do the same procedure as before and expect to bound the canonical energy E(t) = (M0H~, H~). We have the basic energy equality of Friedrich
∂t M0H~, H~ = ΓH~, H~ + 2
e
L(H~), H~+ 2 F, H~ , (3.64)
where Γ = ∂tM0+ ∂x2Mf1+ ∂x2Mf2. On the right side of (3.64), the first term is bounded by
the assumption, and the second term vanishes due to the antisymmetry of eL. Applying Cauchy-Schwarz’s inequality leads to
2 F, H~ ≤ CkH~k2
L2. (3.65)
Therefore,
∂t M0H~, H~ ≤ (kΓkL∞+ C) kH~k2
L2 ≤ k (kΓkL∞+ C) M0H~, H~ , (3.66)
for some k > 1. By Gronwall’s inequality, it follows that for t ∈ [0, T ],
M0H~, H~ ≤ exp [k (kΓkL∞ + C) t] M0H0~, H0~ = 0. (3.67)
Since this holds for all t ∈ [0, T ], H~ = 0; that is, (U~
ω)1 = (Uω~)2. Thus, the classical
3.2. CLASSICAL SOLUTIONS 29
Remark 3.3. Under the assumptions of Remark 3.1, in fact, there exists U~belonging
to C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)) such that Up tends to U~ as p tends to ∞.
However, U~
ω belongs to C([0, T ]; H1(R2)).
We have proven the local existence and uniqueness of classical solutions of the disper-sive perturbation of the quasilinear symmetric hyperbolic system. The result of Proposi-tion 3.2 is pulled back to (1.1) which is equivalent to (3.12) for smooth soluProposi-tions.
Theorem 3.4. Assume the hypotheses of Proposition 3.2. Then the initial value problem of (1.1) and (3.2) has a unique classical solution in C1([0, T ]; C2(R2)) of the form
ψ~(t, x) = A~(t, x) exp i
~
S~(t, x)
on the time interval [0, T ]. Moreover, A~ and ∇S~ are bounded in L∞([0, T ]; Hs(R2))
uniformly in ~, and 1 m∇S
~− ωx⊥ is bounded in L∞([0, T ]; H1(R2)) uniformly in ~.
Proof. The finer insight in Remark 3.1 and Remark 3.3 gives us more information at a more detailed level. Since U~
0 ∈ Hs(R2) and Uω,0~ ∈ H1(R2), we have (A~0, S0~) ∈ Hs(R2) × Hs+1(R2) and 1 m∇S ~ 0 − ωx ⊥ ∈ H1
(R2). Because of the expression (3.1) of ψ~
in the initial value problem for the rotating nonlinear Schr¨odinger equation, ψ~ has the
same regularity as A~. Hence, we will observe the properties of A~ = a~+ ib~ and S~. It
follows from (3.59) that
A~ ∈ C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)). (3.68)
Further, the Sobolev embedding theorem implies that
C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)) ,→ C1([0, T ]; C2(R2)) (3.69)
if s ≥ 5. Since 1 m∇S
~= u~∈ C([0, T ]; Hs
(R2)) ∩ C1([0, T ]; Hs−1(R2)), we have S~(t, ·) ∈
Hs+1(R2) and ∂tS~(t, ·) ∈ Hs(R2); that is,
S~ ∈ C([0, T ]; Hs+1
Once again, the Sobolev embedding theorem implies that Hs(R2) ,→ C2(R2) if s ≥ 3. Hence, we obtain C([0, T ]; Hs+1(R2)) ∩ C1([0, T ]; Hs(R2)) ,→ C1([0, T ]; C2(R2)). (3.71) Moreover, 1 m∇S ~− ωx⊥ = u~ ω ∈ C([0, T ]; H 1 (R2)). (3.72) The initial value problem of (1.1) and (3.2) for the rotating nonlinear Schr¨odinger equation is equivalent to the dispersive quasilinear hyperbolic system (3.12) due to the existence of classical solutions. Applying this equivalent relation, we complete it.
3.2.3. The Properties of ρ~. When we expect that the Euler equation (2.12) and
(2.16) tends to the limiting Euler equation (1.7) as ~ tends to zero, ρ~ must be restricted
to ensure that there is not a singularity in the O(~2) dispersive term appearing in (2.16).
