Integral representations of the solutions to the homogenized transport equations

¹ Univ Rennes, INSA Rennes, CNRS, IRMAR – UMR 6625, France
² Dpto. de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 17 June 2025
Accepted: 18 January 2026

Abstract

This paper provides two general representations of the weak limit u(t, x) of the solution u_ɛ(t, x) for (t, x) ∈ (0, T) × ℝ^d, to the linear transport equation with an oscillating regular velocity b(x/ɛ), an initial datum u_ε⁰(x) and a right-hand side f_ɛ(t, x). Our main assumption is the existence of a function w ∈ C¹(ℝ^d) 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 ℝ^d, 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 Σ^{^}/R, where Σ^{^} 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 Σ^{^}/R and the other one over the d-dimensional torus 𝕋^d, which involve the two-scales limits of the sequences of data u_ε⁰(x), f_ɛ(t, x) 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 ∈ 𝕋^d.