Work

Integral representations of the solutions to the homogenized transport equations

Year: 2026

Type: article

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 ε 0 (x) and a right-hand side f ɛ ( t , x ). Our main assumption is the existence of a function w ∈ C 1 (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 ε 0 (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 .

Source: ESAIM Control Optimisation and Calculus of Variations

DOI: 10.1051/cocv/2026006

Authors: Marc Briane, Juan Casado-Díaz

Institutions: Centre National de la Recherche Scientifique, Institut National des Sciences Appliquées de Rennes, Institut de recherche mathématique de Rennes, Université de Rennes, Universidad de Sevilla

Language: English

Topic: Advanced Mathematical Modeling in Engineering

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid:

Funder: Ministerio de Ciencia, Innovación y Universidades