Abstract: 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.
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