COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 4, Number 2, June 2005 pp. 297–311 ON HOMOGENIZATION OF NONLINEAR HYPERBOLIC EQUATIONS Y. Efendiev Department of Mathematics Texas A&M University, College Station, TX 77843-3368 B. Popov Department of Mathematics Texas A&M University, College Station, TX 77843-3368 (Communicated by Hector Ceniceros) Abstract. In this paper we study homogenization of nonlinear hyperbolic equations. The weak limit of the solutions is investigated by approximating the flux functions with piecewise linear functions. We study mostly Riemann problems for layered velocity fields as well as for the heterogeneous divergence free velocity fields. 1. Introduction. The homogenization of transport phenomena plays an impor- tant role in many applications such as flow in porous media, turbulent diffusion [9, 20]. In many practical applications, the velocity field that transports the sub- stance varies over a wide range of length scales, and, numerical flow models cannot in general resolve all of the scales of variations. Therefore, upscaling (numerical homogenization) approaches are needed for representing the effects of small scale variations on larger scale. The homogenization of hyperbolic equations with periodic velocity field has been studied previously [21, 12] by considering the limit of the solutions as the period size approaches to zero. In [6] the homogenization of nonlinear hyperbolic equation using the two-scale convergence concept is studied, where strong two-scale convergence of the solutions was investigated. The homogenized equation is obtained at the expense of the velocity “averaging” along the trajectories. For example, for the layered velocity field this approach yields the same homogenized equation as the underlying fine-scale equation. In this work our objective is to derive the homogenized equation for “the av- erages” of the solutions. A motivation for our paper stems from porous media applications, where the upscaled models for the solution on the coarse grid are needed. In a recent work [7], a coarse scale model, generalized convection-diffusion method is proposed. The starting point of this approach is the description of the homogenized equations. Once this description is postulated the calculations of the coarse scale quantities can be carried out. Our present paper helps to understand 1991 Mathematics Subject Classification. 35B40. Key words and phrases. Homogenization, nonlinear, hyperbolic, Riemann. 297