Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal time-decay estimates for an Oldroyd-B model with zero viscosity

Jinrui Huang, Yinghui Wang, Huanyao Wen, Ruizhao Zi

Abstract

In this work, we consider the Cauchy problem for a diffusive Oldroyd-B model in three dimensions. Some optimal time-decay rates of the solutions are derived via analysis of upper and lower time-decay estimates provided that the initial data are small and that the absolute value of Fourier transform of the initial velocity is bounded below away from zero in a low-frequency region. It is worth noticing that the optimal rates are independent of the fluid viscosity or the diffusive coefficient, which is a different phenomenon from that for incompressible Navier-Stokes equations.

Optimal time-decay estimates for an Oldroyd-B model with zero viscosity

Abstract

In this work, we consider the Cauchy problem for a diffusive Oldroyd-B model in three dimensions. Some optimal time-decay rates of the solutions are derived via analysis of upper and lower time-decay estimates provided that the initial data are small and that the absolute value of Fourier transform of the initial velocity is bounded below away from zero in a low-frequency region. It is worth noticing that the optimal rates are independent of the fluid viscosity or the diffusive coefficient, which is a different phenomenon from that for incompressible Navier-Stokes equations.

Paper Structure

This paper contains 15 sections, 14 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Ingredients of the proof
  4. Analysis of a linearized system
  5. Fourier transform of the Green matrix
  6. The low-frequency part
  7. Global existence and uniqueness
  8. A priori estimate on the basic energy
  9. Supplement of dissipation
  10. Total energy
  11. Decay estimates for the nonlinear system
  12. Upper time-decay estimates
  13. Lower time-decay estimates
  14. Appendix
  15. More time-decay estimates for vanishing center-of-mass diffusion

Key Result

Theorem 1.1

(Global existence) Assume that $(u_0,\tau_0)\in H^3(\mathbb{R}^3)$. For any given $\epsilon$ and $\mu$ satisfying the Case I or Case II, then there exists a sufficiently small constant $\varepsilon_0>0$ such that the Cauchy problem (system) admits a unique global solution $(u^{\epsilon,\mu},\tau^{\e for $t\geq0$, provided that where $\varepsilon_0$ is a constant depending on $\mu$ (or $\epsilon$)

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • ...and 19 more