Initial layer for the Homogenization of a Conservation Law with Vanishing Viscosity

We study the limit as ε → 0 of the solutions of the equation . This problem has already been addressed in a previous article in the case of well-prepared initial data, i.e. when the microscopic profile of the solution is adapted to the medium at time t = 0. Here, we prove that when the initial data is not well prepared, there is an initial layer during which the solution adapts itself to match the profile dictated by the environment. The typical size of the initial layer is of order ε. The proof relies strongly on the parabolic form of the equation; in particular, no condition of nonlinearity on A is required.