Introduction
Let $u$ be a function of several variables $u(x_1,x_2,\ldots,x_n)$. We say $u$ is smooth provided $u$ is infinitely differentiable with respect to all of the variables. We denote partial derivative of $u$ with respect to $x_k$ by $$ u_{x_k} := \frac{\partial u}{\partial x_k} $$ We sometimes also write $\partial_{x_k}$ to denote the differential operator $\frac{\partial}{\partial x_k}$. With this notation, we may express mixed partial derivatives as $$ u_{x_j x_k} = \frac{\partial^2 u}{\partial x_k \partial x_j} = \partial_{x_k} \partial_{x_j} u. $$ We will for the most part be working with functions that have continuous partial derivatives and hence (by Clairaut-Schwarz’s Theorem) assume that the partial derivatives commute: $$ u_{x_j x_k}=u_{x_k x_j}. $$ This assumption also provides us with a well-defined notation to express high-order derivatives. The multi-index notation due to L. Schwartz.
Multi-index Notation
A multi-index is a vector $\alpha=(\alpha_1,\ldots, \alpha_n)$ where each component $\alpha_j$ is a nonnegative integer. The order of a multi-index $\alpha$ is denoted by $|\alpha|$ and is defined by: $$ |\alpha|:= \alpha_1 + \cdots + \alpha_n. $$
Given a multi-index $\alpha$, we define $$ D^{\alpha} u:=\frac{\partial^{|\alpha|} u}{\partial x_{1}^{\alpha_{1}} \cdots \partial x_{n}^{\alpha_{n}}}=\partial_{x_{1}}^{\alpha_{1}} \cdots \partial_{x_{n}}^{\alpha_{n}} u $$
Example
Let $u$ be a function of 3 variables $(x,y,z)$ and $\alpha=(1,3,2)$. Then $$ D^{\alpha} u= \partial_{x}\partial_{y}^3\partial_{z}^2 u = u_{xyyyzz} $$ where both of the equalities uses the commutativity of the partial derivatives.
Example
If $u=u\left(x_{1}, \ldots, x_{n}\right)$, then $D^1u=\left\{u_{x_{i}}\colon i=1, \ldots, n \right\}$.
In practice, for $k$ non-negative integer, $D^k u$ denotes the set of all partial derivatives of $u$ of order $k$. Assigning a certain ordering we can regard $D^k u(x)$ for $x\in\mathbb{R}^n$ as a point in $\mathbb{R}^{n^k}$.
Special Cases
Let $x=(x_1,\ldots,x_n)\in\mathbb{R}^n$.
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$k=1$: In this case we have: $$ D^1 u(x)=D u(x) =:(u_{x_1}(x),u_{x_2}(x),\ldots,u_{x_n}(x)), $$ which is the gradient vector $\nabla u(x)$, hence $Du(x)\in\mathbb{R}^n$.
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$k=2$: In this case we regard the elements of $D^2u(x)$ as they are arranged in an $n\times n$ matrix: $$ D^2u(x):= \begin{bmatrix} u_{x_1 x_1}(x) & &\cdots& & u_{x_1 x_n}(x) \\ u_{x_2 x_1}(x) & &\cdots& & u_{x_2 x_n}(x)\\ & &\cdots& & \\ \vdots & &\cdots& & \vdots \\ & &\cdots& & \\ u_{x_n x_1}(x) & &\cdots& & u_{x_n x_n}(x) \end{bmatrix} $$ which is the Hessian matrix of $u$ at $x$, and hence $D^2u(x)\in S(n)$, the space of real symmetric $n\times n$ matrices. Note that in this case $\mathrm{tr}(D^2u(x))$ gives the Laplacian of $u$ at $x$ (in rectangular coordinates).
Partial Differential Equations
A partial differential equation (PDE) is an equation involving a(n unknown) function $u$ of several variables and its partial derivatives. The order of a PDE is the order of the highest-order derivative in the PDE.
Examples
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Transport equation: $u_t + b\cdot Du = 0$, $b\in\mathbb{R}^n$; first order.
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Laplace’s equation: $u_{xx}+ u_{yy} =0$; second order.
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Poisson’s equation: $u_{xx}+ u_{yy} =f(x,y)$; second order.
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Heat equation: $u_t - u_{xx} =0$; second order.
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Wave equation: $u_{tt} - u_{xx} =0$; second order.
