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A SIMPLE PROOF OF WELL-POSEDNESS FOR THE

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Well-posedness

Well-posedness

Absห้องสมุดไป่ตู้ract
1 Introduction
In part, our purpose in the rst sections (2{7) of the paper will be to show that even the simplest phase- eld gradient ows for a free energy functional do not always provide well-posed initial value problems. Speci cally, our examples exhibit nonuniqueness|we prove the local existence of many solutions. A special feature of this nonuniqueness is the phenomenon of spontaneous generation of a new phase (Section 7) in a manner akin to nucleation, but with no apparent nucleus. We also look at traveling waves in Section 8 and nd that they exist for these same systems, but again are not unique. The second part of the paper (Sections 9{10) shows that by postulating an additional type of free energy, namely one concentrated at discontinuities, the initial value problem becomes well-posed. We prove the local existence of a unique solution, under natural assumptions on the data. Nonuniqueness phenomena in traveling wave solutions for gradient ows is familiar from Fisher's equation, for which one can choose the wave speed arbitrarily, subject only to an inequality. But in that case, the nonuniqueness is associated with the fact that the traveling wave converts an unstable to a stable state, and the various wave pro les are characterized by their rates of decay in space as one approaches the unstable state ahead of the wave front. This circumstance is not present in the models we are investigating here; in fact traveling waves represent a transition between two stable states. The multiplicity arises from quite a di erent source.

Scattering and Inverse Scattering on ACH Manifolds

Scattering and Inverse Scattering on ACH Manifolds

dt2 + h(t) t2 ∞ ¯ The function ρ := ψ∗ t is where h(t), t ∈ [0, ǫ) is a C 1-parameter family of C ∞ metrics on ∂ X. ¯ ¯ ¯ Note that a boundary defining function in X near ∂ X, which can be extended smoothly to X. n the boundary represents the geometric infinity of X , as does the sphere S for the hyperbolic space Hn+1 the Laplacian of (X, g ) was studied in [27]; it consists a finite pure point spectrum σpp (∆), which is the set of L2 (X ) eigenvalues, and an absolutely continuous spectrum σac (∆) satisfying The resolvent σac (∆) = n2 /4, ∞ and σpp (∆) ⊂ 0, n2 /4 .
R(λ) = (∆g − λ(n − λ))−1 ,
2000 Mathematics Subject Classification. Primary 58J50, Secondary 35P25. Key words and phrases. scattering, pseudoconvex domains, asymptotically complex hyperbolic manifolds. The first author was partially supported by French ANR grants no. [JC05-52556] and [JC05-46063], and an Australian National University postdoctoral fellowship. Both authors were supported in part by the National Science Foundation Grant DMS 0500788.

Fourier Analysis Methods for PDE

Fourier Analysis Methods for PDE

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Introduction
Since the 80’s, Fourier analysis methods have known a growing interest in the study of linear and nonlinear PDE’s. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient. Littlewood-Paley decomposition has been introduced more than fifty years ago in harmonic analysis but its systematic use in the PDE’s framework is rather recent. Paradifferential calculus, as for it, has been introduced in 1981 by J.-M. Bony for the study of the propagation of microlocal singularities in nonlinear hyperbolic PDE’s (see [4]). In the present notes, we aim at giving a survey of those techniques and a few examples of how they may used to solve PDE’s. We focus on two linear models: the heat equation and the transport equation. For each of them, an example of related nonlinear problem is given. Although most of the results we present here belong to the mathematical folklore, we want to point out that Fourier analysis methods are very efficient to tackle most of well-posedness problems for evolutionary PDE’s in the whole space or in the torus. The notes are structured as follows. The first chapter deals with Fourier analysis. We introduce Littlewood-Paley decomposition and show how it may used to characterize functional spaces. We also give a (non so) short insight into the theory of homogeneous spaces which turn out to be well adapted to the study of many PDE’s and are omitted in most of textbooks on functional analysis. Finally, we introduce some smatterings of paradifferential calculus and prove estimates for the product of two temperate distributions, when it makes sense. In the second chapter, we focus on the heat equation. We state a priori estimates in Besov spaces with optimal gain of derivatives. As an application, we prove global well-posedness in Besov spaces with critical regularity for the incompressible Navier-Stokes equations with small data. The third chapter is devoted to the study of transport equations associated to Lipschitz vectorfields. We state a priori estimates in Besov spaces then apply our results to the study of a shallow water equation. In the last chapter, we give an example of coupling between heat equation and transport equation, namely the compressible barotropic Navier-Stokes system. Well-posedness in Besov spaces with critical regularity is stated. Acknowledgments The author is grateful to J.-Y. Chemin for supplying most material for the first chapter, to Zhouping Xin, Chiaojang Xu and Yinbin Deng for the invitation to deliver this course at Wuhan Normal University, to Ping Zhang for his kind invitation at the Chinese Academy of Sciences in Beijing and to Yuxin Ge for helping me to communicate between France and China.

