Integral representations of the solutions to the homogenized transport equations

This paper provides two general representations of the weak limit u(t,x) of the solution uε(t,x) for (t,x) ∈ (0,T)×Rd, to the linear transport equation with an oscillating regular velocity b(x/ε), an initial datum u0ε(x) and a right-hand side fε(t,x). Our main assumption is the existence of a function w ∈ C1(Rd) such that b · ∇w is bounded from below by a positive constant. As a consequence, the dynamic flow Φ(t,y) associated with the vector field b(y) induces a one-to-one mapping from R×Σ onto Rd, where Σ is the equipotential hypersurface {w = 0}. This assumption allows us to finely characterize the kernel of the differential operator b(y) · ∇y(·) thanks to some quotient set Σb/R, where Σb is a suitable compact subset of Σ. Then, using a two-scale procedure we establish two integral formulas of the limit u(t,x), one over the quotient set and the other one over the d-dimensional torus Td, which involve the two-scales limits of the sequences of data and the projection of b onto the kernel of b(y) · ∇y(·). Finally, we also derive two alternative expressions of the asymptotics of the flow lim∞ Φ(t,y)/t for a.e. y ∈ Td.