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\markboth{\hfil Diagonalization of quasilinear systems \hfil EJDE--1999/25}
{EJDE--1999/25\hfil De-xing Kong \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 1999}(1999), No.~25, pp. 1--14. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)}
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A necessary and sufficient condition for the diagonalization
of multi-dimensional quasilinear systems
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35F20, 35L40.
\hfil\break\indent
{\em Key words and phrases:} Quasilinear systems, diagonalization,
strict Riemann invariants, \hfil\break\indent
conservation laws.
\hfil\break\indent
\copyright 1999 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted July 1, 1999. Published July 29, 1999.} }
\date{}
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\author{De-xing Kong}
\maketitle
\begin{abstract}
In this paper, the author obtains a necessary and sufficient condition on
the diagonalization of multi-dimensional quasilinear systems of first order,
and gives some physical applications.
\end{abstract}
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\newtheorem{Definition}{Definition}[section]
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\newtheorem{Corollary}{Corollary}[section]
\newtheorem{Remark}{Remark}[section]
\section{Introduction}
Consider the following multi-dimensional quasilinear system
\begin{equation}
\frac{\partial u}{\partial t}+\sum_{i=1}^m A_i(u)
\frac{\partial u}{\partial x_i}=0\,,
\end{equation}
where $u=(u_1,\dots, u_n)^T$ is the unknown vector function, $A_i(u)=\big(
a_{jk}^i(u)\big)$ is an $n\times n$ matrix with suitable smooth elements
$a_{jk}^i(u)\;(i=1,\dots,m;\ j,k=1,\dots,n)$, $m$ and $n$ are two integers
$\geq 1$.
We say system (1.1) is {\it diagonalizable}, if there exists a
smooth transformation $w=\left(w_1(u),\cdots,w_n(u)\right)^T$ with
non-vanishing Jacobian such
that (1.1) can be equivalently rewritten as the following coupled system
\begin{equation}
\frac{\partial w_j}{\partial t}+
\sum_{i=1}^n \lambda_j^i(w)\frac{\partial w_j}{\partial x_i}=0
\quad (j=1,\dots,n),
\end{equation}
where $\lambda_j^i(w)\; (i=1,\dots,m;\ j=1,\dots,n)$ are smooth functions
of $w$.
Such functions $w_j=w_j(u)\; (j=1,\dots,n)$ are called {\it strict
Riemann invariants}, and $\lambda_j^i(w)$ is called {\it the $x_i$-directional
speed of $w_j$} $(i=1,\dots,m;\ j=1,\dots,n)$.
The diagonal quasilinear system (1.2) is special and important
system possessing many good properties. For instance, it is easier
to find exact solutions and to establish the existence and uniqueness
theory of solutions, etc. ([CH], [Ho], [L1]).
However, not all quasilinear systems are diagonalizable.
Hence, it is interesting and significant to discuss the following two kinds of
problems
\begin{description}
\item{(I)} Under which conditions is system (1.1)
diagonalizable? If it is diagonalizable, what are the Riemann invariants?
\item{(II)} Under which conditions is system (1.1) not diagonalizable?
\end{description}
In this paper, we consider these problems for the multi-dimensional quasilinear
system (1.1) and give a necessary and sufficient condition for its
diagonalization. Moreover, in order to illustrate that our criterion is
effective, we give some applications.
In the case of one space dimension, it is well known that every
quasilinear hyperbolic system with two partial differential equations and with
two unknown functions is always diagonalizable ([CH], [L1]).
