There is a large literature on the theory and the numerics of hyperbolic systems of conservation laws. As an example, consider the shallow water equation: $$ h_t + m_x = 0 \\ m_t + (\frac{m^2}{h} + \frac12 g h^2)_x = 0, $$ where $_t$ and $_x$ denote partial derivative with respect to the time and space variables, and $g$ is a positive constant. The equation above is sometimes called the conservative form of the shallow water equation, since it is possible to identify the conserved variables and the fluxes. Alternatively, it is possible to write the advective form of the equation: $$ \begin{pmatrix} h_t \\ m_t \end{pmatrix} + \begin{pmatrix} 0 & 1 \\ gh - \frac{m^2}{h^2} & 2\frac{m}{h} \end{pmatrix} \begin{pmatrix} h_x \\ m_x \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}. $$ The eigenvalues of the matrix appearing in the equation above are: $$ \frac{m}{h}\pm\sqrt{gh} $$ and since $g, h$ are positive numbers, the eigenvalues are always real, and the system of conservation laws is called hyperbolic.
I was wondering if it has any sense to consider elliptic systems of conservation laws (i.e. systems for which the eigenvalues are imaginary numbers). Is there a physical example? In standard books and on the web I have not found any reference about this, and I would like to learn more on this topic. I would like also to know what are the important properties to keep in mind for the elliptic case (e.g. what about characteristics? Riemann invariants? boundary conditions? or is the theory completely different from the hyperbolic case?)