0模英语作文最有效的一句

0模英语作文最有效的一句

0模英语作文最有效的一句【中英文实用版】The most effective sentence in the 0-model English essay can be summarized as follows: "Practice makes perfect, and consistency is key."在0模英语作文中，最有效的一句话概括起来就是：“熟能生巧，持之以恒是关键。

”In this sentence, the power of diligence and perseverance is highlighted, emphasizing the importance of regular practice and continuous effort in mastering the English language.这句话凸显了勤奋与坚持的力量，强调了在日常练习和持续努力中对掌握英语的重要性。

Moreover, it serves as a motivational mantra for learners, reminding them that slow and steady progress is better than a rapid yet fleeting improvement.此外，这句话也为学习者提供了一个激励性的口头禅，提醒他们稳扎稳打地进步比短暂快速的提高更为有效。

This simple yet profound statement encapsulates the essence of language learning, making it an invaluable piece of advice for anyone striving to excel in English composition.这句简单却深刻的话语概括了语言学习的精髓，对于任何努力在英语作文中取得优异成绩的人来说，这都是一条宝贵的建议。

美国习惯用语theproofisinthepudding实践是真正的检验，空谈不如实证

美国习惯用语theproofisinthepudding实践是真正的检验，空谈不如实证

美国习惯用语theproofisinthepudding实践是真正的检验，空谈不如实证proof就是证明，pudding是西菜里的一种甜点，中文翻成布丁。

the proof is in the pudding的意思是说“布丁好不好，吃了才知道”，只有通过体验才能判断事物的好坏，也就是中国人常说的“实践是真正的检验，空谈不如实证”。

比如说，你的朋友推荐你用一种洗发液，你怎么知道这个产品好用不好用呢？看广告或是听推销员的介绍并不可靠。

这种洗发液适合不适合你，只有亲自用一下才知道。

这就是：•The proof is in the pudding.下面是一位公司主管在对销售部门的雇员讲话。

•I have to say I didn't really like your sexy approach to selling our product, but the proof is in the pudding, and our market share has grown. So I have to conclude that you know exactly what you're doing, and keep up the good work!•我得说我原来并不喜欢你们那种浮夸的推销方式。

不过销售的结果证明你们的做法很好；我们的市场份额增加了。

所以我得出的结论就是，你们是知道该怎么作的，各位就再接再励吧。

The proof is in the pudding这个习惯用语在14世纪就出现了，不过当时人们说的是：the proof of the pudding is in the eating。

后来才逐渐演变成现在的说法：the proof is in the pudding。

不过两种说法的意思是一样的，都是说只有实践才能说明好不好。

下面我们再来听一个例句。

这是一个法学院在它的网站上说，要看这个法学院的教学质量，只要看看这个学校培养出来的学生就知道了。

Global Well--posedness of the Three-dimensional Viscous Primitive Equations of Large Scale

Global Well--posedness of the Three-dimensional Viscous Primitive Equations of Large Scale
boundary conditions:
τ
· n = 0,
∂τ ∂n
× n = 0,
on ∂M.
(13)
∂T ∗ ∂n
=
0
on ∂M.
(14)
PRIMITIVE EQUATIONS
3
In addition, we supply the system with the initial condition:
on Γu :
∂v ∂z
= h τ,
w = 0,
∂T ∂z
= −α(T − T ∗);
(10)
on Γb :
∂v ∂z
= 0,
w = 0,
∂T ∂z
= 0;
(11)
on Γs : v · n = 0,
∂v ∂n
×
n
=
0,
∂T ∂n
= 0,
(12)
where τ (x, y) is the wind stress on ocean surface, n is the normal vector to Γs, and T ∗(x, y) is typical temperature distribution of the top surface of the ocean. For simplicity we assume here that τ and T ∗ are
arXiv:math/0503028v2 [math.AP] 16 Nov 2005
GLOBAL WELL-POSEDNESS OF THE THREE-DIMENSIONAL VISCOUS PRIMITIVE EQUATIONS OF LARGE SCALE OCEAN AND ATMOSPHERE

