Asymptotic analysis of an advection-diffusion equation involving interacting boundary and internal layers

Youcef Amirat (1, 2) , Arnaud Munch (1, 2)

1 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]
2 LMBP - Laboratoire de Mathématiques Blaise Pascal

As ε goes to zero, the unique solution of the scalar advection-diffusion equation y ε t −εy ε xx +M y ε x = 0, (x, t) ∈ (0, 1) × (0, T) submitted to Dirichlet boundary conditions exhibits a boundary layer of size O(ε) and an internal layer of size O(√ ε). If the time T is large enough, these thin layers where the solution y ε displays rapid variations intersect and interact each other. Using the method of matched asymptotic expansions, we show how we can construct an explicit approximation P ε of the solution y ε satisfying y ε − P ε L ∞ (0,T ;L 2 (0,1)) = O(ε 3/2) and y ε − P ε L 2 (0,T ;H 1 (0,1)) = O(ε), for all ε small enough.

Mots clés: Asymptotic analysis; Singular perturbation; Internal and boundary layers; Sobolev estimates

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Youcef Amirat, Arnaud Munch. Asymptotic analysis of an advection-diffusion equation involving interacting boundary and internal layers. 2019. ⟨hal-02110729v1⟩