Before exploring the properties of ρ~, we obtain more information about the phase function
from the modified Madelung transformation. We employ the polar coordinates :
A~= a~+ ib~=pρ~exp iθ~ =pρ~ cos θ~+ i sin θ~ . (3.73)
We use the equality
a~∆b~− b~∆a~ = ∇ · ρ~∇θ~ , (3.74)
and then from (3.9a)–(3.9d), we derive the system
∂tρ~+ ∇ · ρ~ u~ ω+ ~ m∇θ ~ = 0, (3.75a) ∂tθ~+ u~ω· ∇θ~+ ~ 2m|∇θ ~|2 = ~ 2m ∇pρ~ p ρ~ , (3.75b) ∂tu~ω + u~ω· ∇ u~ω+ ∇ g mρ ~+ 1 mV = ω2x − 2ω u~ ω ⊥ . (3.75c) Equation (3.75a) has an extra term of order O(~) in comparison with the usual continuity equation. Moreover, this system (3.75a)–(3.75c) is of order O(~), but not of order O(~2)
in comparison with both (2.12) and (2.14). Consider the limiting equation of (3.75b)
∂tθ + uω· ∇θ = 0 (3.76)
with initial data θ(0, x) = 0. It follows immediately that
3.2. CLASSICAL SOLUTIONS 31
along the characteristic differential equation dx
dt = uω(t, x) subject to the initial condition x(0) = x0. We conclude that the limiting system of (3.75a)–(3.75c) are the same as the
limiting equations of (2.12) and (2.14) when ~ tends to zero.
Proposition 3.5. Assume the hypotheses of Proposition 3.2. If ρ~
0(x) = (a~0)2+(b~0)2 >
0, then ρ~(t, x) > 0 for all t ≥ 0; if ρ~
0 has a compact support, then ρ~(t, ·) does too for
any t ∈ [0, T ], and
Rρ~(t, ·) ≤ R ρ~
0 + (1 + ~)CT
where R{u} ≡ sup{|x| : u(x) 6= 0}.
Proof. Let (τ, ξ) be an arbitrary fixed time-space point in [0, T ] × R2. Since
u~
ω(t, x) +
~ m∇θ
~(t, x) ∈ C1([0.T ]; Hs−2(R2)) ∩ L∞([0.T ]; Hs(R2)), (3.78)
the Existence-Uniqueness theorem for ordinary differential equations guarantees that the problem dx dt = u ~ ω(t, x) + ~ m∇θ ~(t, x), x(τ ) = ξ, (3.79)
has a unique and continuous solution x = Ψ(t) ∈ C1([0, T ]; R2). Moreover, equation
(3.75a) is equivalent to an ODE
d dtρ ~(t, Ψ(t)) = ∂ tρ~(t, Ψ(t)) + ∇ρ~(t, Ψ(t)) · u~ ω(t, x) + ~ m∇θ ~(t, x) = −ρ~(t, Ψ(t)) ∇ · u~ ω(t, x) + ~ m∇θ ~(t, x) . (3.80)
Integrating the above equality over a time interval [0, τ ], we obtain
ρ~(τ, ξ) = ρ~(0, Ψ(0)) exp − Z τ 0 ∇ · u~ ω(t, x) + ~ m∇θ ~(t, x) dt . (3.81)
Thus, ρ~(τ, ξ) ≥ 0 if ρ~(0, Ψ(0)) = ρ~
0(Ψ(0)) ≥ 0. Denote R{u} ≡ sup{|x| : u(x) 6= 0} for
u ∈ C(R2). When ρ~(τ, ξ) 6= 0, ρ~ 0(Ψ(0)) 6= 0, so |Ψ(0)| ≤ R{ρ~0}, and |ξ| = |Ψ(τ )| = Ψ(0) + Z τ 0 u~ ω(t, x) + ~ m∇θ ~(t, x) dt ≤ |Ψ(0)| + Z τ 0 u~ ω(t, x) L∞+ ~ m ∇θ~(t, x) L∞dt ≤ Rρ~ 0 + (1 + ~)CT. (3.82) Hence, we obtain Rρ~(t, ·) ≤ R ρ~ 0 + (1 + ~)CT. (3.83) 3.3. Semiclassical Limit
Let Uω = (a, b, u1+ ωx2, u2− ωx1)T. The limiting system of (3.12) is the quasilinear
hyperbolic system ∂tUω+ M1(Uω)∂x1Uω+ M2(Uω)∂x2Uω+ G = L2(Uω), Uω(0, x) = Uω,0(x) = (a0(x), b0(x), u1,0(x) + ωx2, u2,0(x) − ωx1)T , (3.84) where L2(Uω) = 0 0 0 0 0 0 0 0 0 0 0 2ω 0 0 −2ω 0 a b u1+ ωx2 u2− ωx1 , (3.85)
and (3.84) is equivalent to the compressible rotational Euler equation (1.7) as long as solutions are smooth. We will show that it is possible to pass to the limit ~ → 0 in (3.12).