高英logicfallacy八大逻辑谬误

No!
The cause has no connection with the result.
原因和结果没有必然联系
Contradictory premises 矛盾前提
He decided to give up all his decisions.
No!
When the premises of an argument contradict
No!
Making an analogy between different things.
Hypothesis contrary to fact
与事实相反的假设
If I had studied harder, I would definitely have passed that test .
No!
Exercise is good，so everybody should exercise.
It means an argument based on an unqualified generalization.
即将普遍规则运用于有特殊情况发生 的个别情形之中。
Hasty generalization 草率结论
It applies a special case to general rule.
由个别情形来推断普遍规则.
Post hoc 牵强附会
McDonald's and KFC offer foods with little nutrition, and thus we cannot expect any fast food restaurant to provide us with nutritious foods.
Eight Types of Fallacy

Recovery of the Linearized 4D AdS and dS Schwarzschild Metric in the Karch-Randall Branewor


(5)
∗
E-mail: giannak@
1
where prime and dot denote differentiation with respect to r and y . Note that we have omitted a brane stress energy µ tensor, Tν δ (y ), as a source. The interested reader should consult reference [3] for a detailed analysis in the presence of matter on the brane. The brane tension is accounted for by the kink in the warp factor, Eq. (2). By integrating Eq. (5) across the brane, one obtains the Israel matching conditions on the metric functions a, b and c. In the absense of sources on the brane the Israel matching conditions become the Neumann boundary conditions. These equations constitute the starting point of our investigation. The gauge symmetries of these equations, the coordinate transformations that respect the axially symmetric, static form of the metric, Eq. (4), are generated by the functions v and u that obey the following relations to linear order

在细微处见功夫 作文

在细微处见功夫作文英文回答：In the details lies the effort. This phrase has always resonated with me as it emphasizes the importance of paying attention to the small things in life. Whether it's in our personal relationships, work, or even hobbies, putting effort into the finer details can make a significant difference.One area where I have seen the impact of paying attention to details is in my job as a graphic designer. When creating a design, it's not just about the overall composition, but also the small elements that come together to create a cohesive and visually appealing piece. For example, choosing the right font, color scheme, and spacing can greatly enhance the overall design. By meticulously considering each element, I am able to create designs that are not only aesthetically pleasing but also effectively convey the intended message.Another aspect where I have witnessed the power of paying attention to details is in my relationships with others. It's the small gestures, like remembering someone's birthday or listening attentively when they speak, that can make someone feel valued and appreciated. These seemingly insignificant actions can strengthen the bond between individuals and foster a sense of trust and connection.Furthermore, in my personal life, I have found that paying attention to details can greatly improve my overall well-being. For instance, taking the time to prepare a nutritious meal rather than opting for fast food can have a positive impact on my physical health. Similarly, being mindful of my thoughts and emotions allows me to better understand myself and make necessary adjustments for personal growth.中文回答：在细微之处见功夫。

什么是积极的方式英语作文

As a high school student, Ive often pondered the essence of positivity and its impact on our lives. The concept of a positive way is not just an abstract idea but a tangible force that can shape our experiences and attitudes. Its a mindset that encourages us to approach life with an optimistic outlook, focusing on the brighter side of things even in the face of adversity.Growing up, I was surrounded by a family that valued optimism. My parents always emphasized the importance of maintaining a positive attitude, no matter the circumstances. They would often say, A smile can change your day, and a positive attitude can change your life. This simple yet profound statement has been a guiding principle for me.One of the most vivid examples of positivity in action occurred during a challenging period in my life. It was during my sophomore year when I faced a series of setbacks, including a disappointing grade on a major exam and a strained relationship with a close friend. These events weighed heavily on me, and for a time, I found myself enveloped in a cloud of negativity.However, it was during this time that I truly understood the power of a positive way. I decided to take control of my thoughts and shift my focus. Instead of dwelling on what went wrong, I started to list down what I was grateful for and what I could learn from the situation. This simple act of reflection and gratitude helped me to see the silver lining in my challenges.I began to approach my studies with a renewed sense of purpose, and I made an effort to communicate openly with my friend, which eventuallyled to reconciliation.The positive way is not just about ignoring the negative aspects of life its about acknowledging them and choosing to respond in a constructive manner. Its about cultivating resilience and finding the strength to bounce back from setbacks. For instance, when I failed to make the school soccer team, instead of feeling defeated, I used it as an opportunity to improve my skills and fitness. I joined a local club, trained harder, and eventually, my efforts paid off when I was selected for the team the following year.Moreover, adopting a positive way has a ripple effect on those around us. When we radiate positivity, it can inspire others to do the same. Ive witnessed this firsthand in my interactions with classmates and friends. A simple compliment or a word of encouragement can lift someones spirits and create a more supportive and uplifting environment.In conclusion, the positive way is a powerful tool that can transform our lives. Its about embracing gratitude, resilience, and optimism. Its about turning obstacles into opportunities and viewing failures as stepping stones towards success. By choosing to approach life with a positive mindset, we can navigate through challenges with grace and emerge stronger and more determined. The positive way is not just a philosophy its a practical approach to life that can lead to personal growth, better relationships, and a more fulfilling existence.。

