On Finite Closures of Homogenized Solutions of Nonlinear Hyperbolic Equations

We study nonlinear equations subject to oscillatory initial data. The oscillatory solution of such problems tends to a homogenized weak limit that is characterized by the corresponding homogenized equations. Those equations usually involve an additional independent variable, so that the weak limit is an average of infinitely many functions. In certain cases, however, there is an alternative description to the weak limit via a closed finite system of equations that the weak limit and some of its moments satisfy. We study the question of an existence of such finite closures in the context of semilinear Boltzmann type equations and the quasilinear Euler equations and show that, in most cases, finite closures do not exist. Mailing address: Department of Applied Mathematics, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Telephone: +972-53-646540 Fax: +972-3-6409357 Email address: tamir [email protected]