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

Abstract

We perform an asymptotic analysis with respect to the parameter $\mathit{\epsilon }\mathrm{>}0$ of the solution of the scalar advection–diffusion equation ${\mathit{y}}_{\mathit{t}}^{\mathit{\epsilon }}+\mathit{M}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}{\mathit{y}}_{\mathit{x}}^{\mathit{\epsilon }}-\mathit{\epsilon }{\mathit{y}}_{\mathit{x}\mathit{x}}^{\mathit{\epsilon }}=0$ , $\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\in \mathrm{(}0\mathrm{,}1\mathrm{)}\times \mathrm{(}0\mathrm{,}\mathit{T}\mathrm{)}$ , supplemented with Dirichlet boundary conditions. For small values of ε, the solution ${\mathit{y}}^{\mathit{\epsilon }}$ exhibits a boundary layer of size $\mathcal{O}\mathrm{(}\mathit{\epsilon }\mathrm{)}$ in the neighborhood of $\mathit{x}=1$ (assuming $\mathit{M}\mathrm{>}0$ ) and an internal layer of size $\mathcal{O}\mathrm{(}{\mathit{\epsilon }}^{1\mathrm{/}2}\mathrm{)}$ in the neighborhood of the characteristic starting from the point $\mathrm{(}0\mathrm{,}0\mathrm{)}$ . Assuming that these layers interact each other after a finite time $\mathit{T}\mathrm{>}0$ and using the method of matched asymptotic expansions, we construct an explicit approximation ${\mathit{P}}^{\mathit{\epsilon }}$ satisfying $\Vert {\mathit{y}}^{\mathit{\epsilon }}-{\mathit{P}}^{\mathit{\epsilon }}{\Vert }_{{\mathit{L}}^{\infty }\mathrm{(}0\mathrm{,}\mathit{T};{\mathit{L}}^2\mathrm{(}0\mathrm{,}1\mathrm{)}\mathrm{)}}=\mathcal{O}\mathrm{(}{\mathit{\epsilon }}^{1\mathrm{/}2}\mathrm{)}$ . We emphasize the additional difficulties with respect to the case M constant considered recently by the authors.