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Introduction to Partial Differential Equations
偏微分方程导论

Dongfen Bian 1 1 ^(1){ }^{1} and Emmanuel Grenier 1 1 ^(1){ }^{1}
董芬·边 1 1 ^(1){ }^{1} 和埃马纽埃尔·格雷尼耶 1 1 ^(1){ }^{1}

September 9, 2024 2024 年 9 月 9 日
1 1 ^(1){ }^{1} School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China
北京理工大学数学与统计学院,中国北京

Chapter 1 第一章

Introduction 介绍

1.1 Objectives of these book
1.1 本书的目标

This book is an introduction to the theory of partial differential equations with two main axis
这本书是关于偏微分方程理论的介绍,主要有两个轴心
  • the computation of simple explicit solutions that can be studied in details in order to understand the properties of the corresponding partial differential equations,
    简单显式解的计算可以详细研究,以便理解相应偏微分方程的性质,
  • the study of the existence, uniqueness and qualitative properties of solutions of simple partial differential equations, relying on methods which do not use explicit solutions. These methods are general and can be applied to many different equations.
    对简单偏微分方程解的存在性、唯一性和定性性质的研究,依赖于不使用显式解的方法。这些方法是一般性的,可以应用于许多不同的方程。
The reader should be familiar with calculus, integration theory, implicit function theorem and elementary Fourier analysis.
读者应熟悉微积分、积分理论、隐函数定理和基础傅里叶分析。

1.2 Notations 1.2 符号

1.2.1 Derivatives 1.2.1 导数

In R n R n R^(n)\mathbb{R}^{n}, we will use the following notations
R n R n R^(n)\mathbb{R}^{n} 中,我们将使用以下符号
t = t , j = x j t = t , j = x j del_(t)=(del)/(del t),quaddel_(j)=(del)/(delx_(j))\partial_{t}=\frac{\partial}{\partial t}, \quad \partial_{j}=\frac{\partial}{\partial x_{j}}
where j = 1 , 2 , , n j = 1 , 2 , , n j=1,2,dots,nj=1,2, \ldots, n and x j x j x_(j)x_{j} is the j th j th  j^("th ")j^{\text {th }} coordinate.
其中 j = 1 , 2 , , n j = 1 , 2 , , n j=1,2,dots,nj=1,2, \ldots, n x j x j x_(j)x_{j} j th j th  j^("th ")j^{\text {th }} 坐标。

In R R R\mathbb{R}, we will denote the partial derivative with respect to x x xx by x x del_(x)\partial_{x}.
R R R\mathbb{R} 中,我们将用 x x del_(x)\partial_{x} 表示对 x x xx 的偏导数。
Moreover, grad\nabla is the operator “nabla”
此外, grad\nabla 是算子“nabla”
= ( 1 , , n ) = 1 , , n grad=(del_(1),cdots,del_(n))\nabla=\left(\partial_{1}, \cdots, \partial_{n}\right)
In particular, if V ( t , x ) = ( V 1 ( t , x ) , , V n ( t , x ) ) V ( t , x ) = V 1 ( t , x ) , , V n ( t , x ) V(t,x)=(V_(1)(t,x),dots,V_(n)(t,x))V(t, x)=\left(V_{1}(t, x), \ldots, V_{n}(t, x)\right), 特别是,如果 V ( t , x ) = ( V 1 ( t , x ) , , V n ( t , x ) ) V ( t , x ) = V 1 ( t , x ) , , V n ( t , x ) V(t,x)=(V_(1)(t,x),dots,V_(n)(t,x))V(t, x)=\left(V_{1}(t, x), \ldots, V_{n}(t, x)\right)
V = j = 1 n j V j V = j = 1 n j V j grad*V=sum_(j=1)^(n)del_(j)V_(j)\nabla \cdot V=\sum_{j=1}^{n} \partial_{j} V_{j}
is the divergence of V V VV
V V VV 的散度