For general quasilinear systems in one space dimension, this result is obtained
by making use of Nijenhuis tensor $N^i_{jk}$ of the matrix $A_1(u)$:
\begin{equation}
N^i_{jk}={ \sum^n_{l=1} }\left[ a^1_{lj}
\frac{\partial a^1_{ik}}{\partial u_j}-
a^1_{lk}\frac{\partial a^1_{ij}}{\partial u_l}-a^1_{il}\left(
\frac{\partial a^1_{lk}}{\partial u_j}-
\frac{\partial a^1_{lj}}{\partial u_k} \right)\right]
\quad (i,j,k=1,\dots,n)\,,
\end{equation}
and introducing the tensor
\begin{equation}
T^i_{jk}={ \sum^n_{p,q=1} }\left(N^i_{pq}a^1_{pj}a^1_{qk}-
N^p_{jq}a^1_{ip}a^1_{qk}-N^p_{qk}a^1_{ip}a^1_{qj}+N^p_{jk}a^1_{iq}a^1_{qp}
\right),
\end{equation}
where $i,j,k=1,\dots,n$. Haantjes [Ha]
proved that (1.1) with $m=1$ is diagonalizable if and only if
\begin{equation}
T^i_{jk}\equiv 0\quad (i,j,k=1,\dots,n).
\end{equation}
Applications of this criterion were discussed in [Fe], [FT] and [T].
The advantage of the criterion is that we do not need to calculate
eigenvalues and eigenvectors of matrix $A_1(u)$. However, it seems to
me that it is not easy to generalize the criterion to multi-dimensional
system. Moreover, we do not know how to solve the
Riemann invariants.
Employing the eigenvalues and the eigenvectors of matrix $A_1(u)$,
Serre [S] considered this problem and gave another necessary
and sufficient condition.
Assuming for simplicity that the Spectrum $\left(A_1(u)\right)$ consists of $n$
distinct real values, he proved that (1.1) (in which $m=1$)
is diagonalizable if and only if the Frobenius conditions $l_i\{r_j,r_k\}=0$
$(i,j,k=1,\dots,n; j,k\neq i)$ hold, where $l_i$ and $r_i$ denote the
left and right eigenvectors related to the eigenvalue $\lambda_i(u)$ of
$A_1(u)$, and $\{\cdot ,\cdot\}$ is the Poisson
bracket of vector fields in $u$-space.
By enlarging the system, Garabedian observed that every quasilinear
system in one space dimension can be diagonalized and hence solved
locally by Picard iteration ([G.Page 100]), although the
diagonalizable system may be inhomogeneous.
For nonlinear systems in several space dimensions, only a few
results are known. Dafermos [D] and Lopes Filho \&
Nussenzweig Lopes [LN] examined the system of two partial differential
equations in several space variables, and obtained some results on
Riemann invariants, characteristic structure and the stability of
admissible $L^{\infty}$ solution. These results were started by an
observation in
[Ra] that only systems with commuting matrices can possess {\it BV}
estimates.
This paper aims to generalize Serre's result to
quasilinear system that might be multi-dimensional, that is to say,
to give a necessary and sufficient condition on diagonalization of (1.1).
The main results of this paper are presented and proved in Section 2.
Section 3 is devoted to the applications of the main results to some physical
systems such as the system of gas dynamics, the system of multicomponent
chromatography, the system of electrophoresis
and the quasilinear hyperbolic system of
conservation laws with rotational invariance.
\section{Main results}
\begin{Lemma}
We can derive from (1.1) a partial differential equation in diagonal form
\begin{equation}
\frac{\partial R(u)}{\partial t}+\sum_{i=1}^m \lambda_i(u)
\frac{\partial R(u)}{\partial x_i}=0\quad (\nabla_u R(u)\neq 0),
\end{equation}
where $\lambda_i(u)\;(i=1,\dots,m)$ are smooth functions, if and only if there
exists a smooth row vector $l=l(u)\;(l(u)\neq 0)$ such that $l(u)$ is a
common left eigenvector of $A_i(u)\;(i=1,\dots,m)$, the corresponding
eigenvalue is $\lambda_i(u)$, and it holds that
\begin{equation}
L(u) \wedge dL(u)=0,
\end{equation}
where
\begin{equation}
L(u)=l(u)du.
\end{equation}
\end{Lemma}
Such a smooth function $R=R(u)$ is called {\it one strict Riemann
invariant}. Obviously, system (1.1) is diagonalizable if and only if there
exist $n$ independent strict Riemann invariants.
