The Nonlocal-to-Local Limit for the Inviscid Leray-α Equations

Technical Summary: The Nonlocal-to-Local Limit for the Inviscid Leray-$\alpha$ Equations

Problem Statement
This paper investigates the inviscid Leray-$\alpha$ equations, a nonlocal regularization of the incompressible Euler equations. The system is defined on a domain $\Omega \subseteq \mathbb{R}^d$ ($d=2,3$) with a length scale parameter $\alpha > 0$. The equations relate the filtered velocity $u_\alpha$ to the physical velocity $v_\alpha$ via the Helmholtz operator relation $v_\alpha = u_\alpha - \alpha^2 \Delta u_\alpha$ (or more generally $u_\alpha = K_\alpha * v_\alpha$), while the momentum equation is $\partial_t v_\alpha + (u_\alpha \cdot \nabla)v_\alpha + \nabla p_\alpha = 0$.

The primary objective is to establish the convergence of solutions of the Leray-$\alpha$ equations to solutions of the Euler equations as the regularization parameter $\alpha \to 0$. The authors address two distinct regimes:

1. Strong Solutions: Convergence in high-regularity Sobolev spaces ($H^s$ with $s > d/2 + 1$) on the whole space $\mathbb{R}^d$.
2. Weak Solutions: Convergence in $L^2$ on bounded domains, assuming initial data in $L^2$ and a specific local scaling property related to turbulence theory, without assuming higher regularity.

Methodology

Part 1: Strong Solutions and General Kernels
The authors adapt techniques from Masmoudi's work on the inviscid limit of Navier-Stokes equations and Linshiz and Titi's work on the Euler-$\alpha$ model. A key methodological distinction is that the proof for the Leray-$\alpha$ model is conducted without utilizing the vorticity structure of the system, unlike the Euler-$\alpha$ proofs which rely heavily on the vorticity equation.

The proof strategy involves a three-part decomposition to estimate the difference between the Leray-$\alpha$ solution $v_\alpha$ and the Euler solution $v$:

1. Regularization of Initial Data: The initial data $v_0$ and $v_\alpha^0$ are smoothed using a cutoff in Fourier space (parameter $\delta$) to create $v_0^\delta$ and $v_\alpha^{0,\delta}$.
2. Triangle Inequality: The error $\|v_\alpha - v\|$ is split into three terms:
   • $\|v_\alpha - v_\alpha^\delta\|$: The difference between the Leray-$\alpha$ solution with original data and the Leray-$\alpha$ solution with smoothed data.
   • $\|v_\alpha^\delta - v^\delta\|$: The difference between the smoothed Leray-$\alpha$ solution and the smoothed Euler solution.
   • $\|v^\delta - v\|$: The difference between the smoothed Euler solution and the original Euler solution.
3. Kernel Generalization: The authors maintain a general kernel $K_\alpha$ throughout the derivation, only specifying the properties required for convergence (boundedness in $H^l$ and convergence to the identity). The specific Helmholtz kernel is only applied at the final step to derive explicit convergence rates.

Part 2: Weak Solutions and Local Scaling
For the weak solution regime, the authors assume initial data in $L^2$ and impose a local scaling property (Equation 12) far away from the boundary. This condition bounds the second-order structure function:
$$\sup_{\alpha>0} \int_0^T \int_K |v_\alpha(x+y, t) - v_\alpha(x, t)|^2 dx dt \leq E_K |y|^{2\gamma}$$
for $|y| \geq \eta(\alpha)$. This assumption is motivated by Kolmogorov's 1941 turbulence theory and the inertial range, but crucially, it does not imply classical regularity.

The convergence proof follows the approach of [19] and [20] for the vanishing viscosity limit. It utilizes a mollification technique with a test function $j(z)$ to decompose the nonlinear term $(u_\alpha \otimes v_\alpha)$. The proof demonstrates that the difference between the nonlinear terms of the Leray-$\alpha$ and Euler equations vanishes by splitting the error into three components (A, B, C) involving mollified fields and utilizing the weak convergence of $v_\alpha$ to $v_\infty$ and the strong convergence of mollified fields.

Key Contributions and Results

1. Convergence of Strong Solutions (Theorem 1):

   • The authors prove that for initial data in $H^s(\mathbb{R}^d)$ ($s > d/2 + 1$), the strong solutions of the Leray-$\alpha$ equations converge to the strong solution of the Euler equations in $L^\infty(0, T; H^s)$ as $\alpha \to 0$.
   • Explicit Rates: They derive convergence rates for lower norms $H^{s'}$ ($0 \leq s' \leq s-1$). The rate depends on the initial data error and a term proportional to $\alpha^\iota t$, where $\iota$ depends on the gap between $s$ and $s'$.
   • General Kernels (Corollary 1): The results are extended to a broad class of regularizing kernels $K_\alpha$, provided they satisfy specific boundedness and approximation properties.

2. Convergence of Weak Solutions (Theorem 2):

   • On a bounded domain, if a sequence of weak solutions satisfies the local scaling property (12) and converges weakly in $L^2$, the limit is a weak solution to the Euler equations.
   • The limit solution $v_\infty$ inherits the scaling property.
   • This result accommodates "wild" weak solutions that may exhibit anomalous energy dissipation (consistent with Kolmogorov's Zero-th Law), distinguishing them from strong solutions which conserve energy.

3. Methodological Distinction:

   • The paper explicitly notes that the proof for Leray-$\alpha$ does not rely on the vorticity formulation, which is a significant departure from the literature on Euler-$\alpha$ models. This allows the investigation of convergence for a more general class of kernels.

Significance and Claims

The authors position this work as addressing fundamental questions regarding the inviscid Leray-$\alpha$ model, which has received comparatively little attention in the literature compared to the Euler-$\alpha$ model or viscous Leray-$\alpha$ models.

• Nonlocal-to-Local Limit: The paper establishes the rigorous mathematical link between the nonlocal Leray-$\alpha$ regularization and the local Euler equations, confirming the model's validity as a regularization scheme.
• Comparison with Other Limits: The authors note that the convergence rates regarding the order of $\alpha$ are comparable to those found in the inviscid limit of Navier-Stokes (Masmoudi [40]) and the Euler-$\alpha$ limit (Linshiz and Titi [34]).
• Turbulence Connection: By utilizing the local scaling property in the weak solution analysis, the paper connects the mathematical convergence to physical turbulence theories (Kolmogorov's structure functions), allowing for the convergence of solutions that may not be smooth.
• Boundary Conditions: The work clarifies the handling of boundary conditions, noting that the scaling assumption is applied far from the boundary to avoid complications with boundary layers, a known difficulty in inviscid limits.

The paper concludes that the inviscid Leray-$\alpha$ equations provide a robust nonlocal regularization of the Euler equations, with convergence properties that hold for both smooth and turbulent (weak) regimes under appropriate assumptions.