# Ngô Quốc Anh

## March 14, 2010

### Classification of a system of n first-order PDEs

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 11:18

The classification of a system of $n$ first-order PDEs is based on whether there are $n$ directions along which the PDEs reduce to $n$ ODEs. To be more precise, assume that we are given a system of $n$ equations in $n$ unknowns $u_1, u_2,...,u_n$ which we write in matrix form as

$\displaystyle \mathbf{u}_t + A(x,t,\mathbf{u})\mathbf{u}_x = \mathbf{b}(x,t,\mathbf{u})$,

where $\mathbf{u}=(u_1,...,u_n)^t$, $\mathbf{b}=(b_1,...,b_n)^t$, and $A=(a_{ij}(x,t,\mathbf{u}))$ is an $n \times n$ matrix.

Now we ask whether there is a family of curves along which the PDEs reduce to a system of ODEs, that is, in which the directional derivative of each $u_i$ occurs in the same direction. We consider a row vector $\gamma = (\gamma_1,...,\gamma_n)^t$ to be determined later. Then

$\displaystyle \mathbf{\gamma}^t\mathbf{u}_t + \mathbf{\gamma}^tA(x,t,\mathbf{u})\mathbf{u}_x = \mathbf{\gamma}^t\mathbf{b}(x,t,\mathbf{u})$.

We want the above system to have the form of a linear combination of total derivatives of the $u_i$ in the same direction $\lambda$, that is, we want our system to have the form

$\displaystyle \mathbf{m}^t \left( {{{\mathbf{u}}_t} + \lambda {{\mathbf{u}}_x}} \right) = \mathbf{\gamma}^t{\mathbf{b}}$

for some $\mathbf{m}$. Consequently, we require

$\displaystyle \mathbf{m}=\gamma, \quad \mathbf{m}^t\lambda=\gamma^tA$

or

$\displaystyle \gamma^t A=\lambda \gamma^t$.

This means that $\lambda$ is an eigenvalue of $A$ and $\gamma^t$ is a corresponding left eigenvector. Note that $\lambda$ as well as $\gamma$ can depend on $x$, $t$, and $\mathbf{u}$. So, if $(\lambda, \gamma^t)$ is an eigenpair, then

$\displaystyle \gamma^t \frac{d\mathbf{u}}{dt}=\gamma^t\mathbf{b}$

along

$\displaystyle \frac{dx}{dt}=\lambda(x,t,\mathbf{u})$

and the system of PDEs is reduced to a single ODE along the family of curves, called characteristics, defined by $\frac{dx}{dt}=\lambda$. The eigenvalue $\lambda$ is called the characteristics direction. Clearly, because there are $n$ unknowns, it would appear that $n$ ODEs are required; but if $A$ has $n$ distinct real eigenvalues, there are $n$ ODEs, each holding along a characteristics direction defined by an eigenvalue. In this case we say that the system is hyperbolic.

Definition. The quasilinear system

$\displaystyle \mathbf{u}_t + A(x,t,\mathbf{u})\mathbf{u}_x = \mathbf{b}(x,t,\mathbf{u})$

is hyperbolic if $A$ has $n$ real eigenvalues and $n$ linearly independent left eigenvectors. Once these eigenvectors are distinct, the system is called stricly hyperbolic.

The system is called elliptic if $A$ has no real eigenvalues, and it is parabolic if $A$ has $n$ real eigenvalues but fewer then $n$ independent left eigenvectors.

No exhaustive classification is made in the case that $A$ has both real and complex eigenvalues. Note that once matrix $A$ has $n$ distinct, real eigenvalues it has $n$ independent left eigenvectors, because distinct eigenvalues have independent eigenvectors.

More general systems of the form

$\displaystyle B(x,t,\mathbf{u})\mathbf{u}_t + A(x,t,\mathbf{u})\mathbf{u}_x = \mathbf{b}(x,t,\mathbf{u})$

can be considered as well. We refer the reader to a book entitled “An introduction to nonlinear partial differential equations” due to J.D. Logan.

We are now in a position to see why a single first-order quasilinear PDE is hyperbolic. The coefficient matrix for the equation

$\displaystyle u_t + c(x,t,u)u_x=b(x,t,u)$

is just the real scalar function $c(x,t,u)$ which has the single eigenvalue $c(x,t,u)$ and its corresponding eigenvector $1$, a constant function. In this direction, once $\frac{dx}{dt}=c(x,t,u)$ the PDE reduces to the ODE $\frac{du}{dt}=b(x,t,u)$. We refer the reader to the following topic, called characteristic curves, where we consider when the equation has constant coefficients and variable coefficients.

We place here three more examples.

Example 1 (The shallow-water equations). The following system

$\displaystyle\begin{gathered} {h_t} + u{h_x} + h{u_x} = 0, \hfill \\ {u_t} + u{u_x} + g{h_x} = 0, \hfill \\ \end{gathered}$

is trictly hyperbolic.

Example 2. The following system

$\displaystyle\begin{gathered} {u_t} - {v_x} = 0, \hfill \\ {v_t} - c{u_x} = 0, \hfill \\ \end{gathered}$

is elliptic if $c<0$ and is hyperbolic if $c>0$.

Example 3 (The diffusion equations). The following equation

$\displaystyle u_t=u_{xx}$

may be written as the first-order system

$\displaystyle\begin{gathered}u_t-v_x=0, \hfill \\u_x-v = 0, \hfill \\ \end{gathered}$

and thus is parabolic.