\noindent{\bf Proof of Lemma 2.1} \hskip 2mm {\bf Necessity}:\hskip 3mm
Suppose that there exists a smooth function
$R=R(u)\; (\nabla_u R\neq 0)$ such that (2.1) holds. Thus, from (2.1) we have
\begin{equation}
\nabla_u R(u)\left( \frac{\partial u}{\partial t}+\sum_{i=1}^m \lambda_i(u)
\frac{\partial u}{\partial x_i}\right)=0.
\end{equation}
On the other hand, multiplying (1.1) by $\nabla_u R(u)$ from the left gives
\begin{equation}
\nabla_u R(u)\left( \frac{\partial u}{\partial t}+\sum_{i=1}^m A_i(u)
\frac{\partial u}{\partial x_i}\right)=0.
\end{equation}
The combination of (2.4) and (2.5) yields
\begin{equation}
\nabla_u R(u)A_i(u)=\lambda_i(u) \nabla_u R(u) \quad (i=1,\dots,m).
\end{equation}
(2.6) shows that $\nabla_u R(u)$ is a common left eigenvector of $A_i(u)\;
(i=1,\dots,m)$ and the corresponding eigenvalue is
$\lambda_i(u)$.
Taking $l(u)=\nabla_u R(u)$, we get (2.2) immediately.
{\bf Sufficiency}:\hskip 3mm
Suppose that $l(u)$ is a common left eigenvector of $A_i(u)\;(i=1,\dots,m)$
and $\lambda_i(u)$ is the corresponding real eigenvalue. Multiplying (1.1) by
$l(u)$ from the left gives
\begin{equation}
l(u)\left(\frac{\partial u}{\partial t}+\sum_{i=1}^m \lambda_i(u)
\frac{\partial u}{\partial x_i}\right)=0.
\end{equation}
On the other hand, (2.2) implies that the equation $L(u)=0$ is completely
integrable. Hence, by the well-known Frobenius Theorem, there exists a smooth
function $R=R(u)\;(\nabla_u R(u)\neq 0)$ such that $l(u) /\!\!/ \nabla_u R(u)$.
Thus, from (2.7) we get (2.1) immediately. This finishes the proof.
$\quad\Box$
\begin{Remark}
The proof of the sufficiency in Lemma 2.1 provides a method to obtain the
strict Riemann invariant. The first integral of a common left
eigenvector of $A_i(u)\, (i=1,\dots,n)$ is a strict Riemann invariant.
\end{Remark}
By Lemma 2.1, we have
\begin{Theorem}
System (1.1) is diagonalizable if and only if there exist $n$ independent row
vectors $l_j(u)=\left( l_{j1}(u),\cdots, l_{jn}(u)\right)\ (j=1,\dots,n)$
such that each
$l_j(u)$ is a common left eigenvector of $A_i(u)\; (i=1,\dots,m)$ and
\begin{equation}
L_j(u) \wedge dL_j(u)=0,
\end{equation}
where
\begin{equation}
L_j(u)=l_j(u)du.
\end{equation}
\end{Theorem}
\begin{Corollary}
If system (1.1) is diagonalizable, then
\begin{equation}
A_i(u)A_{i'}(u)=A_{i'}(u)A_i(u)\quad
(i,i'=1,\dots,m).
\end{equation}
\end{Corollary}
\noindent{\bf Proof.}
By Theorem 2.1, there exist $n$ independent row vectors
$l_j(u)\;(j=1,\dots,n)$ such that
\begin{eqnarray}
&l_j(u)A_i(u)=\lambda^i_j(u) l_j(u), &\nonumber \\
&l_j(u)A_{i'}(u)=\lambda^{i'}_j(u) l_j(u),&
\end{eqnarray}
where $i,i'=1,\dots,m;\ j=1,\dots,n$,
and $\lambda^i_j(u)$ is the eigenvalue of $A_i(u)$ corresponding to the
common left eigenvector $l_j(u)$.