Proposition 3.6. Let U~
ω,0, Uω,0 ∈ Hs(R2), s > 3. Suppose that Uω,0~ (x) converges
to Uω,0(x) in Hs(R2) as ~ tends to zero. Let [0, T ] be the fixed interval determined in
Proposition 3.2. Then as ~ tends to zero, there exists Uω(t, x) ∈ L∞([0, T ]; Hs(R2)) such
that for 0 < δ < 2,
U~
3.3. SEMICLASSICAL LIMIT 33
The function Uω(t, x) belongs to C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)) and is a classical
solution of (3.84) with initial data Uω(0, x) = Uω,0(x).
Proof. Since {Uω~}~ is bounded in H s
(R2) for all t ∈ [0, T ], a weak compactness argument shows that for any fixed time t ∈ [0, T ], there exist a subsequence of {U~
ω}~
(always denoted by {U~
ω}~ due to the uniqueness) and a function Uω ∈ H s
(R2) such that U~
ω * Uω in Hs(R2) as ~ → 0. Similarly, ∂tUω~* ∂tUω in Hs−2(R2) as ~ → 0. We use the
mean value theorem to show that for all ~, for 0 < t < T and ξ ∈ (t, t + h), kU~ ω(t + h) − Uω~(t)kHs−2 = k∂tUω~(ξ)hkHs−2 ≤ h max 0 ≤ t ≤ T k∂tUω~(t)kHs−2 → 0 as h → 0, (3.86)
which denotes the sequence {U~
ω}~ is equicontinuous. The Arzela-Ascoli theorem implies
that there exists Uω ∈ L∞([0, T ]; Hs(R2)) ∩ Lip([0, T ]; Hs−2(R2)) such that
max
0 ≤ t ≤ T
kU~
ω(t) − Uω(t)kHs−2 → 0 (3.87)
as ~ tends to zero. Therefore,
∂tUω ∈ L∞([0, T ]; Hs−2(R2)), (3.88)
and
Uω ∈ C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)). (3.89)
The Sobolev embedding theorem shows that
C([0, T ]; Hs(R2)) ∩ C1([0, T ]; Hs−2(R2)) ,→ C1([0, T ]; C2(R2)) (3.90)
if s > 5. We deduce from the interpolation inequality that
U~
ω → Uω in C([0, T ]; Hs−δ(R2)), (3.91)
where 0 < δ < 2. Therefore,
U~
ω converges strongly in C([0, T ]; Hs−δ(R2)) to a function Uω. (3.92)
Furthermore, from the equation itself, we also have
U~
Since U~
ω,0(x) converges strongly to Uω,0(x) in Hs(R2), this limiting system has the initial
data Uω(0, x) = Uω,0(x). In particular, we note that L(Uω~) is uniformly bounded in
Hs−2(R2), so the perturbation term L(U~
ω) tends to L2(Uω) as ~ tends to zero. The
uniqueness proof of this system is like that in Subsection 3.2.2. Hence, the whole sequence
converges to Uω.
Remark 3.7. The strong convergence of U~
ω,0to Uω,0 implies that U0~converges strongly
to U0. For the same reason, the result of Proposition 3.6 reveals the fact that U~converges
strongly to U .
The relation of equivalence between (3.84) and (1.7) leads us to have the following convergent result, link T to the existence time of a smooth solution of (1.7), and ensure the strong convergence of ψ~ to a classical solution of the compressible rotational Euler
equation (1.7).
