On the Homogenization of Oscillatory Solutions to Nonlinear Convection-diffusion Equations1

We study the behavior of oscillatory solutions to convection-diffusion problems, subject to initial and forcing data with modulated oscillations. We quantify the weak convergence in W−1,∞ to the ’expected’ averages and obtain a sharp W−1,∞-convergence rate of order O(ε) – 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 diffusion, etc. In this context, we show how the regularizing effect 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 filters out the initial oscillations and enables the strong convergence in later times. In memory of Haim Nessyahu, a dearest friend and research colleague.