math lessons - Hyperbolic partial differential equation

A hyperbolic partial differential equation is usually a second-order partial differential equation of the form

A u x x + 2 B u x y + C u y y + D u x + E u y + F = 0

with $\det \begin{pmatrix} A & B \\ B & C \end{pmatrix} = A C - B^2 < 0$ . The wave equation:

$\frac{\partial^2 u}{\partial t^2} - \nabla^2 u = 0$

is such a hyperbolic equation.

This type of second-order hyperbolic partial differential equation may be transformed to a hyperbolic system of first-order differential equations.

Hyperbolic system of partial differential equations

Consider the following system of $s$ first order partial differential equations for $s$ unknown functions $\vec u = (u_1, \ldots, u_s)$ , $\vec u =\vec u (\vec x,t)$ , where $\vec x \in \mathbb{R}^d$

$(*) \quad \frac{\partial \vec u}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} \vec {f^j} (\vec u) = 0,$

$\vec {f^j} \in C^1(\mathbb{R}^s, \mathbb{R}^s), j = 1, \ldots, d$ are once continuously differentiable functions, nonlinear in general.

Now define for each $\vec {f^j}$ a matrix $s \times s$

$A^j:= \begin{pmatrix} \frac{\partial f_1^j}{\partial u_1} & \cdots & \frac{\partial f_1^j}{\partial u_s} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_s^j}{\partial u_1} & \cdots & \frac{\partial f_s^j}{\partial u_s} \end{pmatrix}$ , for each $j = 1, \ldots, d$ .

We say that the system $( * )$ is hyperbolic if for all $\alpha_1, \ldots, \alpha_d \in \mathbb{R}$ the matrix $A := \alpha_1 A^1 + \cdots + \alpha_d A^d$ has only real eigenvalues and is diagonalizable.

If the matrix $A$ has distinct real eigenvalues, it follows it's diagonalizable. In this case the system $( * )$ is called strictly hyperbolic.

Hyperbolic system and conservation laws

There is a connection between a hyperbolic system and a conservation law. Consider a hyperbolic system of one partial differential equation for one unknown function $u = u(\vec x, t)$ . Then the system $( * )$ has the form

$(**) \quad \frac{\partial u}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} {f^j} (u) = 0,$

Now $u$ can be some quantity with a flux $\vec f = (f^1, \ldots, f^d)$ .To show that this quantity is conserved, integrate $( * * )$ over a domain $Ω$

$\int_{\Omega} \frac{\partial u}{\partial t} + \int_{\Omega} \nabla . \vec f(u) = 0$

If $u$ and $\vec f$ are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and $\partial / \partial t$ and we get a conservation law for the quantity $u$ in a common form