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Abstract. This paper contains the lectures presented by Shkoller at the Eleventh School on the Mathematical Theory in Fluid Mechanics at Kacov, Czech republic, in May, 2009, based on the well-posedness theory for the free-surface incompressible Euler equations developed in Coutand & Shkoller [3]. The purpose of those lectures and this paper is to present a new simple proof for the construction of solutions to the moving free-boundary incompressible 3-D Euler equations in vacuum. The method relies on the Lagrangian representation of the fluid, and the anisotropic smoothing operation that we call horizontal convolution-by-layers.
2000 Mathematics Subject Classification. 35L65, 35L70, 35L80, 35Q35, 35R35, 76B03. Key words and phrases. Euler, water-waves, free boundary problems, vacuum.
The open subset Ω(t) ⊂ R3 denotes the changing volume occupied by the fluid, Γ(t) := ∂ Ω(t) denotes the moving free-surface, V (Γ(t)) denotes normal velocity of Γ(t), and n(t) denotes the exterior unit normal vector to the free-surface Γ(t). The vector-field u = (u1 , u2 , u3 ) denotes the Eulerian velocity field, and p denotes the pressure function. We use the notation D = (∂1 , ∂2 , ∂3 ) to denote the gradient operator. We have normalized the equations to have all physical constants equal to 1. This is a free-boundary partial differential equation to determine the velocity and pressure in the fluid, as well as the location and smoothness of the a priori unknown free-surface. A recent explosion of interest in the analysis of the free-boundary incompressible Euler equations, particularly in irrotational form, has produced a number of different methodologies for obtaining a priori estimates, and the accompanying existence theories have mostly relied on the Nash-Moser iteration to deal with derivative loss in linearized equations when arbitrary domains are considered, or complex analysis tools for the irrotational problem with infinite depth. We refer the reader to [1], [3], [5], [6], [7], [8], [11], [12], [13], [14] for a partial list of papers on this topic. The purpose of these lectures is to make our presentation in [3] more accessible, and to explain the basic interaction of the geometry of the free-surface with the motion of the fluid. As is often the case, with the passage of time, a more concise treatment can be presented, and this is indeed the case for the free-surface incompressible Euler problem. To avoid the use of local coordinate charts necessary for arbitrary geometries, for simplicity, we will assume that the initial domain Ω at time t = 0 is given by Ω = {(x1 , x2 , x3 ) ∈ R3 | (x1 , x2 ) ∈ T2 , x3 ∈ (0, 1)} , (2)
where T2 denotes the 2-torus, which can be thought of as the unit square (0, 1)2 with periodic boundary conditions. This permits the use of one global Cartesian coordinate system. We only allow the top boundary Γ = {x3 = 1} to move, while the bottom boundary is fixed with boundary condition u3 = 0 on {x3 = 0} × [0, T ] . We refer the reader to our paper [3] for the case that the initial domain is an arbitrary bounded, open subset of R3 with H 3 -class boundary. 1.2. Einstein’s summation convention. Repeated Latin indices i, j, k,, etc., are 2 summed from 1 to 3; for example, F,ii := i=1,3 ∂x∂ . i ∂xi
1. Introduction. This paper is founded upon the lectures presented by Shkoller at the Eleventh School on the Mathematical Theory in Fluid Mechanics at Kacov, Czech republic, in May, 2009 on the well-posedness of the free-surface incompressible Euler equations as developed in Coutand & Shkoller [3]. We are grateful to the organizers Josef M´ alek and Mirko Rokyta for both the opportunity to present these lectures, as well as for the exceptional summer school that they organized. We are also grateful to the many students who attended this summer school, and whose excitement for the subject matter motivated us greatly to provide a new and concise treatment of this well-posedness theory. These lectures are focused on the incompressible Euler equations with moving free-surface. While we present the material for the case of three space dimensions, the same presentation covers the case of plane flow also. We will not consider the presence of surface tension in this paper, and we will not make any assumptions of irrotationality. 1.1. The Eulerian description. For 0 ≤ t ≤ T , the evolution of a three-dimensional incompressible fluid with a moving free-surface is modeled by the incompressible