MULTIPLE-SCALE HOMOGENIZATION FOR WEAKLY NONLINEAR CONSERVATION-LAWS WITH RAPID SPATIAL FLUCTUATIONS

We consider hyperbolic conservation laws with rapid periodic spatial f luctuations and study initial value problems that correspond to small perturbations about a steady state. Weakly nonlinear solutions are com puted asymptotically using multiple spatial and temporal scales to cap ture the homogenized solution as well as its long-term behavior. We sh ow that the linear problem may be destabilized through interactions be tween two solution modes and the periodic structure. We also show that a discontinuity, either in the initial data or due to shock formation , introduces rapid spatial and temporal fluctuations to leading order in its zone of influence. The evolution equations we derive for the ho mogenized leading-order solution are more general than their counterpa rts for conservation laws having no rapid spatial variations. In parti cular, these equations may be diffusive for certain general flux vecto rs. Selected examples are solved numerically to substantiate the asymp totic results.