
# 3.3.1: Examples

Example 3.3.1.1: Beltrami Equations

\begin{eqnarray}
\label{belt1}\tag{3.3.1.1}
Wu_x-bv_x-cv_y&=&0\\
\label{belt2}\tag{3.3.1.2}
Wu_y+av_x+bv_y&=&0,
\end{eqnarray}

where $$W,\ a,\ b,\ c$$ are given functions depending of $$(x,y)$$, $$W\not=0$$ and the matrix

$$\left(\begin{array}{cc} a&b\\ b&c \end{array}\right)$$

is positive definite.

The Beltrami system is a generalization of Cauchy-Riemann equations. The function $$f(z)=u(x,y)+iv(x,y)$$, where $$z=x+iy$$, is called a quasiconform mapping, see for example [9], Chapter 12, for an application to partial differential equations.

Set

$$A^1=\left(\begin{array}{cc} W&-b\\ 0&a \end{array}\right),\ \ A^2=\left(\begin{array}{cc} 0&-c\\ W&b \end{array}\right).$$

Then the system (\ref{belt1}), (\ref{belt2}) can be written as

$$A^1\left(\begin{array}{c} u_x\\v_x \end{array}\right)+ A^2\left(\begin{array}{c} u_y\\v_y \end{array}\right)=\left(\begin{array}{c}0\\0\end{array}\right).$$

Thus,

\begin{eqnarray*}
C(x,y,\zeta)=\left|\begin{array}{cc}
W\zeta_1&-b\zeta_1-c\zeta_2\\
W\zeta_2&a\zeta_1+b\zeta_2
\end{array}\right|
=W(a\zeta_1^2+2b\zeta_1\zeta_2+c\zeta_2^2),
\end{eqnarray*}

which is different from zero if $$\zeta\not=0$$ according to the above assumptions. Thus the Beltrami system is elliptic.

Example 3.3.1.2: Maxwell Equations

The Maxwell equations in the isotropic case are

\begin{eqnarray}
\label{max1}\tag{3.3.1.3}
c\ \text{rot}_x\ H&=&\lambda E+\epsilon E_t\\
\label{max2}\tag{3.3.1.4}
c\ \text{rot}_x\ E&=&-\mu H_t,
\end{eqnarray}

where

• $$E=(e_1,e_2,e_3)^T$$ electric field strength, $$e_i=e_i(x,t)$$, $$x=(x_1,x_2,x_3)$$,
• $$H=(h_1,h_2,h_3)^T$$ magnetic field strength, $$h_i=h_i(x,t)$$,
• $$c$$ speed of light,
• $$\lambda$$ specific conductivity,
• $$\epsilon$$  dielectricity constant,
• $$\mu$$ magnetic permeability.

Here $$c,\ \lambda,\ \epsilon$$ and $$\mu$$ are positive constants.

Set $$p_0=\chi_t,\ p_i=\chi_{x_i}$$, $$i=1,\ldots 3$$, then the characteristic differential equation is

$$\left|\begin{array}{cccccc} \epsilon p_0/c&0&0&0&p_3&-p_2\\ 0&\epsilon p_0/c&0&-p_3&0&p_1\\ 0&0&\epsilon p_0/c&p_2&-p_1&0\\ 0&-p_3&p_2&\mu p_0/c&0&0\\ p_3&0&-p_1&0&\mu p_0/c&0\\ -p_2&p_1&0&0&0&\mu p_0/c \end{array}\right|=0.$$

The following manipulations  simplifies this equation:

1. multiply the first three columns with $$\mu p_0/c$$,
2. multiply the 5th column with $$-p_3$$ and the the 6th column with $$p_2$$ and add the sum to the 1st column,
3. multiply the 4th column with $$p_3$$ and the 6th column with $$-p_1$$ and add the sum to the 2th column,
4. multiply the 4th column with $$-p_2$$ and the 5th column with $$p_1$$ and add the sum to the 3th column,
5. expand the resulting determinant with respect to the elements of the 6th, 5th and 4th row.

