A Case study in vanishing viscosity

We consider a special 2 x 2 viscous hyperbolic system of conservation laws of the form u_t + A(u)u_x = εu_xx, where A(u) = Df(u) is the Jacobian of a flux function f. For initial data with small total variation, we prove that the solutions satisfy a uniform BV bound, independent of ε. Letting ε → 0, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system u_t + f(u)_x = 0. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.