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Mathematics LibreTexts

3.3.1: Examples

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    Example Beltrami Equations


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


    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.


    \end{array}\right),\ \

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




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

    Example Maxwell Equations

    The Maxwell equations in the isotropic case are

    c\ \text{rot}_x\ H&=&\lambda E+\epsilon E_t\\
    c\ \text{rot}_x\ E&=&-\mu H_t,


    • \(E=(e_1,e_2,e_3)^T\) 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

    \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

    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




    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.,


    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


    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

    Figure \(d'(t)\) 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.


    \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\\

    Example 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)


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

    \rho_t+ \rho\ \text{div}\ v+v\cdot\nabla\rho&=&0,

    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
    \rho\chi_{x_1}& \rho\chi_{x_2}&\rho\chi_{x_3}&\frac{d\chi}{dt}


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

    Evaluating the determinant, we get the characteristic differential equation


    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,
    {\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 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

    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 \(\triangle t=t_1-t_0\).


    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 Definition of relative speed

    Using the above formulas for \(P\) and \(v_n\) it follows
    Then, we obtain from the characteristic equation (\ref{chargas}) that
    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.