Internal layer intersecting the boundary of a domain in a singular advection–diffusion equation

Article type: Research Article

Authors: Amirat, Youcef | Münch, Arnaud^{; *}

Affiliations: Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-Ferrand, France

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We perform an asymptotic analysis with respect to the parameter ε>0 of the solution of the scalar advection–diffusion equation ytε+M(x,t)yxε−εyxxε=0, (x,t)∈(0,1)×(0,T), supplemented with Dirichlet boundary conditions. For small values of ε, the solution yε exhibits a boundary layer of size O(ε) in the neighborhood of x=1 (assuming M>0) and an internal layer of size O(ε1/2) in the neighborhood of the characteristic starting from the point (0,0). Assuming that these layers interact each other after a finite time T>0 and using the method of matched asymptotic expansions, we construct an explicit approximation Pε satisfying ‖yε−Pε‖L∞(0,T;L2(0,1))=O(ε1/2). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.

Keywords: Asymptotic analysis, singular perturbation, internal and boundary layers, Sobolev estimates

DOI: 10.3233/ASY-231836

Journal: Asymptotic Analysis, vol. 134, no. 3-4, pp. 297-343, 2023