On the Homogenization of Oscillatory Solutions to Nonlinear Convection-diffusion Equations Eitan Tadmor and Tamir Tassa

We study the behavior of oscillatory solutions to convection-diiusion problems, subject to initial and forcing data with modulated oscillations. We quantify the weak convergence in W ?1;1 to the 'expected' averages and obtain a sharp W ?1;1-convergence rate of order O(") { the small scale of the modulated oscillations. Moreover, in case the solution operator of the equation is compact, this weak convergence is translated into a strong one. Examples include nonlinear conservation laws, equations with nonlinear degenerate diiusion, etc. In this context, we show how the regularizing eeect built-in such compact cases smoothes out initial oscillations and, in particular, outpaces the persisting generation of oscillations due to the source term. This yields a precise description of the weakly convergent initial layer which lters out the initial oscillations and enables the strong convergence in later times.