We obtain

$$\left|\begin{array}{ccc} q+p_1^2&p_1p_2&p_1p_3\\ p_1p_2&q+p_2^2&p_2p_3\\ p_1p_3&p_2p_3&q+p_3^2 \end{array}\right|=0,$$

where

$$q:=\frac{\epsilon\mu}{c^2}p_0^2-g^2$$

with $$g^2:=p_1^2+p_2^2+p_3^2$$. The evaluation of the above equation leads to $$q^2(q+g^2)=0$$, i. e.,

$$\chi_t^2\left(\frac{\epsilon\mu}{c^2}\chi_t^2-|\nabla_x\chi|^2\right)=0.$$

It follows immediately that Maxwell equations are a hyperbolic system, see an exercise.
There are two solutions of this characteristic equation. The first one are characteristic surfaces $$\mathcal{S}(t)$$, defined by $$\chi(x,t)=0$$, which satisfy $$\chi_t=0$$. These surfaces are called stationary waves The second type of characteristic surfaces are defined by solutions of

$$\frac{\epsilon\mu}{c^2}\chi_t^2=|\nabla_x\chi|^2.$$

Functions defined by $$\chi=f(n\cdot x-Vt)$$ are solutions of this equation.
Here is $$f(s)$$ an arbitrary function with $$f'(s)\not=0$$, $$n$$ is a unit vector and $$V=c/\sqrt{\epsilon\mu}$$.
The associated characteristic surfaces $$\mathcal{S}(t)$$ are defined by

$$\chi(x,t)\equiv f(n\cdot x-Vt)=0,$$

here we assume that $$0$$ is in he range of $$f:\ \mathbb{R}^1\mapsto\mathbb{R}^1$$. Thus, $$\mathcal{S}(t)$$ is defined by $$n\cdot x-Vt=c$$, where $$c$$ is a fixed constant. It follows that the planes $$\mathcal{S}(t)$$ with normal $$n$$ move with speed $$V$$ in direction of $$n$$, see Figure 3.3.1.1.

Figure 3.3.1.1:  $$d'(t)$$ is the speed of plane waves

$$V$$ is called speed of the plane wave $$\mathcal{S}(t)$$.

Remark. According to the previous discussions, singularities of a solution of Maxwell equations are located at most on characteristic surfaces.

A special case of Maxwell equations are the telegraph equations, which follow from Maxwell equations if $$\text{\div}\ E=0$$ and $$\text{div}\ H=0$$ i. e., $$E$$ and $$H$$ are  fields free of sources. In fact, it is sufficient to assume that this assumption is satisfied at a fixed time $$t_0$$ only, see an exercise.

Since

$$\text{rot}_x\ \text{rot}_x\ A=\mbox{grad}_x\ \text{div}_x\ A-\triangle_xA$$

for each $$C^2$$-vector field $$A$$, it follows from Maxwell equations the uncoupled system

\begin{eqnarray*} \triangle_xE&=&\frac{\epsilon\mu}{c^2}E_{tt}+\frac{\lambda\mu}{c^2}E_t\\
\triangle_xH&=&\frac{\epsilon\mu}{c^2}H_{tt}+\frac{\lambda\mu}{c^2}H_t.
\end{eqnarray*}

Example 3.3.1.3: Equations of Gas Dynamics

Consider the following quasilinear equations of first order.

$$v_t+(v\cdot\nabla_x)\ v+\frac{1}{\rho} \nabla_x p =f\ \ \ \mbox{(Euler equations)}.$$

Here is

• $$v=(v_1,v_2,v_3)$$ the vector of speed, $$v_i=v_i(x,t)$$, $$x=(x_1,x_2,x_3)$$,
• $$p$$ pressure, $$p=(x,t)$$,
• $$\rho$$ density, $$\rho=\rho(x,t)$$,
• $$f=(f_1,f_2,f_3)$$ density of the external force, $$f_i=f_i(x,t)$$,

$$(v\cdot\nabla_x)v\equiv (v\cdot\nabla_x v_1,v\cdot\nabla_x v_2,v\cdot\nabla_x v_3))^T$$.

The second equation is

$$\rho_t+v\cdot\nabla_x\rho+\rho\ \text{div}_x\ v=0\ \ \ \mbox{(conservation of mass)}.$$

Assume the gas is compressible and that there is a function (state equation)

$$p=p(\rho),$$

where  $$p'(\rho)>0$$ if $$\rho>0$$. Then the above system of four equations is

\begin{eqnarray}
\label{euler}\tag{3.3.1.5}
v_t+(v\cdot\nabla)v+\frac{1}{\rho}p'(\rho)\nabla\rho&=&f\\
\label{cont}\tag{3.3.1.6}
\rho_t+ \rho\ \text{div}\ v+v\cdot\nabla\rho&=&0,
\end{eqnarray}

where $$\nabla\equiv\nabla_x$$ and $$\text{div}\equiv\text{div}_x$$, i. e., these operators apply on the spatial variables only.