Multiplying the first (resp. second) equation of (2.11) by
$A_{i'}$ (resp. $A_i$) from the right gives
\begin{equation}
l_j(u)A_i(u)A_{i'}(u)=l_j(u)A_{i'}(u)A_i(u)\quad
(j=1,\dots,n).
\end{equation}
Noting the fact that $l_j(u)\;(j=1,\dots,n)$ are independent, from (2.12) we
get (2.10) immediately. This completes the proof. $\quad\Box$
\begin{Remark}
Diagonal system of partial differential equations is hyperbolic. But when
$m>1$, it is not strictly hyperbolic in the sense of Majda ([M]).
In fact, if all $A_i(u)\;(i=1,\dots,m)$ are diagonal matrices, then for any
given $u$ on the domain under consideration, there is at least one unit
vector $\omega=(\omega_1,\cdots,\omega_m)$ such that the matrix ${
\sum_{i=1}^n\omega_i A_i(u)}$ does not have $n$ distinct real eigenvalues.
Not only that, it can not have any non-degenerate wave cones whatsoever
([D], [LN]). This is different from non-diagonal systems ([L2]).
\end{Remark}
\vskip 4mm
In what follows, we discuss three special but important cases.
{\bf Case I.} $\quad$ Hyperbolic systems with at least one-directional
strict hyperbolicity
For simplicity of statement, we introduce
\begin{Definition}
We say an $n\times n$ matrix $A(u)$ is hyperbolic, if $A(u)$ has $n$ real
eigenvalues and is diagonalizable for any given $u$ on the domain under
consideration; $A(u)$ is strictly hyperbolic, if $A(u)$ has $n$ distinct
real eigenvalues.
\end{Definition}
\begin{Lemma}
Suppose that matrix $A(u)$ is strictly hyperbolic and $B(u)$ is hyperbolic.
Then there exist $n$ independent row vectors
$l_j(u)=\left(l_{j1}(u),\cdots,l_{jn}(u)\right)\;
(j=1,\dots,n)$ such that
each $l_j(u)$ is a common left eigenvector of $A(u)$ and $B(u)$, if and only if
\begin{equation}
AB=BA.
\end{equation}
\end{Lemma}
\noindent{\bf Proof.}
The proof of the necessity is the same as that of Corollary 2.1, moreover,
we do not require that $A(u)$ is strictly hyperbolic.
It remains to prove the sufficiency.
Let $\lambda^A_j\;(j=1,\dots,n)$ be the $n$ distinct real eigenvalues of
$A(u)$. Without loss of generality, we may suppose that
\begin{equation}
\lambda^A_1 (u)<\cdots <\lambda^A_n (u).
\end{equation}
Moreover, let $l_j^A(u)$ be the left eigenvector of $A(u)$ corresponding to
$\lambda^A_j (u)\;(j=1,\dots,n)$ and introduce
$${\cal L}^A=\left(\begin{array}{c}
l_1^A(u) \\
\vdots \\
l_n^A(u)
\end{array}\right).$$
Thus, we have
\begin{equation}
{\cal L}^A A\left({\cal L}^A \right)^{-1}={\rm diag} \left(\lambda_1^A(u),\cdots,\lambda^A_n(u)
\right).
\end{equation}
On the other hand, by (2.13) we have
\begin{equation}
\left({\cal L}^A A\left({\cal L}^A \right)^{-1}\right)
\left({\cal L}^A B\left({\cal L}^A \right)^{-1}\right)=
\left({\cal L}^A B\left({\cal L}^A \right)^{-1}\right)
\left({\cal L}^A A\left({\cal L}^A \right)^{-1}\right).
