I am giving a talk in front of my applied PDE research group on hyperbolic conservation laws, the most basic form of which is the PDE $$ u\_t + f(u)\_x = 0 $$ where $u$ is the conserved quantity and $f$ is the flux. I was asked to present "nice applications" of these, and I thought to ask here. Does anyone here know of "nice" or "useful" applications of these in pure or applied mathematics? Maybe something you use in your own research. I thank all contributors.



Examples of applications of hyperbolic conservation laws
I am aware of some real-world applications which I learnt from Chapter 1 of
"Hyperbolic Partial Differential Equations. Theory, Numerics and Applications" by Meister and Struckmeier.
This chapter presents plenty of scenarios which can me modelled using balance laws, which can be reduced to hyperbolic conservation laws when the model is simpliefied enough.


1. *The traffic flow model*.


We model the number of vehicles as $\int\_\mathbb{R} u(t,x) dx$ (using a continuum hypothesis to give a meaning to the traffic density). The simplest model assumes that the veichles are moving with velocity $a(x,t)$ which only depends on the local traffic density, hence $a(x,t)=V(u(t,x))$ with $V(u)$ decreasing (We decelerate when we are in a traffic jam). Assuming that the number of cars on the road is invariant in time we get the nonlinear hyperbolic conservation law
$$
u\_t+f(u)\_x=0, \quad \text{for}\quad (x,t)\in \mathbb{R}\times \mathbb{R}^+,\quad \text{where} \quad f(u)=u V(u)
$$


A good driver prevents an accident by taking into account not only the local density of traffic but also its rate of change; indeed he observes that the velocity of the veihcles in front is increasing and he starts to reduce decelerate. We can model the speed suppoisng a linear change with respect to the rate of change of density, hence $a(x,t)=u V(u)-\alpha u\_x$. This gives a parabolic balance law, which degenerate in the hyperbolic case for $\alpha \to 0$.


**Other applications**


2. The model of *loss of a semiconductor surface caused by bombardment beam of ions*.
3. The model of *flow dynamics of an ideal gas* using the Euler equation


