Hypocoercivity in algebraically constrained partial differential equations with application to Oseen equations

TL;DR

This paper investigates the long-time decay behavior of solutions to isotropic and anisotropic Oseen equations on the 2D torus using hypocoercivity, linking decay rates to the hypocoercivity index.

Contribution

It extends hypocoercivity analysis to algebraically constrained PDEs like Oseen equations, identifying decay characteristics and the hypocoercivity index.

Findings

Hypocoercivity index determines decay rate of solutions.

Similar decay behavior to finite-dimensional differential equations.

Analysis applies to both isotropic and anisotropic Oseen equations.

Abstract

The long-time behavior of solutions to different versions of Oseen equations of fluid flow on the 2D torus is analyzed using the concept of hypocoercivity. The considered models are isotropic Oseen equations where the viscosity acts uniformly in all directions and anisotropic Oseen-type equations with different viscosity directions. The hypocoercivity index is determined (if it exists) and it is shown that similar to the finite dimensional case of ordinary differential equations and differential-algebraic equations it characterizes its decay behavior.