\end{equation}
Noting (2.14) and (2.15), from (2.16) we see that
${\cal L}^A B\left({\cal L}^A \right)^{-1}$ is a diagonal matrix. This finishes the proof. $\quad\Box$
Therefore, by Theorem 2.1 we have
\begin{Corollary}
Suppose that $A_i(u)\;(i=1,\dots,m-1)$ are hyperbolic and $A_m(u)$ is strictly
hyperbolic. Then system (1.1) is diagonalizable if and only if
\begin{equation}
A_i(u)A_m(u)=A_m(u)A_i(u)\quad (i=1,\dots,m-1)
\end{equation}
and
\begin{equation}
L_j^{A_m}(u) \wedge dL_j^{A_m}(u)=0 \quad (j=1,\dots,n),
\end{equation}
where $L_j^{A_m}(u)=l_j^{A_m}(u)du$, in which $l_j^{A_m}(u)$ is a
left eigenvector of $A_m(u)$ corresponding to $\lambda_j^{A_m}(u)$.
\end{Corollary}
Such a system is called {\it the hyperbolic system with at least
one-directional strict hyperbolicity}.
{\bf Case II.} $\quad$ Symmetric systems
The following Lemma is well known.
\begin{Lemma}
Suppose that $A(u)$ and $B(u)$ are $n\times n$ real symmetric matrices. Then
the conclusion of Lemma 2.2 is still valid.
\end{Lemma}
Hence, by Theorem 2.1 we get
\begin{Corollary}
Suppose that $A_i(u)\;(i=1,\dots,m)$ are $n\times n$
real symmetric matrices. Then system
(1.1) is diagonalizable if and only if
\begin{equation}
A_i(u)A_{i'}(u)=A_{i'}(u)A_i(u)\quad
(i,i'=1,\dots,m)
\end{equation}
and (2.8) holds, where $L_j(u)=l_j(u)du$, in which
$l_j(u)$ stands for
a common left eigenvector of $A_i(u)\;(i=1,\dots,m)$ corresponding to
$\lambda_j^{i}(u)$.
\end{Corollary}
{\bf Case III.} $\quad$ Systems with constant multiplicity eigenvalues
Now we turn to consider the case that $A_i(u)\;(i=1,\dots,m)$ have
constant multiplicity eigenvalues. Without loss of generality, we suppose that
\begin{equation}
\lambda^i (u)\stackrel{\triangle}{=}\lambda^i_1 (u)\equiv\cdots\equiv
\lambda^i_{p_i} (u)< \lambda^i_{p_i+1} (u)<\cdots <\lambda^i_n (u)
\quad (i=1,\dots,m),
\end{equation}
where $p_i$ is an integer $> 1$.
Suppose that there exist $n$ independent smooth row vectors \\
$l_j(u)=\left(l_{j1}(u),\cdots,l_{jn}(u)\right)$ $(j=1,\dots,n)$ such that
$A_i(u)\;(i=1,\dots,m)$ can be diagonalized simultaneously, namely,
\begin{equation}
{\cal L}(u)A_{i}(u){\cal L}^{-1}(u)={\rm diag} \left(\lambda^i_1 (u),\cdots,\lambda^i_n (u)
\right)\quad (i=1,\dots,m),
\end{equation}
where ${\cal L}(u)=\left(l_{jk}(u)\right)$ is an $n\times n$ matrix and
$\lambda^i_j (u)$
is the eigenvalue of $A_i(u)$ corresponding to the left eigenvector $l_j(u)$.
We have
\begin{Theorem}
Under the hypotheses (2.20)-(2.21), system (1.1) is diagonalizable if and
only if
\begin{equation}
L_1(u)\wedge \cdots\wedge L_p(u)\wedge dL_{\alpha}(u)=0\quad
(\alpha=1,\cdots,p)
\end{equation}
and
\begin{equation}
L_{\beta}(u)\wedge dL_{\beta}(u)=0\quad
(\beta=p+1,\dots,n),
\end{equation}
where $p={ \min_{i=1,\dots,m}}\left\{p_i\right\}$ and
$L_j(u)=l_j(u)du\;\ (j=1,\dots,n)$.