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Airy’s equation: $u_t + u_{xxx} = 0$; third order.
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Korteweg-de Vries equation: $u_t + u u_x + u_{xxx} = 0$; third order.
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Eikonal equation: $u_x^2 +u_y^2 = c^2$; first order.
We express a general $k^{\text{th}}$ order PDE in the form $$ F(x, u, Du, D^2 u, \ldots, D^k u)=0 $$ for some function $F$.
Classification
A PDE of order $k$ is linear if it is of the form $$ \sum_{|\alpha|\leq k} a_{\alpha}(x)D^{\alpha} u = 0. $$ If $u_1$ and $u_2$ are two solutions of a linear PDE, then so is any linear combination $c_1 u + c_2 u_2$, $c_1$ and $c_2$ are constants.
We say a PDE is nonlinear if it is not linear.
A nonlinear PDE of order $k$ is semilinear if it is linear in the highest-order derivatives. In other words, if it is of the form: $$ \left(\sum_{|\alpha|=k} a_{\alpha}(x) D^{\alpha} u \right) + a_0 \left(D^{k-1}u, D^{k-2}u,\ldots, Du, u, x \right)=0. $$ As you can see, the coefficients of the terms with the highest-order derivatives are allowed to depend only on $x$.
A PDE of order $k$ is quasilinear if it is of the form
$$ \left(\sum_{|\alpha|=k} a_{\alpha}(D^{k-1}u,D^{k-2}u,\ldots, Du,u,x)D^{\alpha} u \right) + a_0 \left(D^{k-1}u, D^{k-2}u,\ldots, Du, u, x \right)=0 $$ but it is not semilinear. This means that in a quasilinear PDE, the highest-order derivatives $D^\alpha$, $|\alpha|=k$ are allowed to be multiplied by terms involving derivatives of lower orders.
A PDE order $k$ is fully nonlinear if it is nonlinear but not quasilinear or semilinear. In this case the highest order derivatives of $u$ themselves appear nonlinearly in the PDE.
Examples
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Airy’s equation: $u_t + u_{xxx} = 0$; linear.
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Korteweg-de Vries equation: $u_t + u u_x + u_{xxx} = 0$; semilinear.
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Inviscid Burger’s equation: $u_t + uu_x=0$; quasilinear.
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Eikonal equation: $u_x^2 +u_y^2 = c^2$; fully nonlinear.
Main Goals and Motivations in Studying PDEs
Just like ordinary differential equations, a partial differential equation may have no solution, or infinitely many solutions.
Example
Consider $u_t - u_x =0$. $u(x,t):=0$ is a solution. So is $u(x,t):=C$ for any constant $C\in \mathbb{R}$. So is $u(x,t):= x+t$, and actually, so is $u(x,t):= C_1(x+t)+C_2$ for any constants $C_1,C_2\in\mathbb{R}$.
We impose auxiliary conditions, e.g. initial conditions and boundary conditions, and pose a problem to pick a particular solution. For example, $$ \left\{ \begin{alignedat}{2} u_t - u_x &= 0 \quad&& x\in\mathbb{R},~t>0\\ u&=g \quad&& x\in\mathbb{R} \end{alignedat} \right. $$ where $g\colon \mathbb{R}\to \mathbb{R}$ is a given function. This is an initial-value problem.
Well-Posedness
Bad news: Most PDEs cannot be solved in explicit form. Thus, most of the efforts are geared towards understanding existence, properties, and behavior of solutions.
Given a PDE problem such as the initial value-problem above, we focus on three main issues:
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Existence: the problem in fact has a solution.
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Uniqueness: this solution is unique.
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Continuous dependence on given data: the solution depends continuously on the data given under some appropriate norm.
If a PDE problem has all of the properties 1, 2, and 3 above, we say that the problem is well-posed.
Notion of a Solution
What do we mean by solution of a PDE in general? Ideally, a solution to a $k^{\text{th}}$-order PDE is required to have continuous partial derivatives of all orders up to and including order $k$. Such a solution is called a classical solution. But it is often the case that one is interested in functions that satisfy a weaker formulation of the PDE, that is, solutions that are not required to be differentiable. Such solutions, loosely speaking, are called weak solutions, distributional solutions, or integral solutions, depending on the context of the treatment. We will give a brief introduction to such solutions and work with them when we cover shock theory for scalar conservation laws.
Full Set of Lecture Notes
The notes for this lecture are available here (12 pages).