Preprint 1998-016

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

Eitan Tadmor and Tamir Tassa

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.

Available as PostScript
On the homogenization of oscillatory solutions to nonlinear convection-diffusion equations
Eitan Tadmor, <>
Tamir Tassa, <>
Publishing information:
Advances in Mathematical Sciences and Applications 7 (1997) 93-117
Submitted by:
<> April 7 1998.

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