# 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.

1. Is there an analogous criteria to determine whether the system is Elliptic or Parabolic?
2. 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.

## 1 Answer

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.