If V ( t , x ) = ( V 1 ( t , x ) , , V n ( t , x ) ) V ( t , x ) = V 1 ( t , x ) , , V n ( t , x ) V(t,x)=(V_(1)(t,x),dots,V_(n)(t,x))V(t, x)=\left(V_{1}(t, x), \ldots, V_{n}(t, x)\right) and if ϕ ( t , x ) ϕ ( t , x ) phi(t,x)\phi(t, x) is a function, then
如果 V ( t , x ) = ( V 1 ( t , x ) , , V n ( t , x ) ) V ( t , x ) = V 1 ( t , x ) , , V n ( t , x ) V(t,x)=(V_(1)(t,x),dots,V_(n)(t,x))V(t, x)=\left(V_{1}(t, x), \ldots, V_{n}(t, x)\right) ,并且如果 ϕ ( t , x ) ϕ ( t , x ) phi(t,x)\phi(t, x) 是一个函数,那么
( V ) ϕ = V ϕ = j = 1 n V j j ϕ ( V ) ϕ = V ϕ = j = 1 n V j j ϕ (V*grad)phi=V*grad phi=sum_(j=1)^(n)V_(j)del_(j)phi(V \cdot \nabla) \phi=V \cdot \nabla \phi=\sum_{j=1}^{n} V_{j} \partial_{j} \phi
is called the transport of ϕ ϕ phi\phi by V V VV.
被称为 V V VV ϕ ϕ phi\phi 的运输。

1.2.2 Functional spaces 1.2.2 功能空间

The measure will always be Lebesgue measure. Let Ω Ω Omega\Omega be R n , R + × R R n , R + × R R^(n),R^(+)xxR\mathbb{R}^{n}, \mathbb{R}^{+} \times \mathbb{R}, or an open set of R n R n R^(n)\mathbb{R}^{n}. The space L ( Ω ) L ( Ω ) L^(oo)(Omega)L^{\infty}(\Omega) is the set of measurable functions such that
该度量将始终是勒贝格测度。设 Ω Ω Omega\Omega R n , R + × R R n , R + × R R^(n),R^(+)xxR\mathbb{R}^{n}, \mathbb{R}^{+} \times \mathbb{R} ,或 R n R n R^(n)\mathbb{R}^{n} 的开集。空间 L ( Ω ) L ( Ω ) L^(oo)(Omega)L^{\infty}(\Omega) 是可测函数的集合,使得
f L = sup x Ω | f ( x ) | < + . f L = sup x Ω | f ( x ) | < + . ||f||_(L^(oo))=s u p_(x in Omega)|f(x)| < +oo.\|f\|_{L^{\infty}}=\sup _{x \in \Omega}|f(x)|<+\infty .
For 1 p < + 1 p < + 1 <= p < +oo1 \leq p<+\infty, the space L p ( Ω ) L p ( Ω ) L^(p)(Omega)L^{p}(\Omega) is the set of measurable function f f ff such that
对于 1 p < + 1 p < + 1 <= p < +oo1 \leq p<+\infty ,空间 L p ( Ω ) L p ( Ω ) L^(p)(Omega)L^{p}(\Omega) 是可测函数 f f ff 的集合,使得
f L p = ( Ω | f ( x ) | p d x ) 1 / p < + f L p = Ω | f ( x ) | p d x 1 / p < + ||f||_(L^(p))=(int_(Omega)|f(x)|^(p)dx)^(1//p) < +oo\|f\|_{L^{p}}=\left(\int_{\Omega}|f(x)|^{p} d x\right)^{1 / p}<+\infty
We recall that, for every p [ 1 , ] , L p ( Ω ) p [ 1 , ] , L p ( Ω ) p in[1,oo],L^(p)(Omega)p \in[1, \infty], L^{p}(\Omega) is complete for the norm L p L p ||*||_(L^(p))\|\cdot\|_{L^{p}} and that L 2 ( Ω ) L 2 ( Ω ) L^(2)(Omega)L^{2}(\Omega) is an Hilbert space with the associated scalar product
我们回忆到,对于每个 p [ 1 , ] , L p ( Ω ) p [ 1 , ] , L p ( Ω ) p in[1,oo],L^(p)(Omega)p \in[1, \infty], L^{p}(\Omega) 在范数 L p L p ||*||_(L^(p))\|\cdot\|_{L^{p}} 下是完备的,并且 L 2 ( Ω ) L 2 ( Ω ) L^(2)(Omega)L^{2}(\Omega) 是一个具有相关标量积的希尔伯特空间
f , g L 2 = Ω f ( x ) g ( x ) d x f , g L 2 = Ω f ( x ) g ( x ) d x (:f,g:)_(L^(2))=int_(Omega)f(x)g(x)dx\langle f, g\rangle_{L^{2}}=\int_{\Omega} f(x) g(x) d x

