Asymptotic Equivalence and Decay Characterization in Second-Grade Fluid Equations

TL;DR

This work analyzes the long-time behavior of solutions to the second-grade fluids equations on $\mathbb{R}^3$, focusing on decay rates in the natural $H^1_\alpha$ norm and the relationship between nonlinear dynamics and the linearized, pseudo-parabolic problem. By employing a decay-character framework and a Fourier-splitting approach, the authors establish decay rates in terms of the initial data's low-frequency behavior and quantify how the nonlinear solution converges to its linear counterpart for sufficiently regular, small data. They prove a sharp upper decay bound for the nonlinear solution, derive an asymptotic equivalence between the nonlinear solution and the linear solution, and obtain lower bounds for decay in a restricted range of decay-characters, thereby clarifying the distinct roles of linear and nonlinear dissipation. The results provide a rigorous, quantitative description of the time-asymptotic behavior of non-Newtonian second-grade fluids and offer insights into how initial regularity and spectral properties govern long-time dynamics.

Abstract

We establish new asymptotic results for the solutions of the second-grade fluids equations and characterize their decay rate in terms of the behavior of the initial data. Moreover, assuming more regularity for the initial data, we study the large-time behavior of these solutions by comparing them to the solutions of the linearized equations. As a consequence, we obtain lower bounds for the solutions. Other auxiliary results of interest are discussed.

Theorems & Definitions (7)

• proof
• proof
• proof
• proof : Proof of Theorem \ref{['Main1-Intro']}
• proof
• proof : Proof of Theorem \ref{['Main2-intro']}
• proof : Proof of Corollary \ref{['lower-bound']}