This course introduces partial differential equations as the language of evolving functions, geometric motion, and physical balance laws. It begins with the basic idea that a PDE relates a function of several variables to its derivatives, then develops the main first-order models that arise in transport, wave propagation, and nonlinear dynamics. The emphasis is on understanding what PDEs mean, how their solutions are organized, and why different classes of equations require different methods.

The central themes are characteristics, Hamilton-Jacobi theory, conservation laws, weak solutions, and admissibility. The course starts with linear first-order equations and then moves to quasilinear equations, where solution curves and nonlinear flow effects become essential. From there it studies conservation laws, the role of shocks and discontinuities, entropy conditions, and the Riemann problem, which together explain how physically meaningful solutions are selected when classical smooth solutions fail.

Later chapters broaden the toolkit by introducing distributions, fundamental solutions, Green kernels, representation formulas, maximum principles, and energy methods. These tools connect classical explicit methods with the modern weak formulation of PDEs and clarify how boundary and initial data enter well-posed problems. The course ends by synthesizing these ideas into a coherent picture: classical formulas, qualitative estimates, and weak-solution concepts all fit into a single framework for understanding first-order PDEs and the foundations of broader PDE theory.

# Introduction

These introductory notes set the viewpoint for the course: a partial differential equation is not only an equation containing derivatives, but a rule that constrains how a function may vary in several independent directions at once. The course begins with first-order equations because they expose the geometry of propagation before the analytic machinery of elliptic, parabolic, and hyperbolic theory is developed. The recurring questions are existence, uniqueness, stability, and the formation of singularities from smooth data.

## What Is a Partial Differential Equation Trying to Determine?

The first problem is to identify the unknown and the information that is being prescribed. In ordinary differential equations, the unknown usually evolves along one independent variable, often time. In a partial differential equation, the unknown depends on several variables, so the equation may constrain spatial variation, temporal variation, or both. The course therefore needs a broad baseline definition before distinguishing steady equations, evolution equations, and boundary-value problems.

[definition: Partial Differential Equation] A partial differential equation is an equation for an unknown function $u: U \to \mathbb R^m$, where $U \subseteq \mathbb R^n$ is open, involving $u$ and finitely many of its partial derivatives on $U$. [/definition]

The definition is intentionally broad. The same formal object can describe a steady state, a time evolution, a conservation law, or a geometric constraint. What turns the formal equation into a mathematical problem is the choice of data imposed on part of the domain or on its boundary.

[example: Three Prototype Meanings] Let $u$ be a scalar field. For a steady field $u:\mathbb R^n\to\mathbb R$, the equation $\Delta u=0$ means \begin{align*} \partial_{x_1x_1}u(x)+\partial_{x_2x_2}u(x)+\cdots+\partial_{x_nx_n}u(x)=0 \end{align*} at each point $x$, so the second-order bending in the coordinate directions balances to zero. For a time-dependent field $u:\mathbb R^n\times(0,\infty)\to\mathbb R$, the [heat equation](/page/Heat%20Equation) $\partial_tu-\Delta u=0$ is the same as \begin{align*} \partial_tu(x,t)=\Delta u(x,t), \end{align*} so the time change is determined by the spatial curvature. For constant transport, fix $b=(b_1,\dots,b_n)\in\mathbb R^n$ and let $u_0\in C^1(\mathbb R^n)$. Define $u(x,t)=u_0(x-bt)$, where $x-bt=(x_1-b_1t,\dots,x_n-b_nt)$. By the chain rule, \begin{align*} \partial_tu(x,t)=\sum_{i=1}^n \partial_{x_i}u_0(x-bt)(-b_i)=-\sum_{i=1}^n b_i\partial_{x_i}u_0(x-bt). \end{align*} For each $j\in\{1,\dots,n\}$, the chain rule also gives \begin{align*} \partial_{x_j}u(x,t)=\sum_{i=1}^n \partial_{x_i}u_0(x-bt)\delta_{ij}=\partial_{x_j}u_0(x-bt). \end{align*} Hence \begin{align*} b\cdot\nabla u(x,t)=\sum_{i=1}^n b_i\partial_{x_i}u(x,t)=\sum_{i=1}^n b_i\partial_{x_i}u_0(x-bt). \end{align*} Combining the two displayed identities, \begin{align*} \partial_tu(x,t)+b\cdot\nabla u(x,t)=-\sum_{i=1}^n b_i\partial_{x_i}u_0(x-bt)+\sum_{i=1}^n b_i\partial_{x_i}u_0(x-bt)=0. \end{align*} Thus $\Delta u=0$ models balance, $\partial_tu-\Delta u=0$ models diffusion driven by curvature, and $\partial_tu+b\cdot\nabla u=0$ models rigid transport of the initial profile along the lines $x-bt=\text{constant}$. [/example]

These examples also signal why PDE theory separates equations by qualitative behaviour. The Laplace equation spreads boundary information throughout a region, the heat equation smooths initial data forward in time, and the transport equation carries data along curves.

