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<p>The Oldroyd-B equations model viscoelastic fluids. Attention here is focused on a three-dimensional (3D) Oldroyd-B system with mixed dissipation, horizontal velocity dissipation and vertical diffusion for the non-Newtonian stress tensor <i>τ</i>. The equation of <i>τ</i> has no damping, a setup relevant to high Weissenberg viscoelastic flows. In this paper, we solve the small-data global well-posedness and the stability problem in the Sobolev space <i>H</i><sup>2</sup>(ℝ<sup>3</sup>). The lack of the horizontal dissipation or damping in the equation of <i>τ</i> makes the problem almost impossible. This paper discovers that the coupling and interaction of the fluid velocity <i>u</i> and <i>τ</i> generates extra smoothing and stabilization. Mathematically, <i>u</i> and ℙ∇· <i>τ</i> satisfy a system of wave equations, which provides the desired enhanced dissipation. Here, ℙ = <i>I</i> − ∇∆<sup>−1</sup>∇· denotes the projection onto divergence-free vector fields. In addition, time-weighted energy functionals are introduced to control low-regularity terms. The second major result of this paper establishes optimal decay rates by making use of the aforementioned enhanced dissipation and the integral representation.</p>

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Stability on a 3D incompressible Oldroyd-B model with mixed partial dissipation

  • 期刊论文

  • 发表日期：

    2025年1月21日

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  • Hongxia Lin,
  • Jiahong Wu,
  • Nicki Boardman

摘要

The Oldroyd-B equations model viscoelastic fluids. Attention here is focused on a three-dimensional (3D) Oldroyd-B system with mixed dissipation, horizontal velocity dissipation and vertical diffusion for the non-Newtonian stress tensor τ. The equation of τ has no damping, a setup relevant to high Weissenberg viscoelastic flows. In this paper, we solve the small-data global well-posedness and the stability problem in the Sobolev space H2(ℝ3). The lack of the horizontal dissipation or damping in the equation of τ makes the problem almost impossible. This paper discovers that the coupling and interaction of the fluid velocity u and τ generates extra smoothing and stabilization. Mathematically, u and ℙ∇· τ satisfy a system of wave equations, which provides the desired enhanced dissipation. Here, ℙ = I − ∇∆−1∇· denotes the projection onto divergence-free vector fields. In addition, time-weighted energy functionals are introduced to control low-regularity terms. The second major result of this paper establishes optimal decay rates by making use of the aforementioned enhanced dissipation and the integral representation.