**Brian H. Gilding, Roberto Natalini, and Alberto Tesei**

## How Parabolic Free Boundaries Approximate Hyperbolic Fronts

**abstract:**

Some recent results concerning existence and qualitative behaviour of the
boundaries of the suppurts of solutions of the Cauchy problem for nonlinear
first--order hyperbolic and second-order parabolic scalar conservation laws
are discussed. Among other properties, it is shown that, under appropriate
assumptions, parabolic interfaces converge to hyperbolic ones in the vanishing
viscosity limit.

**Mathematics Subject Classification:**
35L65, 35K55, 35K65, 35L67, 35R35.

**Key words and phrases:**
Conservation laws, convection--diffusion equations,
shock waves, enthropy solutions, speed of propagation,
vanishing viscosity limit.