Homogenization of scalar conservation laws and transport equations.

Summary This thesis is devoted to the study of the asymptotic behavior of solutions of a class of partial differential equations with strongly oscillating coefficients. In a first step, we focus on a family of nonlinear equations, heterogeneous scalar conservation laws, which are involved in various problems in fluid mechanics or nonlinear electromagnetism. We assume that the flow of this equation is spatially periodic, and that the period of the oscillations tends to zero. We then identify the microscopic and macroscopic asymptotic profiles of the solution, and we prove a strong convergence result. In particular, we show that when the initial condition does not follow the microscopic profile dictated by the equation, an initial layer in time is formed during which the solutions adapt to it. In a second step, we consider a linear transport equation, which models the evolution of the density of a set of charged particles in a random and highly oscillating electric potential. We establish the appearance of microscopic oscillations in time and space in the density, in response to the excitation by the electric potential. Explicit formulas are also given for the homogenized transport operator when the dimension of space is equal to one.