Inertial manifolds for 1D reaction-diffusion-advection systems. Part II: periodic boundary conditions

TL;DR

This paper investigates the existence of inertial manifolds for 1D reaction-diffusion-advection systems with periodic boundary conditions, revealing that such manifolds may not exist for systems but do exist for scalar equations, highlighting the impact of boundary conditions.

Contribution

It demonstrates the dependence of inertial manifold existence on boundary conditions in 1D reaction-diffusion-advection systems, extending previous work to periodic cases.

Findings

Inertial manifolds may not exist for systems with periodic boundary conditions.

Inertial manifolds do exist for scalar reaction-diffusion-advection equations under periodic conditions.

Boundary conditions critically influence the existence of inertial manifolds in these systems.

Abstract

This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in \cite{KZI} and it is devoted to the case of periodic boundary conditions. It is shown that, in contrast to the case of Dirichlet or Neumann boundary conditions, considered in the first part, Inertial Manifolds may not exist in the case of systems endowed by periodic boundary conditions. However, as also shown, inertial manifolds still exist in the case of scalar reaction-diffusion-advection equations. Thus, the existence or non-existence of inertial manifolds for this class of dissipative systems strongly depend on the choice of boundary conditions.