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

We perform an asymptotic analysis with respect to the parameter ε > 0 of the solution of the scalar advection–diffusion equation y t ε + M (x , t) y x ε − ε y x x ε = 0 , (x , t) ∈ (0 , 1) × (0 , T) , supplemented with Dirichlet boundary conditions. For small values of ε, the solution y ε exhibits a boundary layer of size O (ε) in the neighborhood of x = 1 (assuming M > 0) and an internal layer of size O (ε 1 / 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 P ε satisfying ‖ y ε − P ε ‖ L ∞ (0 , T ; L 2 (0 , 1)) = O (ε 1 / 2). We emphasize the additional difficulties with respect to the case M constant considered recently by the authors. [ABSTRACT FROM AUTHOR]