\end{Theorem}
\noindent{\bf Proof.} \hskip 4mm {\bf Necessity}: \hskip 3mm
Noting (2.20)-(2.21), from (1.1) we have
\begin{equation}
l_{\alpha}(u)\left( \frac{\partial u}{\partial t}+\sum_{i=1}^m
\lambda^i(u)\frac{\partial u}{\partial x_i} \right)=0
\quad (\alpha=1,\cdots,p).
\end{equation}
By the fact that (1.1) is diagonalizable, there exists a $p\times p$ smooth
invertible matrix $C(u)=\left( C_{\mu\nu}(u) \right)^p_{\mu,\nu =1}$ and smooth
functions $w_{\alpha}=w_{\alpha}(u)\; (\alpha=1,\cdots,p)$ such that
\begin{equation}
\sum_{\mu=1}^p C_{\alpha\mu}(u)L_{\mu} (u)=dw_{\alpha}(u)\quad
(\alpha=1,\cdots,p).
\end{equation}
Then, it follows from (2.25) that
\begin{equation}
\sum_{\mu=1}^p C_{\alpha\mu}(u)dL_{\mu} (u)+\sum_{\mu=1}^p dC_{\alpha\mu}(u)
\wedge L_{\mu}(u)=0 \quad
(\alpha=1,\cdots,p).
\end{equation}
Thus, we have
\begin{equation}
\sum_{\mu=1}^p C_{\alpha\mu}(u)L_1(u)\wedge\cdots \wedge L_p(u)\wedge dL_{\mu}(u)=0
\quad (\alpha=1,\cdots,p).
\end{equation}
Noting the fact that $C(u)$ is invertible, by (2.27) we get (2.22) immediately.
The proof of (2.23) is the same as that of Theorem 2.1.
The sufficiency can be proved in a manner similar to the proof of the
sufficiency of Lemma 2.1. For brevity, we omit it here.
This completes the proof. $\quad\Box$
\begin{Remark}
When $p=n-1$, (2.22) becomes trivial, namely, (2.22) holds automatically;
When $p=n$, (2.22) and (2.23) always hold, so system (1.1) is always
diagonalizable in this case.
\end{Remark}
\begin{Remark}
When $m=1$, Theorem 2.1 goes back to a Serre's result in [S].
\end{Remark}
\section {Applications}
In this section, we give the applications of the results presented in
Section 2 to some physical systems.
\subsection*{System of gas dynamics}
Consider the system of gas dynamics in three space dimensions ([CH])
\begin{equation}
\frac{\partial U}{\partial t}+A(U)\frac{\partial U}{\partial x}+
B(U)\frac{\partial U}{\partial y}+C(U)\frac{\partial U}{\partial z}=0,
\end{equation}
where
\begin{equation}
U=\left(\begin{array}{c}
\rho\\u\\v\\w\\S
\end{array}\right),\quad
A(U)=\left(\begin{array}{ccccc}
u & \rho & 0 & 0 & 0 \\
\frac{1}{\rho}\frac{\partial p}{\partial {\rho}} & u & 0 & 0 & \frac{1}{\rho}\frac{\partial p}{\partial S} \\
0 & 0 & u & 0 & 0 \\
0 & 0 & 0 & u & 0 \\
0 & 0 & 0 & 0 & u
\end{array}\right),
\end{equation}
$$B(U)=\left(\begin{array}{ccccc}
v & 0 & \rho & 0 & 0 \\
0 & v & 0 & 0 & 0 \\
\frac{1}{\rho}\frac{\partial p}{\partial {\rho}} & 0 & v & 0 & \frac{1}{\rho}\frac{\partial p}{\partial S} \\
0 & 0 & 0 & v & 0 \\
0 & 0 & 0 & 0 & v
\end{array}\right),\
C(U)=\left(\begin{array}{ccccc}
w & 0 & 0 & \rho & 0 \\
0 & w & 0 & 0 & 0 \\
0 & 0 & w & 0 & 0 \\
\frac{1}{\rho}\frac{\partial p}{\partial {\rho}} & 0 & 0 & w & \frac{1}{\rho}\frac{\partial p}{\partial S} \\
0 & 0 & 0 & 0 & w
\end{array}\right),
$$
$\rho >0$ is the density, $(u,v,w)$ is the velocity, $S$ is the entropy,
$p$ is the pressure and the state equation is
\begin{equation}
p=p(\rho,S)>0,
\end{equation}
in which $p(\rho,S)$ satisfies that on each finite domain of $\rho>0$,
\begin{equation}\frac {\partial p}{\partial \rho}(\rho,S)>0.\end{equation}
By a simple calculation, we observe that there is one and only
one independent row vector
\begin{equation}
l_5(U)\stackrel{\triangle}{=}(0,\ 0,\ 0,\ 0,\ 1)
\end{equation}
such that $l_5(U)$ is a non-zero common left eigenvector of $A(U), B(U)$ and
$C(U)$, and it holds that
\begin{equation}L_5(U)\wedge dL_5(U)=0,
\end{equation}
where $L_5(U)=l_5(U)dU$. Hence, by Lemma 2.1 we obtain
\begin{Theorem}
From system (3.1), one and only one non-trivial
partial differential equation in diagonal
form can be reduced, namely, the
entropy equation of conservation law.
