# Stationary convection-diffusion equation in an infinite cylinder

## TL;DR

This paper investigates the existence and uniqueness of solutions to a stationary convection-diffusion equation in an infinite cylinder composed of two semi-infinite parts with different periodic regimes, revealing conditions for solution existence.

## Contribution

It provides a comprehensive analysis of solution existence, uniqueness, and multiplicity for the convection-diffusion equation in complex cylindrical geometries with varying regimes.

## Key findings

- Unique solutions under certain convection directions
- Existence of a one-parameter family of solutions in some cases
- Necessary and sufficient conditions for non-existence

## Abstract

We study the existence and uniqueness of a solution to a linear stationary convection-diffusion equation stated in an infinite cylinder, Neumann boundary condition being imposed on the boundary. We assume that the cylinder is a junction of two semi-infinite cylinders with two different periodic regimes. Depending on the direction of the effective convection in the two semi-infinite cylinders, we either get a unique solution, or one-parameter family of solutions, or even non-existence in the general case. In the latter case we provide necessary and sufficient conditions for the existence of a solution.