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.

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.