\end{Theorem}
\begin{Remark} Similarly, it is easy to check that the system of isentropic
flow in two space dimensions is not diagonalizable. However,
it is well known that the system of isentropic flow in one space dimension
is always diagonalizable ([CH]).
\end{Remark}
\subsection*{System of multicomponent chromatography}
The following system arises in multicomponent chromatography (see [RA] or
[T] for the case of one space dimension)
\begin{equation}
\frac{\partial u_i}{\partial t}+{\sum_{j=1}^m}\frac{\partial}
{\partial x_j}\bigg(\frac{a_i^ju_i}{1+{\sum^n_{k=1}}u_k}\bigg)=0
\quad (i=1,\cdots ,n),
\end{equation}
where $u_i=u_i(t,x)\;(i=1,\dots,n)$ are the non-negative unknown functions,
$a_i^j\; (i=1,\dots,n;\; j=1,\dots,m)$ are positive constants
satisfying
\begin{equation}
00,$$
and $a_i^j\;(i=1,\cdots ,n;\; j=1,\cdots ,m)$ are positive constants satisfying
$$00, \quad\forall\;r>0,
\end{equation}
$m$ is an integer $\geq 1$. System (3.19) can be used to describe the
propagation of waves in various situations in mechanics (such as the reactive
flows, magneto-hydrodynamics and elasticity theory, etc.) at least for
the case that $m=1$ ([B], [F1], [KK], [LW]). It is no longer strictly
hyperbolic, and possesses the eigenvalues with constant multiplicity even for
the case that $m=1$. When $m=1$ and $n=2$, system (3.19) was first studied
by [KK] and [LW]. Freist\"{u}hler [F1]-[F2] considered the Riemann problem
and the Cauchy problem for system (3.19) with $m=1$ and $n \ge 1$.
Rewrite (3.19) as
\begin{equation}
\frac{\partial u}{\partial t}+\sum_{i=1}^m A_i(u)\frac{\partial u}
{\partial x_i}=0,
\end{equation}
where
\begin{equation}
A_i(u)=\left(\begin{array}{cccc}
f_i(r)+\frac{f_i'(r)}{r}u_1^2 & \frac{f_i'(r)}{r}u_1u_2
& \cdots & \frac{f_i'(r)}{r}u_1u_n \\[3pt]
\frac{f_i'(r)}{r}u_1u_2 & f_i(r)+\frac{f_i'(r)}{r}u_2^2
& \cdots & \frac{f_i'(r)}{r}u_2u_n \\[2pt]
\vdots & \vdots & \ddots & \vdots \\[2pt]
\frac{f_i'(r)}{r}u_1u_n & \frac{f_i'(r)}{r}u_2u_n &
\cdots & f_i(r)+\frac{f_i'(r)}{r}u_n^2
\end{array}\right),
\end{equation}
in which $r=|u|>0$.