## What Counts as a Solution?

A second problem appears before any equation is solved: the derivatives in the equation may not exist after the dynamics develops. The course therefore begins with classical solutions because this is the setting where the equation can be read literally, then returns in Chapters 5 and 6 to weak and entropy solutions when shocks and conservation laws force a larger solution class.

[definition: Classical Solution] Let $U \subseteq \mathbb R^n$ be open. A classical solution of a partial differential equation of order $k$ on $U$ is a function $u \in C^k(U;\mathbb R^m)$ whose derivatives satisfy the equation at every point of $U$. [/definition]

Classical solutions are the natural first notion because the equation can be checked pointwise. Their limitation is that first-order nonlinear equations can create discontinuities even from smooth initial data, so the pointwise definition is sometimes too restrictive for the long-time problem.

[example: Smooth Transport Solution] Fix $b=(b_1,\dots,b_n)\in\mathbb R^n$ and let $u_0\in C^1(\mathbb R^n)$. Define $u:\mathbb R^n\times\mathbb R\to\mathbb R$ by $u(x,t)=u_0(x-bt)$, where $x-bt=(x_1-b_1t,\dots,x_n-b_nt)$. We verify that $u$ satisfies $\partial_tu+b\cdot\nabla u=0$ pointwise. By the chain rule applied to the map $t\mapsto x-bt$, \begin{align*} \partial_tu(x,t)=\sum_{i=1}^n \partial_{x_i}u_0(x-bt)\,\partial_t(x_i-b_it). \end{align*} Since $\partial_t(x_i-b_it)=-b_i$, this becomes \begin{align*} \partial_tu(x,t)=\sum_{i=1}^n \partial_{x_i}u_0(x-bt)(-b_i)=-\sum_{i=1}^n b_i\partial_{x_i}u_0(x-bt). \end{align*} For each $j\in\{1,\dots,n\}$, the chain rule applied to the map $x\mapsto x-bt$ gives \begin{align*} \partial_{x_j}u(x,t)=\sum_{i=1}^n \partial_{x_i}u_0(x-bt)\,\partial_{x_j}(x_i-b_it). \end{align*} Here $\partial_{x_j}(x_i-b_it)=\delta_{ij}$, so \begin{align*} \partial_{x_j}u(x,t)=\sum_{i=1}^n \partial_{x_i}u_0(x-bt)\delta_{ij}=\partial_{x_j}u_0(x-bt). \end{align*} Therefore \begin{align*} b\cdot\nabla u(x,t)=\sum_{j=1}^n b_j\partial_{x_j}u(x,t)=\sum_{j=1}^n b_j\partial_{x_j}u_0(x-bt). \end{align*} Combining the time derivative with this spatial derivative, \begin{align*} \partial_tu(x,t)+b\cdot\nabla u(x,t)=-\sum_{i=1}^n b_i\partial_{x_i}u_0(x-bt)+\sum_{j=1}^n b_j\partial_{x_j}u_0(x-bt)=0. \end{align*} Thus the first-order equation describes rigid motion of the initial values along the lines $x-bt=\text{constant}$, rather than local averaging. [/example]

The preceding computation is the model for the [method of characteristics](/page/Method%20of%20Characteristics), and it also motivates a theorem because existence is only half of the Cauchy problem. We also need to know that the initial profile determines no other classical solution.

[quotetheorem:6169]

[citeproof:6169]

The $C^1$ hypothesis is doing real work: the proof differentiates both the proposed solution and an arbitrary competing solution along characteristic curves. If $n=1$, $b=1$, and $u_0(x)=|x|$, then the formula gives $u(x,t)=|x-t|$, which transports the initial profile but is not a classical $C^1$ solution across the line $x=t$. Thus the formula may still describe propagation beyond the classical category, but the theorem as stated only proves classical existence and uniqueness.