3.3: Systems of First Order

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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.

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 $$ D(x,u(x),\eta_1,\ldots,\eta_{n-1},\kappa)=C(x,u(x),x,Q\eta).$$
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.
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 and Attributions

Prof. Dr. Erich Miersemann (Universität Leipzig)
Integrated by Justin Marshall.

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