Singularly perturbed profiles

In the current paper the so-called REaction-DIffusion Manifold (REDIM) method of model reduction is discussed within the framework of standard singular perturbation theory. According to the REDIM a reduced model for the system describing a reacting flow (accounting for chemical reaction, advection and molecular diffusion) is represented by a low-dimensional manifold, which is embedded in the system state space and approximates the evolution of the system solution profiles in space and in time. This pure geometric construction is reviewed by using Singular Perturbed System (SPS) theory as the only possibility to formalize, to justify and to verify the suggested methodology. The REDIM is studied as a correction by the diffusion of the slow invariant manifold defined for a pure homogeneous system. A main result of the study is an estimation of this correction to the slow invariant manifold. A benchmark model of Michaelis-Menten is extended to the system with the standard diffusion described by the Laplacian and used as an illustration and for validation of analytic results. Let us remark that the Michaelis-Mentens model is not explicit SPS. We use so-called Global Quasi-Linearization (GQL) method to find a linear change of variable that rearranges the Michaelis-Mentens model as an explicit SPS.