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Stability and large-time behavior for the N-Dimensional Euler-FENE dumbbell model near an equilibrium

Zheng-an Yao, Ruijia Yu

TL;DR

The paper tackles the stability and long-time behavior of the N-dimensional Euler-FENE dumbbell model in the dissipationless velocity regime, near the equilibrium $(0,\psi_\infty)$. By uncovering a wave structure induced by the equilibrium and the coupling between the fluid and polymer microstructure, the authors establish global existence for small data in $H^s$ via a two-tier energy framework and bootstrap arguments. They subsequently derive decay rates for $u$ and the perturbation $\psi$ that, under an $L^1$-type smallness assumption, mirror diffusion-driven decay and align with results for viscoelastic FENE models, using Fourier splitting to achieve optimal rates. The work illuminates how wave effects from the coupling compensate for the lack of velocity dissipation and yield decay identical to the heat equation, providing a rigorous stabilization mechanism for this fluid-structure interaction system.

Abstract

This paper studies the N-dimensional FENE dumbbell model without velocity dissipation, focusing on the stability and decay of perturbations near the steady solution $(0,\pin)$. Due to the lack of velocity dissipation, the above problems are highly challenging. In fact, without coupling, the corresponding N-dimensional Euler equation near u=0 is well known to be unstable. To overcome this difficulty, we analyze the wave structure arising in the system governing perturbations around the steady state, which originates from the equilibrium configuration and the coupling effects. This wave structure enables us to establish the global stability in the $H^s$-type Sobolev norms. Also, we highlight the critical role of wave structure in the decay estimates of the Euler-FENE dumbbell model. By combining this property with the Fourier splitting method, we derive the decay rate, which is identical to that of the general FENE dumbbell with velocity dissipation.

Stability and large-time behavior for the N-Dimensional Euler-FENE dumbbell model near an equilibrium

TL;DR

The paper tackles the stability and long-time behavior of the N-dimensional Euler-FENE dumbbell model in the dissipationless velocity regime, near the equilibrium . By uncovering a wave structure induced by the equilibrium and the coupling between the fluid and polymer microstructure, the authors establish global existence for small data in via a two-tier energy framework and bootstrap arguments. They subsequently derive decay rates for and the perturbation that, under an -type smallness assumption, mirror diffusion-driven decay and align with results for viscoelastic FENE models, using Fourier splitting to achieve optimal rates. The work illuminates how wave effects from the coupling compensate for the lack of velocity dissipation and yield decay identical to the heat equation, providing a rigorous stabilization mechanism for this fluid-structure interaction system.

Abstract

This paper studies the N-dimensional FENE dumbbell model without velocity dissipation, focusing on the stability and decay of perturbations near the steady solution . Due to the lack of velocity dissipation, the above problems are highly challenging. In fact, without coupling, the corresponding N-dimensional Euler equation near u=0 is well known to be unstable. To overcome this difficulty, we analyze the wave structure arising in the system governing perturbations around the steady state, which originates from the equilibrium configuration and the coupling effects. This wave structure enables us to establish the global stability in the -type Sobolev norms. Also, we highlight the critical role of wave structure in the decay estimates of the Euler-FENE dumbbell model. By combining this property with the Fourier splitting method, we derive the decay rate, which is identical to that of the general FENE dumbbell with velocity dissipation.

Paper Structure

This paper contains 9 sections, 4 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Main difficulties and strategies
  3. Global existence and stability
  4. Decay
  5. Preliminaries
  6. Notations
  7. inequalities in $R$-variable
  8. Global Existence
  9. Decay estimates

Key Result

Theorem 1.1

Suppose that $k>1$, $s >\frac{N}{2}+1$, there exists a small constant $\varepsilon>0$ such that for $u_0 \in H^s(\mathbb{R}^N)$, $\text{div}\, u_0 = 0$ and $\psi_0 \in H^s(\mathbb{R}^N; \mathcal{L}^2)$, if then (fene) has a unique global solution $(u,\psi)$. In addition, for all $t > 0$, the following estimates hold:

Theorems & Definitions (8)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Lemma 2.1: masmoudi-cpam
  • Lemma 2.2: masmoudi-cpam
  • proof : Proof of Theorem \ref{['thm-exist']}
  • proof : Proof of Theorem \ref{['thm-decay']}