Theorem 3.8. Assume that (ρ, µω) is a solution of the compressible rotational Euler
equation (1.7) for 0 ≤ t ≤ T and belongs to C([0, T ]; Hs(R2)), s > 3, with initial condition
ρ0(x) = ρ(0, x) = |A0(x)|2, µω,0(x) = µω(0, x) = |A0(x)|2 1 m∇S0(x) − ωx ⊥ .
Then there exists a critical value of ~, ~c depending on T , such that under the hypotheses
(1) 1
m∇V (x) − ω
2x ∈ Hs
(R2), (2) A~
0(x) converges strongly to A0(x) in Hs(R2) as ~ tends to zero,
(3) (ρ0, µω,0) ∈ L∞([0, T ]; Hs(R2)),
(4) 0 < ~ < ~c,
the initial value problem of (1.1) and (3.2) has a unique classical solution ψ~ of the
form (3.1), where A~ and ∇S~ are bounded in L∞([0, T ]; Hs(R2)) uniformly in ~, and
1 m∇S
~ − ωx⊥ is bounded in L∞([0, T ]; H1(R2)) uniformly in ~, on [0, T ]. Moreover,
(ρ~, µ~
ω) converges strongly to the solution (ρ, µω) of (1.7) in C([0, T ]; Hs−2(R2)) as ~
3.3. SEMICLASSICAL LIMIT 35
Proof. Assume that there exists a solution (ρ, µω) in L∞([0, T ]; Hs(R2)) of (1.7) on
a time interval [0, T ] with s > 3 for the initial data
ρ0(x) = |A0(x)|2 = lim ~→0 A~ 0(x) 2 , µω,0(x) = |A0(x)|2 1 m∇S0(x) − ωx ⊥ = lim ~→0 A~ 0(x) 2 lim ~→0 1 m∇S ~ 0(x) − ωx ⊥ , (3.94) satisfying kρ0(·)kHs < C, kµω,0(·)kHs < C. It makes sense since kUω,0~ (·)kHs < C1, and
U~
ω,0(x) converges strongly to Uω,0(x) in Hs(R2) as ~ tends to zero.
The existence time T of solutions of (1.7) coincides with that in Proposition 3.2. There will be no confliction. Assume that the limiting system (3.84) admits a solution on a maximal time interval [0, T∗]. Let us prove that T∗ > T . If T∗ ≤ T , then ρ and µω are
in L∞([0, T∗]; Hs(R2)), so u
ω ∈ L∞([0, T∗]; Hs(R2)). By using (3.9c) and (3.9d), we get
that a and b are in L∞([0, T∗]; Hs−1(R2)), which is impossible since T∗ is set to be the
maximal time of existence. Hence, T∗ > T .
Along the lines of the proof of Proposition 3.6, we consider the difference of (3.12) and (3.84). Set H~= U~ ω− Uω. Then ∂tH~+ M1 H~+ Uω ∂x1H ~+ M 2 H~+ Uω ∂x2H ~= L H~ + F~, (3.95a) where F~ = (L − L 2) (Uω) −M1 H~+ Uω − M1(Uω) ∂x1Uω −M2 H~+ Uω − M2(Uω) ∂x2Uω. (3.95b)
Since M0 is positive definite for all (H~+ Uω), (3.95a) becomes
M0∂tH~+ fM1 H~+ Uω ∂x1H
~+ fM
2 H~+ Uω ∂x2H
~ = eL H~ + M
0F~, (3.96)
where fM1 = M0M1, fM2 = M0M2, and eL = M0L. Here the matrices fM1 H~+ Uω and
f
M2 H~+ Uω are symmetric. The energy associated with (3.96) is
E(t) = M0H~, H~ =
Z Z
H~T
and the Friedrich energy equality is written as d dtE(t) = Γ ~H~, H~ + 2 e L H~ + M 0F~, H~ , (3.98) where Γ~= ∂
tM0+ ∂x1Mf1+ ∂x2Mf2. Since eL is antisymmetric, we have
( eL(H~), H~) = 0. (3.99)
Applying Cauchy-Schwarz’s inequality and Young’s inequality leads to e L − fL2 (Uω), H~ ≤ ~C + ~C H~ 2 L2, (3.100) and for i = 1, 2, −hMfi(H~+ Uω) − fMi(Uω) i ∂xiUω, H ~≤ C H~ 2 L2. (3.101) Hence, d dtE(t) ≤ (kΓkL∞ + ~C + C) H~ 2 L2 + ~C ≤ k(kΓkL∞+ ~C + C) M0H~, H~ + ~C (3.102)
for some k > 1. By applying Gronwall’s inequality and the strict positive of M0, we
deduce that for t ∈ [0, T ], H~ 2 L2 ≤ exp[k(kΓkL∞+ ~C + C)t] U~ ω,0(x) − Uω,0(x) L2 + ~Ct kM0kL∞ = C(~) → exp[k(kΓkL∞+ C)t] · (0 + 0) = 0 (3.103)
as ~ tends to zero. We complete the proof.
