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

Technical Summary: Convergence of a Two-Parameter Hyperbolic Relaxation System Toward the Incompressible Navier-Stokes Equations

Problem Statement
The paper investigates the rigorous convergence of a two-parameter hyperbolic relaxation system (1.2) toward the incompressible Navier-Stokes equations (1.1) in both two-dimensional ($T^2$) and three-dimensional ($T^3$) periodic domains. The approximation system (1.2) integrates two classical paradigms: the artificial compressibility (AC) method, which relaxes the divergence-free constraint via a time-evolution equation for pressure, and a first-order relaxation model for a stress-like tensor $U$. The system is governed by small parameters $\epsilon$ and $\delta$, representing the relaxation times for pressure and the stress tensor, respectively.

The primary mathematical challenge addressed is the simultaneous limit $(\epsilon, \delta) \to 0$. While single-parameter limits (either $\epsilon \to 0$ for AC or $\delta \to 0$ for relaxation models) are well-documented, the simultaneous limit presents difficulties in controlling pressure errors and handling initial data perturbations. Specifically, the authors aim to establish convergence for two distinct regimes of initial velocity perturbations:

1. Vanishing perturbations: Where the initial difference $\|u_{\epsilon,\delta}^0 - u^0\|$ is of order $O((\epsilon+\delta)^\alpha)$.
2. Order-one ($O(1)$) perturbations: Where the initial difference is $O(1)$, a regime crucial for numerical robustness but traditionally difficult for energy-based methods due to the limited dissipation in the relaxation system.

Methodology
The authors employ distinct analytical strategies depending on the magnitude of the initial perturbations and the dimensionality of the domain.

• Intermediate Affine System (Vanishing Perturbations): To derive convergence rates for the pressure in the vanishing perturbation regime, the authors construct an intermediate affine system. This system acts as a bridge between the original relaxation system (1.2) and the target Navier-Stokes system (1.1). By bounding the discrepancies between the intermediate system and both the limit and approximation systems, the authors obtain necessary error estimates for the pressure, overcoming the limitation where direct energy methods typically yield only $O(1)$ bounds for pressure errors.

• Modulated Energy and Bootstrap Arguments ($O(1)$ Perturbations): For the $O(1)$ perturbation regime, standard symmetric hyperbolic energy estimates are insufficient because the dissipation provided by the damping of the tensor $U_{\epsilon,\delta}$ is not strong enough to suppress nonlinear effects over long time intervals. Instead, the authors utilize a modulated energy functional (introduced in prior work [4]) involving the vorticity $\omega_{\epsilon,\delta} = \text{curl } u_{\epsilon,\delta}$.

  • This approach leverages the dissipation of $\nabla \omega_{\epsilon,\delta}$ to control the velocity field.
  • A bootstrap argument is implemented to close the estimates. This requires specific scaling relations between parameters (specifically $\epsilon = o(\delta)$) and constraints on the initial data where the initial divergence of velocity is sufficiently small relative to the initial curl.
  • Initial Layer Correction: To handle the stress-like tensor $U_{\epsilon,\delta}$ without requiring it to be $o(1)$ close to the limit initial data, the authors introduce initial layer corrections (exponential decay terms) in the energy estimates.

• Dimensional Distinctions:

  • In 3D ($T^3$), the vorticity is a vector field. The analysis identifies two distinct "extreme cases" for convergence rates based on the order of the $L^2$-norm of the initial velocity perturbation ($O(\delta^{3/4})$ vs. $O(\delta^{1/4})$), leading to Theorems 2.6 and 2.7.
  • In 2D ($T^2$), the vorticity is a scalar. The analysis reveals only one viable construction for the convergence order of initial perturbations (Theorem 2.8).

Key Results
The paper establishes the following rigorous convergence results:

1. Vanishing Perturbation Regime (Theorems 2.1, 2.5, Corollary 2.3):

   • For sufficiently small initial perturbations, the solution to the relaxation system exists on the same time interval as the smooth Navier-Stokes solution (or a logarithmically extended interval if the Navier-Stokes solution is global).
   • Convergence of velocity $u_{\epsilon,\delta} \to u$ is proven in $L^\infty([0,T]; H^1)$ and $L^\infty([0,T]; L^2)$.
   • Crucially, under stricter conditions (specifically $\delta = o(\sqrt{\epsilon})$), the pressure $p_{\epsilon,\delta}$ is shown to converge to $p$ in $L^\infty([0,T]; H^1)$.

2. Order-One Perturbation Regime (Theorems 2.6, 2.7, 2.8):

   • The authors prove global-in-time (or up to the lifespan of the limit solution) velocity convergence even when initial velocity perturbations are $O(1)$.
   • 3D Results: Two theorems provide convergence bounds for velocity in $L^2$ and vorticity in $H^1$ under different scaling assumptions for the initial data. The bounds depend on $\delta$ and the ratio $\epsilon/\delta$.
   • 2D Result: A similar global velocity convergence is established for the 2D case with a specific scaling of initial perturbations.
   • In all $O(1)$ cases, the divergence of the velocity field $\text{div } u_{\epsilon,\delta}$ is shown to remain small (controlled by $\epsilon$ and $\delta$), ensuring the solution stays close to the incompressible manifold.

Significance and Claims
The paper claims to provide a foundational theoretical validation for the two-parameter hyperbolic relaxation model (1.2) in numerical simulations. Its specific contributions include:

• Simultaneous Two-Parameter Limit: It addresses the gap in existing literature regarding the simultaneous limit $(\epsilon, \delta) \to 0$, a scenario less explored than single-parameter limits.
• Pressure Convergence: By introducing the intermediate affine system, the work rigorously derives error estimates for the pressure in the vanishing perturbation regime, a step often missing in previous relaxation model analyses.
• Robustness to Large Perturbations: The extension of velocity convergence to the $O(1)$ perturbation regime demonstrates the model's robustness against significant initial inconsistencies, which are common in numerical start-ups. This is achieved by overcoming the limitations of standard energy estimates through the modulated energy method and bootstrap arguments.
• Structural Insights: The analysis highlights the structural richness of the 3D case compared to the 2D case, identifying distinct convergence rate regimes in three dimensions.

The authors note that while the velocity convergence is established for $O(1)$ perturbations, the convergence of pressure in this specific regime remains an open challenge for future research. The work is intended to support the application of this model in downstream tasks, such as the statistics of turbulent flows, by ensuring the mathematical validity of the approximation.