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Diffusion wave phenomena and optimal time decay for incompressible viscoelastic flows

Shenghan Li, Yong Wang

TL;DR

This work analyzes diffusion-wave phenomena in three-dimensional incompressible viscoelastic flows near the rest state. It blends a representation-formula approach for the linearized wave equation with stationary-phase techniques and a robust energy framework to derive $L^p$ decay estimates for all $p\ge1$, revealing a hyperbolic-diffusive balance induced by the elastic stress. The results show diffusion waves dominate for $1\le p<2$ while faster decay occurs for $p\ge2$, and they provide sharp decay rates for higher derivatives, along with global well-posedness for small data via negative Sobolev/Besov methods. Overall, the paper advances understanding of the parabolic-hyperbolic coupling in incompressible viscoelastic flows and delivers precise time-decay profiles with potential implications for long-time dynamics and stability analysis.

Abstract

Motivated by the work of D. Hoff and K. Zumbrun (Indiana Univ. Math. J. 44: 603-676, 1995), we investigate the diffusion wave phenomena in three-dimensional incompressible viscoelastic flows. By employing the representation formula of the wave equation and the stationary phase methods on the sphere $\mathbb{S}^{d-1}$, we establish $L^p$ decay estimates for the solution over the whole range $1\leq p \leq \infty$, which reveals the hyperbolic nature of the incompressible viscoelastic flows.

Diffusion wave phenomena and optimal time decay for incompressible viscoelastic flows

TL;DR

This work analyzes diffusion-wave phenomena in three-dimensional incompressible viscoelastic flows near the rest state. It blends a representation-formula approach for the linearized wave equation with stationary-phase techniques and a robust energy framework to derive decay estimates for all , revealing a hyperbolic-diffusive balance induced by the elastic stress. The results show diffusion waves dominate for while faster decay occurs for , and they provide sharp decay rates for higher derivatives, along with global well-posedness for small data via negative Sobolev/Besov methods. Overall, the paper advances understanding of the parabolic-hyperbolic coupling in incompressible viscoelastic flows and delivers precise time-decay profiles with potential implications for long-time dynamics and stability analysis.

Abstract

Motivated by the work of D. Hoff and K. Zumbrun (Indiana Univ. Math. J. 44: 603-676, 1995), we investigate the diffusion wave phenomena in three-dimensional incompressible viscoelastic flows. By employing the representation formula of the wave equation and the stationary phase methods on the sphere , we establish decay estimates for the solution over the whole range , which reveals the hyperbolic nature of the incompressible viscoelastic flows.
Paper Structure (8 sections, 41 theorems, 265 equations)

This paper contains 8 sections, 41 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['diffusion waves-infty-202512201451']}
  3. Proof of Theorem \ref{['diffusion waves-1-202512201453']}
  4. Energy Estimates
  5. Negative Sobolev/Besov estimates
  6. Proof of Theorem \ref{['pure-energy-202512221702']} \ref{['202512221715-globalsolution-incompressible viscoelastic flows']}-\ref{['pure-energy-202512221703']}
  7. Proof of Theorem \ref{['pure-energy-202512221702']} \ref{['N-pure-energy-202512221703']}
  8. Auxiliary lemmas

Key Result

Proposition 1.1

(Hu-Wu2015) Suppose that $n=3$ and the initial data $u_0, \mathbb{E}_0 \in L^1\left(\mathbb{R}^3\right) \cap H^k\left(\mathbb{R}^3\right)$ ( $k \geq 2$ being an integer) fulfill the assumptions important properties1-202512201228-important properties3-202512201228. If the initial data satisfy $\left\ for all integer $j$ and multi-index $\alpha$ satisfying $2 j+|\alpha| \leq k$. For all $t \geq 0$,

Theorems & Definitions (72)

  • Proposition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.4
  • Remark 1.5
  • ...and 62 more