One Dimensional Hyperbolic Systems of Conservation Laws

Abstract

These notes are meant to provide a survey of some recent results and techniques in the theory of conservation laws. In one space dimension, a system of conservation laws can be written as u_t + f(u) _x = 0.

Here u = (u₁, ... , u_n) is the vector of conserved quantities while the components of f = (f₁, ... , f_n) are called the fluxes. Integrating over the interval [a, b] one obtains

d/dt ?_a^b u(t,x) dx = ?_a^b u_t(t,x) dx = - ?_a^b f( u(t,x))_x dx = f(u(t,a)) - f(u(t,b)) = [inflow at a ] - outflow at b].

In other words, each component of the vector u represents a quantity which is neither created nor destroyed: its total amount inside any given interval [a, b] can change only because of the flow across boundary points.