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

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.

💡 Research Summary

The paper introduces a novel two‑parameter hyperbolic relaxation system (equations (1.2)) that combines the artificial compressibility (AC) method with a first‑order relaxation for the stress‑like tensor U. The parameters ε > 0 and δ > 0 control, respectively, the relaxation of the divergence constraint and the relaxation of the tensor U. Formally, as (ε, δ)→(0, 0) the system recovers the incompressible Navier‑Stokes equations (1.1).

The authors address two distinct regimes of initial data.

1. Vanishing‑amplitude perturbations: When the initial velocity perturbation satisfies ‖u₍ε,δ₎⁰−u⁰‖{L∞}=O((ε+δ)^α) for some α>0, they prove uniform local existence of the relaxed solution on a time interval independent of (ε, δ) and obtain convergence of the velocity in H¹ with rate O(ε+δ) (Theorem 2.1). By employing standard Sobolev interpolation inequalities (Lemma 2.2) they control the L∞‑norm and derive a logarithmic lower bound for the lifespan T{ε,δ} (formula (2.3)).

   The pressure convergence is more delicate. Direct energy estimates only give an O(1) bound for ‖p_{ε,δ}−p‖_{L²}. To obtain H¹‑convergence, the authors construct an intermediate affine system that bridges (1.2) and (1.1). This auxiliary system allows simultaneous error control for both velocity and pressure. Under the additional scaling δ≤C₀√ε and refined H¹‑regularity assumptions on the initial data (see (2.9)–(2.11)), they prove that
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