1.2.3 Miscellaneous 1.2.3 杂项

If f ( t , x ) f ( t , x ) f(t,x)f(t, x) is a function of two (or more) variables, f ( t , ) f ( t , ) f(t,*)f(t, \cdot) denotes the function of x x xx defined by x f ( t , x ) x f ( t , x ) x rarr f(t,x)x \rightarrow f(t, x).
如果 f ( t , x ) f ( t , x ) f(t,x)f(t, x) 是一个两个(或更多)变量的函数, f ( t , ) f ( t , ) f(t,*)f(t, \cdot) 表示由 x f ( t , x ) x f ( t , x ) x rarr f(t,x)x \rightarrow f(t, x) 定义的 x x xx 的函数。
Let a R a R a inRa \in \mathbb{R}, then 1 x e a 1 x e a 1_(x∣ea)1_{x \mid e a} is the function which equals 1 if x a x a x <= ax \leq a and 0 if x > a x > a x > ax>a. Similarly, 1 x a 1 x a 1_(x >= a)1_{x \geq a} equals 1 if x a x a x >= ax \geq a and 0 if x < a x < a x < ax<a. More generally, if I I II an interval, or a set, 1 I ( x ) 1 I ( x ) 1_(I)(x)1_{I}(x) equals 1 if x I x I x in Ix \in I and 0 if x I x I x!in Ix \notin I.
a R a R a inRa \in \mathbb{R} ,则 1 x e a 1 x e a 1_(x∣ea)1_{x \mid e a} 是一个函数,当 x a x a x <= ax \leq a 时等于 1,当 x > a x > a x > ax>a 时等于 0。类似地, 1 x a 1 x a 1_(x >= a)1_{x \geq a} x a x a x >= ax \geq a 时等于 1,在 x < a x < a x < ax<a 时等于 0。更一般地,如果 I I II 是一个区间或集合, 1 I ( x ) 1 I ( x ) 1_(I)(x)1_{I}(x) x I x I x in Ix \in I 时等于 1,在 x I x I x!in Ix \notin I 时等于 0。

Chapter 2 第二章

Transport equations 运输方程

2.1 A first example in R R R\mathbb{R}
2.1 第一个例子在 R R R\mathbb{R}

2.1.1 Introduction 2.1.1 介绍

We first consider the transport equation in dimension 1 with constant speed c R c R c inRc \in \mathbb{R}. Let ϕ 0 C 1 ( R ) ϕ 0 C 1 ( R ) phi_(0)inC^(1)(R)\phi_{0} \in C^{1}(\mathbb{R}). We want to “transport” the values of ϕ 0 ϕ 0 phi_(0)\phi_{0}.
我们首先考虑一维常速 c R c R c inRc \in \mathbb{R} 的传输方程。设 ϕ 0 C 1 ( R ) ϕ 0 C 1 ( R ) phi_(0)inC^(1)(R)\phi_{0} \in C^{1}(\mathbb{R}) 。我们想要“传输” ϕ 0 ϕ 0 phi_(0)\phi_{0} 的值。
We define the function ϕ ( t , x ) ϕ ( t , x ) phi(t,x)\phi(t, x) by: the value of ϕ ϕ phi\phi at time t t tt and position x + c t x + c t x+ctx+c t equals the value of ϕ 0 ϕ 0 phi_(0)\phi_{0} at position x x xx, namely
我们通过以下方式定义函数 ϕ ( t , x ) ϕ ( t , x ) phi(t,x)\phi(t, x) :在时间 t t tt 和位置 x + c t x + c t x+ctx+c t 时, ϕ ϕ phi\phi 的值等于在位置 x x xx ϕ 0 ϕ 0 phi_(0)\phi_{0} 的值,即
ϕ ( t , x + c t ) = ϕ 0 ( x ) , t R + , x R ϕ ( t , x + c t ) = ϕ 0 ( x ) , t R + , x R phi(t,x+ct)=phi_(0)(x),quad AA t inR^(+),AA x inR\phi(t, x+c t)=\phi_{0}(x), \quad \forall t \in \mathbb{R}^{+}, \forall x \in \mathbb{R}
This defines a function ϕ ϕ phi\phi by
这通过定义一个函数 ϕ ϕ phi\phi
ϕ ( t , x ) = ϕ 0 ( x c t ) , t R + , x R ϕ ( t , x ) = ϕ 0 ( x c t ) , t R + , x R phi(t,x)=phi_(0)(x-ct),quad AA t inR^(+),AA x inR\phi(t, x)=\phi_{0}(x-c t), \quad \forall t \in \mathbb{R}^{+}, \forall x \in \mathbb{R}
If ϕ 0 C 1 ( R ) ϕ 0 C 1 ( R ) phi_(0)inC^(1)(R)\phi_{0} \in C^{1}(\mathbb{R}), then ϕ C 1 ( R + × R ) ϕ C 1 R + × R phi inC^(1)(R^(+)xxR)\phi \in C^{1}\left(\mathbb{R}^{+} \times \mathbb{R}\right). We observe that
如果 ϕ 0 C 1 ( R ) ϕ 0 C 1 ( R ) phi_(0)inC^(1)(R)\phi_{0} \in C^{1}(\mathbb{R}) ,那么 ϕ C 1 ( R + × R ) ϕ C 1 R + × R phi inC^(1)(R^(+)xxR)\phi \in C^{1}\left(\mathbb{R}^{+} \times \mathbb{R}\right) 。我们观察到
t ϕ + c x ϕ = 0 , t R + , x R ϕ ( 0 , x ) = ϕ 0 ( x ) , x R t ϕ + c x ϕ = 0 , t R + , x R ϕ ( 0 , x ) = ϕ 0 ( x ) , x R {:[del_(t)phi+cdel_(x)phi=0","quad AA t inR^(+)","AA x inR],[phi(0","x)=phi_(0)(x)","quad AA x inR]:}\begin{aligned} \partial_{t} \phi+c \partial_{x} \phi & =0, \quad \forall t \in \mathbb{R}^{+}, \forall x \in \mathbb{R} \\ \phi(0, x) & =\phi_{0}(x), \quad \forall x \in \mathbb{R} \end{aligned}
The equation (2.3) is called the transport equation. Equation (2.4) provides an initial data. Let us introduce some vocabulary
方程 (2.3) 被称为输运方程。方程 (2.4) 提供了初始数据。让我们引入一些词汇。
  • in equation (2.1), we “follow” the values of ϕ ϕ phi\phi. It is called a Lagrangian point of view.
    在方程(2.1)中,我们“跟随” ϕ ϕ phi\phi 的值。这被称为拉格朗日视角。
  • in equation (2.3), we do not “move”. It is called an Eulerian point of view.
    在方程(2.3)中,我们不“移动”。这被称为欧拉视角。

