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Mathematics LibreTexts

3.3: Systems of First Order

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.

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