BV Solutions to Hyperbolic Systems by Vanishing Viscosity

Aim of these notes is to provide a self-contained presentation of recents results on hyperbolic systems of conservation laws, based on the vanishing viscosity approach. A system of conservation laws in one space dimension takes the form u t + f (u) x = 0. (1.1) Here u = (u 1 ,. .. , u n) is the vector of conserved quantities while the components of f = (f 1 ,. .. , f n) are called the fluxes. Integrating (1.1) over a fixed interval [a, b] we find d dt b a u(t, x) dx = b a u t (t, x) dx = − b a f u(t, x) x dx = f u(t, a) − f u(t, b) = [inflow at a] − [outflow at b]. Each component of the vector u thus 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. Systems of the form (1.1) are commonly used to express the fundamental balance laws of continuum physics, when small viscosity or dissipation effects are neglected. For a comprehensive 1