In what follows, we consider the case that $r>0$.
Without loss of generality, we may suppose that $u_1\neq 0$. It is easy to
calculate that
\begin{equation}
\lambda^i(u)\stackrel{\triangle}{=}\lambda^i_1 (u)\equiv \cdots\equiv
\lambda^i_{n-1} (u)=f_i(r)
\end{equation}
is an eigenvalue of $A_i(u)$ with constant multiplicity $n-1$.
$A_i(u)\;(i=1,\dots,m)$ have $n-1$ independent common left eigenvectors
corresponding to the eigenvalue $\lambda^i(u)$:
\begin{equation}\begin{array}{ll}
l_1(u)=(-u_2,u_1,0,\cdots,0), \qquad\;\, l_2(u)=(-u_3,0,u_1,0,\cdots,0),
\nonumber \\
\hskip 4.5cm \cdots\cdots, \nonumber \\
l_{n-2}(u)=(-u_{n-1},0,\dots,u_1,0),\; l_{n-1}(u)=(-u_n,0,\cdots,0,u_1).
\end{array}\end{equation}
Moreover,
\begin{equation}
\lambda^i_{n} (u)\stackrel{\triangle}{=}f_i(r)+rf_i'(r)
\end{equation}
is another eigenvalue of $A_i(u)$, and
\begin{equation}
l_{n} (u)=(u_1,\dots,u_n)
\end{equation}
is a common left eigenvector of $A_i(u)$ corresponding to
the eigenvalue $\lambda^i_{n} (u)$. Obviously,
$l_{j} (u)\;(j=1,\dots,n)$
given by (3.24) and (3.26) are independent. On the other hand, when $r>0$, we
have
\begin{equation}
\lambda^i_{n} (u)>\lambda^i (u)\quad (i=1,\dots,m).
\end{equation}
It is easy to check that system (3.19) satisfies all conditions required by
Theorem 2.2. Hence, by Theorem 2.2 we have
\begin{Theorem}
Consider system (3.19) on the domain of $r>0$. If (3.20) holds,
then system (3.19) is always diagonalizable.
\end{Theorem}
In fact, let
\begin{equation}
u=rs,
\end{equation}
where $r=|u|$, $s=(s_1,\cdots,s_n)^T\in S^{n-1}$. Then system (3.19) can be
rewritten as
\begin{equation}
\frac{\partial s}{\partial t}+\sum_{i=1}^m f_i(r)\frac{\partial s}
{\partial x_i}=0,\quad\;
\end{equation}
\begin{equation}
\frac{\partial r}{\partial t}+\sum_{i=1}^m\frac{\partial }{\partial x_i}
\left(rf_i(r)\right)=0.
\end{equation}
\begin{Remark}
A method for finding exact solutions to system (3.29)-(3.30)
was given by [KN] at least for the case that $m=1$.
\end{Remark}
\begin{Remark}
In this paper, we introduce the multi-dimensional system (3.7) of multicomponent
chromatography, the multi-dimensional system (3.18) of electrophoresis and the
multi-dimensional system (3.19) of conservation laws with rotational invariance.
When $m=1$, they go back to the classical one-dimensional systems.
Therefore, systems (3.7), (3.18) and (3.19) can be regarded as the
generalization of the classical one-dimensional systems. The further
study for systems (3.7), (3.18) and (3.19) remains to be done.
\end{Remark}
\paragraph{Acknowledgments}
This work was partially supported by the Grant-in-Aid for
Scientific Research for JSPS Postdoctoral Fellowship for Foreign
Researcher in Japan,
provided by the Japan Ministry of Education, Science and Culture.
The author thanks the referee for his/her pertinent comments
and valuable suggestions.
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\noindent{\sc De-xing Kong }\\
Department of Mathematics\\
Kyoto Sangyo University \\
Kyoto 603-8555, Japan \\
E-mail address: dkong@cc.kyoto-su.ac.jp
\end{document}