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Mathematics LibreTexts

3.3: Systems of First Order

  • Page ID
    2144
  • [ "article:topic", "showtoc:no" ]

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    Consider the quasilinear system

    \begin{equation}
    \label{syst1}
    \sum_{k=1}^nA^k(x,u)u_{u_k}+b(x,u)=0,
    \end{equation}

    where \(A^k\) are \(m\times m\)-matrices, sufficiently regular with respect to their arguments, and

    $$
    u=\left(\begin{array}{c}
    u_1\\ \vdots\\u_m
    \end{array}\right),\ \
    u_{x_k}=\left(\begin{array}{c}
    u_{1,x_k}\\ \vdots\\u_{m,x_k}
    \end{array}\right),\ \
    b=\left(\begin{array}{c}
    b_1\\ \vdots\\b_m
    \end{array}\right).
    $$

    We ask the same question as above: can we calculate all derivatives of \(u\) in a neighborhood of a given hypersurface \(\mathcal{S}\) in \(\mathbb{R}\) defined by \(\chi(x)=0\), \(\nabla\chi\not=0\), provided \(u(x)\) is given on \(\mathcal{S}\)?

    For an answer we map \(\mathcal{S}\) onto a flat surface \(\mathcal{S}_0\)  by using the mapping \(\lambda=\lambda(x)\) of Section 3.1 and write equation (\ref{syst1}) in new coordinates. Set \(v(\lambda)=u(x(\lambda))\), then

    $$\sum_{k=1}^nA^k(x,u)\chi_{x_k}v_{\lambda_n}=\mbox{terms known on}\ \mathcal{S}_0.$$

    We can solve this system with respect to \(v_{\lambda_n}\), provided that

    $$\det\left(\sum_{k=1}^nA^k(x,u)\chi_{x_k}\right)\not=0$$

    on \(\mathcal{S}\).

    Definition. Equation

    $$\det\left(\sum_{k=1}^nA^k(x,u)\chi_{x_k}\right)=0$$

    is called characteristic equation associated to equation (\ref{syst1}) and a surface \({\mathcal{S}}\): \(\chi(x)=0\), defined by a solution \(\chi\), \(\nabla\chi\not=0\), of this characteristic equation is said to be characteristic surface.

    Set

    $$C(x,u,\zeta)=\det\left(\sum_{k=1}^nA^k(x,u)\zeta_k\right)$$

    for \(\zeta_k\in\mathbb{R}\).

    Definition.

    1. The system (\ref{syst1}) is hyperbolic at \((x,u(x))\) if there is a regular linear mapping \(\zeta=Q\eta\), where \(\eta=(\eta_1,\ldots,\eta_{n-1},\kappa)\), such that there exists \(m\) {\it real} roots \(\kappa_k=\kappa_k(x,u(x),\eta_1,\ldots,\eta_{n-1})\), \(k=1,\ldots,m\), of $$ D(x,u(x),\eta_1,\ldots,\eta_{n-1},\kappa)=0 $$ for all \((\eta_1,\ldots,\eta_{n-1})\), where $$ D(x,u(x),\eta_1,\ldots,\eta_{n-1},\kappa)=C(x,u(x),x,Q\eta).$$
    2. System (\ref{syst1}) is parabolic if there exists a regular linear mapping \(\zeta=Q\eta\) such that \(D\) is independent of \(\kappa\), that is, \(D\) depends on less than \(n\) parameters.
    3. System (\ref{syst1}) is elliptic if \(C(x,u,\zeta)=0\) only if \(\zeta=0\).

    Remark. In the elliptic case all derivatives of the solution can be calculated from the given data and the given equation.

    Contributors