These results indicate that the regularity (3.59) of solutions of the quasilinear hyper-bolic system (3.12) controls that of solutions of the quantum hydrodynamic equations of the rotating nonlinear Schr¨odinger equation (1.1).
CHAPTER 4
WKB Expansion
We must be content with approximate solutions of the system (3.12) obtained by perturbation expansion : U~ ω = U (0) ω + ~U (1) + ~2U(2)+ · · · + ~NU(N )+ · · · , (4.1)
where Uω(0) = U(0) − ω (0, 0, −x2, x1)T. We write M1(Uω~) as the Taylor series expansion
around Uω(0) M1(Uω~) = M1 Uω(0)+ ~U(1)+ ~2U(2)+ · · · + ~NU(N )+ · · · = M1(U (0) ω ) + DM1(U (0) ω ) ~U(1)+ ~2U(2)+ · · · +D 2M 1(U (0) ω ) 2! ~U (1) + ~2U(2)+ · · ·2+D 3M 1(U (0) ω ) 3! ~U (1) + ~2U(2)+ · · ·3 + · · · + D NM 1(U (0) ω ) N ! ~U (1) + ~2U(2)+ · · ·N + · · · . (4.2) Similarly, we do the same to M2(Uω~). We present the hierarchy by the order of ~ as
follows : ∂tUω(0)+ M1(Uω(0))∂x1U (0) ω + M2(Uω(0))∂x2U (0) ω + G = L2(Uω(0)), (4.3a) ∂tU(1)+ M1(U (0) ω )∂x1U (1)+ M 2(U (0) ω )∂x2U (1) +DM1(U (0) ω )U(1)∂x1U (0) ω + DM2(U (0) ω )U(1)∂x2U (0) ω = L1(U (0) ω ) + L2(U(1)), (4.3b) 37
∂tU(2)+ M1(U (0) ω )∂x1U (2)+ M 2(U (0) ω )∂x2U (2) +DM1(U (0) ω ) h U(1)∂ x1U (1)+ U(2)∂ x1U (0) ω i + DM2(U (0) ω ) h U(1)∂ x2U (1)+ U(2)∂ x2U (0) ω i +D 2M 1(U (0) ω ) 2! U (1)2 ∂x1U (0) ω + D2M 2(U (0) ω ) 2! U (1)2 ∂x2U (0) ω = L1(U(1)) + L2(U(2)), (4.3c) .. .
We have the general formula
∂tU(N ) + 2 X j=1 N X k = 0 0 ≤ ni≤ k, ni∈ N, i = 1, · · · , N n1+ n2+ · · · + nN= k n1· 1 + n2· 2 + · · · + nN· N + nxj = N 0 < nxj ≤ N, nxj ∈ N DkM j(U (0) ω ) k! ·U(1)n1 U(2)n2 · · · · ·U(N )nN ∂xjU (nxj) + 2 X j=1 N X k = 0 0 ≤ ni≤ k, ni∈ N, i = 1, · · · , N n1+ n2+ · · · + nN= k n1· 1 + n2· 2 + · · · + nN· N = N DkM j(U (0) ω ) k! ·U(1)n1 U(2)n2 · · · · ·U(N )nN ∂xjU (0) ω = L1(U(N −1)) + L2(U(N )), N = 2, 3, · · · (4.4)
where L = ~L1+ L2, L2 is as that in (3.85), and
L1 = 0 − 1 2m∆ 0 0 1 2m∆ 0 0 0 0 0 0 0 0 0 0 0 . (4.5)
