Determining whether a System of PDEs is Parabolic or Elliptic Given a set of PDEs (linear or nonlinear) of the form:
$$ \begin{bmatrix} \dfrac{\partial\rho}{\partial t} \\ \dfrac{\partial v}{\partial t} \end{bmatrix} + 
\begin{bmatrix} A & B \\ C & D \end{bmatrix}
\begin{bmatrix} \dfrac{\partial\rho}{\partial x} \\ \dfrac{\partial v}{\partial x}\end{bmatrix} = \begin{bmatrix} F \\ G  \end{bmatrix} $$
where we have to solve for $\rho (x,t)$ and $v(x,t)$ and $A,B,C,D,F,G$ are functions of $x,\rho,v$. If the matrix $\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ has real and distinct eigenvalues, then I know the system is called a hyperbolic PDE. 


*

*Is there an analogous criteria to determine whether the system is Elliptic or Parabolic?

*In particular what type of system will it be if it has two real but repeated eigenvalues?


$\textbf {P.S.}$ I did try searching online but most results referred to a single PDE and the few that did refer to a system of PDEs were in a formal mathematical language I could not understand.
 A: We are considering the one-dimensional conservation law
$$
\frac{\partial\mathbf{U}}{\partial t} + \frac{\partial\mathbf{F}(\mathbf{U})}{\partial x} =\mathbf{0}\quad\text{where}\quad\mathbf{U}\equiv\begin{bmatrix} \rho\\ v \end{bmatrix}
$$
or in quasi-linear form
$$
\frac{\partial\mathbf{U}}{\partial t} + \mathbf{A}\frac{\partial\mathbf{U}}{\partial x}=\mathbf{0}
\quad\text{where}\quad\mathbf{A}\equiv\frac{\partial\mathbf{F}}{\partial\mathbf{U}}.
$$
Following Toro [ Riemann Solvers and Numerical Methods in Fluid Dynamics ] and LeVeque [ Numerical Methods for Conservation Laws ] 
"The system is hyperbolic [...], if $\mathbf{A}$ has $m$ real Eigenvalues $\lambda_1,...,\lambda_m$ […]. The system is said to be strictly hyperbolic if the eigenvalues $\lambda_i$ are all distinct."
and 
"The system is said to be elliptic […], if none of the eigenvalues $\lambda_i$ of $\mathbf{A}$ are real [thus all eigenvalues $\lambda_i$ need to be complex]."
2. In particular what type of system will it be if it has two real but repeated eigenvalues?
The eigenvalues $\lambda_i$ are real, but not distinct - the system is hyperbolic, even tough it is not strictly hyperbolic. Therefor $\mathbf{A}$ is only diagonizable, if $m$ linear independent eigenvectors exist.
1. Is there an analogous criteria to determine whether the system is elliptic or parabolic?
If all eigenvalues are complex the system is elliptic. Unfortunately for a parabolic system, I am not aware of a analogous criteria; From a physical point of view, a parabolic equation corresponds to a second derivative in space (e.g. transient head conduction). Therefor the flux $\mathbf{F}$ is not only dependent in $\mathbf{U}$, but also in its gradient;
$$
\mathbf{F}(\mathbf{U},\nabla\mathbf{U})
$$
and the quasi-linear form (which is the basis of our definition) does not hold anymore. If someone has more insight for this case, please leave a comment.
