Hyperbolic Conservation Laws

Question

Why the name Hyperbolic Conservation law for $u_t+f(u)_x=0$ Is there any parabolic or elliptic conservation laws?

Answer 1

Definition: A system of $n$ equations of the form $U_t+\operatorname{div} f(U)=0$ is called hyperbolic if jacobian matrix of $f$ has $n$ linearly independent eigenvectors.

In the scalar case $u_t+f(u)_x=0$, where $f$ is a real valued function obviously satisfies the above definition and hence its hyperbolic.

In general a conservation law need not be hyperbolic. For example heat equation $u_t -\Delta u=0$ is a conservation law, because it can be written as $u_t -\operatorname{div} \cdot \operatorname{grad} u=0$. Heat equation is a parabolic PDE.

Hence equations of the form $u_t+f(u)_x=0$ are called hyperbolic conservation laws to emphasise that the equation hyperbolic and is a conservation law.