Decomposition of the solution to a two-dimensional singularly perturbed convection–diffusion equation with variable coefficients in a square and estimates in Hölder norms

The Dirichlet boundary value problem for a linear stationary singularly perturbed convection–diffusion equation with variable coefficients in a unit square of the $Oxy$ plane is considered. For a given convection coefficient, the problem is assumed to have one regular and two characteristic boundary layers, each located near one of the square sides. A decomposition of the solution to the problem is constructed, and a priori estimates in Hölder norms are obtained for the regular component of the decomposition.