A few model PDEs

Transport Equation

For $x \in \R^3$, $t \in \R$, let $\rho(x, t)$ represents the concentration of a solute at point $x$ and time $t$. Suppose the ambient solution is moving with velocity $v$ (which could depend on $x$ and $t$). Given $v$, how does $\rho$ evolve with time (ignoring diffusion, which we will address later)?

Let $R$ be any region, and $c(t)$ be the amount of the solute in the region $R$ at time $t$. Then $$ \partial_t c = \partial_t \int_R \rho \, dV = \int_R \partial_t \rho \, dV \,. $$ Also $$ \partial_t c = -\int_{\partial R} \rho v \cdot \hat n \, dS = -\int_R \dv (\rho v) \, dV. $$

Since this is true for all regions $R$, we must have \begin{equation}\label{e:T}\tag{T} \partial_t \rho + \dv (\rho v) = 0 \end{equation} This is the transport equation.

Heat equation

Consider a conductor (thin rod, metal plate, solid ball) and let $\theta(x, t)$ be the temperature of the rod at point $x$ and time $t$. Fourier’s law tells us the heat flux is proportional to the gradient of temperature.

If $R$ is any region in the conductor, $Q(t)$ is the heat contained in $R$ at time $t$, then $$ \int_R \rho c \partial_t \theta(x, t) \, dV = \partial_t Q = \int_{\partial R} \kappa \grad \theta \cdot \hat n \, dS = \int_R \dv (\kappa \grad \theta) \, dV \,. $$ Here $\kappa$ is the conductivity, $c$ is the specific heat, $\rho$ is the density.

Both $c$ and $\kappa$ are material properties. When the material is homogenous, they are constant, and we obtain $$ \rho c \partial_t \theta = \dv( \kappa \grad \theta) = \kappa \lap \theta \,, $$ which is the heat equation.

Wave Equation

One Dimensional Version

Hold a thin string taught at two points. Let $u(x,t)$ be the displacement of the string from it’s mean position at point $x$ and time $t$. Our goal is to find an equation determining the evolution of $u$.

Let $T$ be the magnitude of the tension in the string, $\rho$ be the density, and $A$ be the area of cross section. (Recall, the tension is a force that’s always directed tangential to the string.)

Let $x_1 \lt x_2$, $R$ be the region of the string between $x_1$ and $x_2$, and $F$ be the total force (in the vertical direction) exerted on the region $R$.

• By Newton’s second law, $F = \int_{x_1}^{x_2} A \rho \partial_t^2 u(x,t) \, dx$.

• Upward vertical force on right end point of the string: $T \sin\tan^{-1} (\partial_x u(x_2, t)$.

• Assuming $\partial_x u$ is small, this is roughly $T \partial_x u(x_2, t)$.

• Similarly, downward vertical force on the left endpoint of the string is $T \partial_x u(x_1, t)$.

Thus $$ \int_{x_1}^{x_2} A \rho \partial_t^2 u(x,t) \, dx = T (\partial_x u(x_2, t) - \partial_x u(x_1, t)) \,, $$ which implies $$ \partial_t^2 u = c^2 \partial_x^2 u, \quad \text{where } c = \sqrt{\frac{T}{A\rho}}. $$

Two Dimensional Version

Now let $(x, y) \in \R^2$, and $u(x, y,t)$ be the displacement of the surface of a drum from it’s mean position. Let $h$ be the thickness of the drum surface, and $R$ be any region in the drum surface. Balancing forces gives $$ \int_R h \rho \partial_t^2 u \, dx \, dy = T \int_{\partial R} \hat n \cdot \grad u \, dl. $$ Using the usual argument we get the PDE $$ \partial_t^2 u = c^2 \lap u $$ in two dimensions.

The same PDE governs the propagation of waves in higher dimensions.

Electro Magnetic Waves

Maxwel’s equations govern the evolution of Electric and Magnetic fields: \begin{gather*} \dv E = \frac{\rho}{\epsilon_0},\cr \dv B = 0,\cr \curl E = -\partial_t B,\cr \curl B = \mu_0 J + \mu_0 \epsilon_0 \partial_t E, \end{gather*} where:

• $E$ is the Electric field,
• $B$ is the Magnetic field,
• $\rho$ is the charge density (a scalar),
• $J$ is the current density (a vector!), and
• $\epsilon_0$ the electric constant (a universal constant, measuring the “permittivity” of free space)
• $\mu_0$ is the magnetic constant (a universal constant, measuring the “permeability” of free space).

In vacuum, the charge density and current density are both $0$. Thus Maxwell’s equations give $$ \partial_t^2 E = \frac{1}{\mu_0 \epsilon_0} \curl \partial_t B = -\frac{1}{\mu_0 \epsilon_0} \curl \curl E = +\frac{1}{\mu_0 \epsilon_0} (\lap E - \grad \dv E) = \frac{1}{\mu_0 \epsilon_0} \lap E, $$ and hence $$ \partial_t^2 E - c^2 \lap E = 0, \quad \text{where } c = \sqrt{\frac{1}{\mu_0 \epsilon_0}}. $$ This is the 3D wave equation, which governs the propagation of light and radio waves. (The propagation of sound is also governed by this equation, though the mechanism is very different.)

Poisson’s equation

Consider a conductor with a (possibly non-uniform) stationary (i.e. time independent) charge density $\rho$. Then the electric and magnetic fields are constant in time and Maxell’s equations give $$ \curl E = - \partial_t B = 0. $$ On simply connected domains), any curl free vector field can be written as a gradient. Hence there exists a scalar function $V$ such that $$ E = -\grad V. $$ The negative sign is introduced for convenience. Physically, this means that a positively charged particle will experience a force directed ‘’along’’ the electric field, and “want to move” from regions of high electric potential to low potential.

By Maxwell’s (or Gauss’s) law, $\dv E = \rho / \epsilon_0$, and hence the potential $V$ satisfies. $$ -\lap V = \frac{\rho}{\epsilon_0}. $$ This is the Poisson equation.

The Poisson equation also arises as steady states to the (forced) heat and wave equations.

Harmonic Functions

A function $u$ is said to be harmonic if it satisfies Laplace’s equation: $$ \lap u = 0. $$ For example, if you put a point charge in vacuum (at the origin), the charge density is $0$ on $\R^3-\{0\}$. Consequently, the electric potential associated to this point charge is a harmonic function on $\R^3 - \{0\}$.

The potential associated with a point charge of strength $\rho / \epsilon_0$ in $3$ dimensions turns out to be $$ V(x) = \frac{1}{4 \pi \abs{x}}, $$ something we will encounter later on.

Minimal surfaces

Suppose $\Gamma \subseteq \R^3$ is a simple closed curve. What is the surface of smallest area whose boundary is $\Gamma$? This is one version of the Plateau problem; while proving existence smoothness is usually tricky, one can quickly show that if such a surface exist, it must have zero mean curvature.