The characteristic differential equation is here
$$\left|\begin{array}{cccc} \frac{d\chi}{dt}&0&0&\frac{1}{\rho}p'\chi_{x_1}\\ 0&\frac{d\chi}{dt}&0&\frac{1}{\rho}p'\chi_{x_2}\\ 0&0&\frac{d\chi}{dt}&\frac{1}{\rho}p'\chi_{x_3}\\ \rho\chi_{x_1}& \rho\chi_{x_2}&\rho\chi_{x_3}&\frac{d\chi}{dt} \end{array}\right|=0,$$

where

$$\dfrac{d\chi}{dt}:=\chi_t+(\nabla_x\chi)\cdot v.$$

Evaluating the determinant, we get the characteristic differential equation

\label{chargas}\tag{3.3.1.7}
\left(\frac{d\chi}{dt}\right)^2\left(\left(\frac{d\chi}{dt}\right)^2-p'(\rho)|\nabla_x\chi|^2\right)=0.

This equation implies consequences for the speed of the characteristic surfaces as the following consideration shows.

Consider a family $$\mathcal{S}(t)$$ of surfaces in $$\mathbb{R}^3$$ defined by $$\chi(x,t)=c$$, where
$$x\in\mathbb{R}^3$$ and $$c$$ is a fixed constant. As usually, we assume that $$\nabla_x\chi\not=0$$.
One of the two normals on $$\mathcal{S}(t)$$ at a point of the surface $$\mathcal{S}(t)$$ is given by, see an exercise,

\label{surfnormal}\tag{3.3.1.8}
{\bf n}=\frac{\nabla_x\chi}{|\nabla_x\chi|}.

Let $$Q_0\in\mathcal{S}(t_0)$$ and let $$Q_1\in\mathcal{S}(t_1)$$ be a point on the line defined by $$Q_0+s{\bf n}$$, where $${\bf n}$$ is the normal (\ref{surfnormal}) on $$\mathcal{S}(t_0)$$ at $$Q_0$$ and $$t_0<t_1$$, $$t_1-t_0$$ small, see Figure 3.3.1.2.

3.3.1.2: Definition of the speed of a surface

Definition. The limit
$$P=\lim_{t_1\to t_0}\frac{|Q_1-Q_0|}{t_1-t_0}$$
is called speed of the surface $$\mathcal{S}(t)$$.

Proposition 3.2. The speed of the surface $$\mathcal{S}(t)$$ is

\label{speedsurf}
P=-\frac{\chi_t}{|\nabla_x\chi|}.

Proof. The proof follows from $$\chi(Q_0,t_0)=0$$ and $$\chi(Q_0+d{\bf n},t_0+\triangle t)=0$$, where $$d=|Q_1-Q_0|$$ and $$\triangle t=t_1-t_0$$.

$$\Box$$

Set $$v_n:=v\cdot{\bf n}$$ which is the component of the velocity vector in direction $${\bf n}$$.
From ({\ref{surfnormal}) we get
$$v_n=\frac{1}{|\nabla_x\chi|}v\cdot \nabla_x\chi.$$

Definition. $$V:=P-v_n$$, the difference of the speed of the surface and the speed of liquid particles, is called relative speed.

Figure 3.3.1.3: Definition of relative speed

Using the above formulas for $$P$$ and $$v_n$$ it follows
$$V=P-v_n=-\frac{\chi_t}{|\nabla_x\chi|}-\frac{v\cdot\nabla_x\chi}{|\nabla_x\chi|}=-\frac{1}{|\nabla_x\chi|}\frac{d\chi}{dt}.$$
Then, we obtain from the characteristic equation (\ref{chargas}) that
$$V^2|\nabla_x\chi|^2\left(V^2|\nabla_x\chi|^2-p'(\rho)|\nabla_x\chi|^2\right)=0.$$
An interesting conclusion is that there are two relative speeds: $$V=0$$ or $$V^2=p'(\rho)$$.

Definition. $$\sqrt{p'(\rho)}$$ is called speed of sound.

### Contributors

• Integrated by Justin Marshall.