Asymptotic analysis for singularly perturbed convection-diffusion equations with a turning point

Abstract

Turning points occur in many circumstances in fluid mechanics. When the viscosity is small, very complex phenomena can occur near turning points, which are not yet well understood. A model problem, corresponding to a linear convection-diffusion equation (e.g., suitable linearization of the Navier-Stokes or B́nard convection equations) is considered. Our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the Navier-Stokes or B́nard convection equations. Of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems. In our case, at this stage, the full theoretical (asymptotic) analysis is provided. This study is totally new to the best of our knowledge. Numerical treatment and more complex problems will be considered elsewhere.