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On the hydrostatic approximation of 3D Oldroyd-B model

Marius Paicu, Tianyuan Yu, Ning Zhu

TL;DR

This work analyzes the hydrostatic approximation for the 3D Oldroyd-B model in a thin strip, with $\operatorname{Re}=\varepsilon^{-2}$ and $\operatorname{We}=\varepsilon$, establishing global well-posedness of the hydrostatic limit under small analytic-in-$x$ data. It then rigorously justifies the hydrostatic limit from the rescaled Oldroyd-B system to the hydrostatic system, providing an explicit convergence rate by a detailed weighted-energy analysis in anisotropic Sobolev spaces and using phase-based analytic control. The authors derive and bound the velocity and stress-field dynamics, including a nonlinear-terms analysis via para-linearization, to close the estimates. The results supply a rigorous foundation for hydrostatic regimes in viscoelastic fluids and offer a methodological blueprint for singular-limit problems in complex rheological models.

Abstract

In this paper, we study the hydrostatic approximation for the 3D Oldroyd-B model. Firstly, we derive the hydrostatic approximate system for this model and prove the global well-posedness of the limit system with small analytic initial data in horizontal variable. Then we justify the hydrostatic limit strictly from the re-scaled Oldroyd-B model to the hydrostatic Oldroyd-B model and obtain the precise convergence rate.

On the hydrostatic approximation of 3D Oldroyd-B model

TL;DR

This work analyzes the hydrostatic approximation for the 3D Oldroyd-B model in a thin strip, with and , establishing global well-posedness of the hydrostatic limit under small analytic-in- data. It then rigorously justifies the hydrostatic limit from the rescaled Oldroyd-B system to the hydrostatic system, providing an explicit convergence rate by a detailed weighted-energy analysis in anisotropic Sobolev spaces and using phase-based analytic control. The authors derive and bound the velocity and stress-field dynamics, including a nonlinear-terms analysis via para-linearization, to close the estimates. The results supply a rigorous foundation for hydrostatic regimes in viscoelastic fluids and offer a methodological blueprint for singular-limit problems in complex rheological models.

Abstract

In this paper, we study the hydrostatic approximation for the 3D Oldroyd-B model. Firstly, we derive the hydrostatic approximate system for this model and prove the global well-posedness of the limit system with small analytic initial data in horizontal variable. Then we justify the hydrostatic limit strictly from the re-scaled Oldroyd-B model to the hydrostatic Oldroyd-B model and obtain the precise convergence rate.

Paper Structure

This paper contains 10 sections, 8 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Global well-posedness of the equations \ref{['Oldroyd-B-Prandtl']}: Proof of Theorem \ref{['th1.1']}
  3. Proof of Theorem \ref{['th1.1']}: Estimate on $u$
  4. Proof of Theorem \ref{['th1.1']}: Estimate on $\partial_yu$
  5. Completing the proof of Theorem \ref{['th1.1']}
  6. Proof of Lemma \ref{['nonlinear']}
  7. Justification for the hydrostatic limit: Proof of Theorem \ref{['th1.2']}
  8. Proof of Lemmas \ref{['time-tau']}--\ref{['K1+cdots+K16']}
  9. Derivation of \ref{['tau13+tau23']}
  10. Proof of some technical lemmas

Key Result

Theorem 1.1

Let $a>0$ and $s_1>\frac{3}{2},s_2>\frac{1}{2}$. Then there exists a constant $c_1>0$ sufficiently small such that the following conclusion holds. If the initial data $u_0$ in Oldroyd-B-Prandtl satisfies and the compatibility condition then the hydrostatic Oldroyd-B equations Oldroyd-B-Prandtl admit a unique global-in-time solution $u$ satisfying that where $e^{\frac{a}{2} \langle D_x \rangle}$

Theorems & Definitions (15)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof : Proof of Lemma \ref{['time-tau']}
  • proof : Proof of Lemma \ref{['K1+cdots+K16']}
  • Lemma B.1
  • ...and 5 more