Preprint 1998-016

On the Homogenization of Oscillatory Solutions to Nonlinear Convection-Diffusion Equations

Abstract: We study the behavior of oscillatory solutions to convection-diffusion problems, subject to initial and forcing data with modulated oscillations. We {\it quantify} the weak convergence in $W^{-1,\infty}$ to the 'expected' averages and obtain a sharp $W^{-1,\infty}$-convergence rate of order ${\cal O}(\ep)$ -- 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.

Paper:

Available as PostScript

Title:

On the homogenization of oscillatory solutions to nonlinear convection-diffusion equations

Author(s):

Eitan Tadmor, <tadmor@math.ucla.edu>

Tamir Tassa, <tamir@arx.com>

Publishing information:

Advances in Mathematical Sciences and Applications 7 (1997) 93-117

Comments:

Submitted by:

<tadmor@math.ucla.edu> April 7 1998.