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Convergence of a two-parameter hyperbolic relaxation system toward the incompressible Navier-Stokes equations

Qian Huang, Christian Rohde, Ruixi Zhang

Abstract

We investigate a two-parameter hyperbolic relaxation approximation to the incompressible Navier-Stokes equations, incorporating a first-order relaxation and the artificial compressibility method. With vanishingly small perturbations of initial velocity, we rigorously prove the simultaneous convergence of fluid velocity and pressure toward the Navier-Stokes limit in the three-dimensional case by constructing an intermediate affine system to obtain the necessary error estimates for the pressure. Furthermore, we extend the velocity convergence analysis to the case of $\mathcal O(1)$ initial velocity perturbations, and establish the global-in-time recovery of the velocity field using a modulated energy structure and delicate bootstrap arguments in both two- and three-dimensional settings.

Convergence of a two-parameter hyperbolic relaxation system toward the incompressible Navier-Stokes equations

Abstract

We investigate a two-parameter hyperbolic relaxation approximation to the incompressible Navier-Stokes equations, incorporating a first-order relaxation and the artificial compressibility method. With vanishingly small perturbations of initial velocity, we rigorously prove the simultaneous convergence of fluid velocity and pressure toward the Navier-Stokes limit in the three-dimensional case by constructing an intermediate affine system to obtain the necessary error estimates for the pressure. Furthermore, we extend the velocity convergence analysis to the case of initial velocity perturbations, and establish the global-in-time recovery of the velocity field using a modulated energy structure and delicate bootstrap arguments in both two- and three-dimensional settings.
Paper Structure (14 sections, 9 theorems, 202 equations)

This paper contains 14 sections, 9 theorems, 202 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Vanishing amplitude of initial velocity perturbations on $\mathbb T^3$
  4. Order-one initial velocity perturbations
  5. Preliminaries
  6. Vanishing amplitude of initial velocity perturbations on $\mathbb T^3$
  7. Proof of Theorem \ref{['thm:uconverg_H1']}
  8. Proof of Corollary \ref{['cor:convergence']}
  9. Proof of Theorem \ref{['thm:pressure_3d']}
  10. Order-one initial velocity perturbations
  11. Proof of Theorem \ref{['thm:critical_3d']} on $\mathbb T^3$
  12. Proof of Theorem \ref{['thm:critical_3d_inhomo']} on $\mathbb T^3$
  13. Proof of Theorem \ref{['thm:critical_2d']} on $\mathbb T^2$
  14. Conclusions

Key Result

Theorem 2.1

Assume that there exist constants $a \in [0,2)$ and $C_0>0$, both independent of $\epsilon$ and $\delta$, such that Then, if $T_0<\infty$, the solution $(u^{\epsilon,\delta},p^{\epsilon,\delta},U^{\epsilon,\delta})$ of the initial value problem for eq:relax exists on $[0,T_0]$ for sufficiently small $\epsilon$ and $\delta$. Otherwise, if $T_0=\infty$, the existing time $T_{\epsilon,\delta}$ is un

Theorems & Definitions (15)

  • Theorem 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Lemma 2.9: Lemma 2.8 of HRYZ25
  • Proposition 4.1
  • ...and 5 more