Optimal Tracing of Viscous Shocks in Solutions of Viscous Conservation Laws

This paper contains a qualitative study of a scalar conservation law with viscosity: ut + f(u)x = uxx . We consider the problem of identifying the location of viscous shocks, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law. We introduce a nonlinear functional whose minimizers yield the viscous travelling profiles which “optimally fit” the given solution. We prove that, outside an initial time interval and away from times of shock interactions, our functional remains very small, i.e. the solution can be accurately represented by a finite number of viscous travelling waves.