2.1.2 Solution of the transport equation
2.1.2 运输方程的解

Theorem 2.1.1. Let ϕ 0 C 1 ( R ) ϕ 0 C 1 ( R ) phi_(0)inC^(1)(R)\phi_{0} \in C^{1}(\mathbb{R}). Then there exists a unique solution ϕ C 1 ( R + × R ) ϕ C 1 R + × R phi inC^(1)(R^(+)xxR)\phi \in C^{1}\left(\mathbb{R}^{+} \times \mathbb{R}\right) to 2.3 [2.4). Moreover, ϕ ϕ phi\phi is given by
定理 2.1.1. 设 ϕ 0 C 1 ( R ) ϕ 0 C 1 ( R ) phi_(0)inC^(1)(R)\phi_{0} \in C^{1}(\mathbb{R}) 。则存在唯一解 ϕ C 1 ( R + × R ) ϕ C 1 R + × R phi inC^(1)(R^(+)xxR)\phi \in C^{1}\left(\mathbb{R}^{+} \times \mathbb{R}\right) 对于 2.3 [2.4)。此外, ϕ ϕ phi\phi 由以下给出
ϕ ( t , x ) = ϕ 0 ( x c t ) ϕ ( t , x ) = ϕ 0 ( x c t ) phi(t,x)=phi_(0)(x-ct)\phi(t, x)=\phi_{0}(x-c t)
Proof. First we note that ϕ ( t , x ) = ϕ 0 ( x c t ) ϕ ( t , x ) = ϕ 0 ( x c t ) phi(t,x)=phi_(0)(x-ct)\phi(t, x)=\phi_{0}(x-c t) satisfies
证明。首先我们注意到 ϕ ( t , x ) = ϕ 0 ( x c t ) ϕ ( t , x ) = ϕ 0 ( x c t ) phi(t,x)=phi_(0)(x-ct)\phi(t, x)=\phi_{0}(x-c t) 满足
t ϕ = c x ϕ 0 = c x ϕ t ϕ = c x ϕ 0 = c x ϕ del_(t)phi=-cdel_(x)phi_(0)=-cdel_(x)phi\partial_{t} \phi=-c \partial_{x} \phi_{0}=-c \partial_{x} \phi
Thus ϕ ϕ phi\phi is a solution. It is C 1 C 1 C^(1)C^{1} since ϕ 0 ϕ 0 phi_(0)\phi_{0} is C 1 C 1 C^(1)C^{1}.
因此 ϕ ϕ phi\phi 是一个解决方案。它是 C 1 C 1 C^(1)C^{1} ,因为 ϕ 0 ϕ 0 phi_(0)\phi_{0} C 1 C 1 C^(1)C^{1}