Inertial manifolds for 1D reaction-diffusion-advection systems. Part I: Dirichlet and Neumann boundary conditions

TL;DR

This paper establishes the existence of inertial manifolds for 1D reaction-diffusion-advection systems with Dirichlet or Neumann boundary conditions by applying a non-local change of variables, addressing a key spectral gap challenge.

Contribution

It demonstrates that the spectral gap property can be achieved through a non-local transformation, enabling inertial manifold construction for these boundary conditions.

Findings

Spectral gap property is satisfied after a non-local change of variables.

Inertial manifolds exist for Dirichlet and Neumann boundary conditions.

Periodic boundary conditions may not admit inertial manifolds.

Abstract

This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions. Although this problem does not initially possess the spectral gap property, it is shown that this property is satisfied after the proper non-local change of the dependent variable. The case of periodic boundary conditions where the situation is principally different and the inertial manifold may not exist is considered in the second part of our study.