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2D Navier-Stokes with viscosity-dependent Navier slip: strong convergence for unbounded initial velocity

This paper establishes the strong vorticity convergence of two-dimensional incompressible Navier-Stokes solutions to Euler solutions in the vanishing viscosity limit for initial data in $L^p$ ( $p>1$ ) with viscosity-dependent Navier slip, proving that the boundary-generated vorticity vanishes under specific growth conditions on the friction coefficients and that kinetic energy is conserved in the limit.

Original authors: Josef Demmel

Published 2026-11-04

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CC BY 4.0

Original authors: Josef Demmel

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Fluids are everywhere, from the air we breathe to the water in our rivers, and understanding how they move is one of the oldest challenges in physics. When a fluid flows, it can do so in two very different ways. Sometimes it moves smoothly and predictably, like a calm river; other times it becomes chaotic and turbulent, churning with swirling eddies. Scientists describe this motion using two main sets of rules. The first set, known as the Euler equations, describes an ideal fluid that has no internal friction, or viscosity. In this perfect world, the fluid slides effortlessly past itself and any surface it touches. The second set, the Navier-Stokes equations, describes real fluids that do have friction. This internal friction acts like a thick syrup, resisting motion and causing the fluid to slow down, especially when it rubs against a solid wall.

For decades, a central question has been how these two descriptions connect. If you take a real fluid with a tiny amount of friction and gradually reduce that friction until it disappears, does the fluid's behavior smoothly transform into the ideal, frictionless motion described by the Euler equations? In open spaces without walls, the answer is often yes. But when a fluid is confined inside a container, the problem becomes much harder. The walls introduce a new complication: the way the fluid sticks or slips against the surface. In the real world, fluids usually slip a little bit along a wall rather than stopping dead, a behavior known as Navier slip. As the fluid's internal friction vanishes, the way it interacts with the wall can change in complex ways, potentially creating a thin, chaotic layer of swirling motion right at the boundary. This boundary layer is the main obstacle to proving that the real fluid smoothly becomes the ideal one.

In a recent study, mathematician Josef Demmell tackled this difficult problem for fluids moving in two dimensions, such as a thin layer of water or air. The research focuses on a specific type of fluid motion where the friction coefficient at the wall is not fixed but can change as the fluid's internal viscosity decreases. In many physical situations, as a fluid becomes less viscous, the effective distance it slips along the wall—the slip length—can shrink, making the wall act more like a sticky surface. Demmell investigated whether the swirling motion, or vorticity, of the fluid would still converge to the ideal pattern even if the wall's behavior changed in this way. The study proves that it does, but only under specific conditions regarding how fast the wall's friction can grow as the fluid becomes thinner.

The core of the work involves splitting the fluid's swirling motion into two distinct parts. The first part comes from the initial swirls the fluid had when it started moving. The second part is generated entirely by the interaction with the boundary wall. In the past, it was known that if the wall's friction stayed constant, the boundary-generated swirls would eventually vanish as the fluid's viscosity disappeared, leaving only the initial swirls to dictate the flow. However, if the wall's friction grows too quickly as the viscosity drops, these boundary swirls could become too strong and disrupt the smooth transition to the ideal flow. Demmell's work establishes a precise mathematical limit for this growth. He showed that as long as the friction at the wall does not increase too rapidly relative to the initial smoothness of the fluid's swirls, the boundary-generated motion will still fade away.

To reach this conclusion, the author developed a new method to track how the fluid moves near the wall. By carefully analyzing the speed of the fluid in the tiny region right next to the boundary, the study demonstrated that the fluid's motion remains controlled enough to prevent the boundary swirls from taking over. This control allowed the researcher to prove that the swirling motion of the real fluid converges strongly to the swirling motion of the ideal fluid. This is a significant improvement over previous results, which could only guarantee that the average behavior matched or required the fluid to be much smoother to begin with. The new proof works for a much wider range of initial conditions, including fluids that start with less smooth, more chaotic swirls.

A crucial finding of the study is that this convergence happens even when the slip length at the wall shrinks to zero, effectively turning the wall into a no-slip surface where the fluid stops completely. This was a regime where convergence was previously uncertain and difficult to prove. The research confirms that even in this extreme case, as long as the friction does not grow too fast, the fluid's internal structure remains stable. The study also showed that the total energy of the fluid is conserved in the final, ideal state. This means that the ideal fluid retains the same amount of kinetic energy it started with, without losing any to the boundary, which is a fundamental property expected of frictionless flow.

The results provide a clearer picture of how real fluids transition into ideal ones, resolving a long-standing question about the role of the boundary. By proving that the boundary effects can be controlled under realistic growth conditions, the work strengthens our understanding of fluid dynamics in confined spaces. It suggests that the complex, chaotic layers that form near walls in real fluids do not necessarily prevent the fluid from behaving like an ideal system in the limit, provided the interaction at the wall does not become too extreme. This insight helps bridge the gap between the messy reality of viscous fluids and the elegant simplicity of ideal fluid theory, offering a more robust foundation for predicting how fluids will behave in engineering and natural systems.

Technical Summary: Strong Convergence of Vorticity for 2D Navier–Stokes with Viscosity-Dependent Navier Slip

Problem Statement
This paper addresses the vanishing viscosity limit ( $\nu \to 0$ ) for the two-dimensional incompressible Navier–Stokes equations on a smooth, bounded, simply connected domain $\Omega \subset \mathbb{R}^2$ . The system is subject to Navier (friction) boundary conditions, where the friction coefficient $\alpha_\nu$ is allowed to depend on the viscosity $\nu$ . Specifically, the study investigates the regime where $\alpha_\nu$ may grow as $\nu \to 0$ (corresponding to a vanishing slip length), governed by the growth bound $\|\alpha_\nu\|_{L^\infty(\partial\Omega)} \leq C\nu^{-q}$ for $q \geq 0$ .

The primary objective is to establish strong convergence of the vorticity $\omega_\nu$ to the vorticity of a weak solution of the Euler equations, given initial vorticity $\omega_0 \in L^p(\Omega)$ with $p > 1$ . This extends previous results which were restricted to viscosity-independent friction or initial data with higher integrability ( $p > 2$ ). A central challenge in this problem is controlling the vorticity generated at the boundary, which does not vanish automatically in the inviscid limit due to the mismatch between Navier–Stokes and Euler boundary conditions.

Methodology
The proof strategy relies on a decomposition of the vorticity $\omega_\nu$ into two distinct components:

Initial Contribution ( $a_\nu$ ): Solves the advection-diffusion equation with homogeneous Dirichlet boundary conditions ( $a_\nu|_{\partial\Omega} = 0$ ) and initial data $\omega_0$ .
Boundary Contribution ( $b_\nu$ ): Solves the same equation with zero initial data but non-homogeneous boundary conditions driven by the friction term: $b_\nu|_{\partial\Omega} = (2\kappa - \alpha_\nu)u_\nu \cdot \tau$ .

The core technical innovation involves establishing uniform bounds on the boundary contribution $b_\nu$ in the vanishing viscosity limit. The author employs the following steps:

Renormalization and Drift Control: A key lemma (Lemma 4.1) combines renormalization techniques with estimates on the inward normal component of the velocity drift near the boundary. By assuming a bound of the form $|u_\nu \cdot \nabla d| \leq B(s)d(x)^\gamma$ (where $d(x)$ is the distance to the boundary), the author controls the diffusive boundary flux in the renormalization identity.
Parabolic Smoothing and Biot–Savart Estimates: To satisfy the drift condition, the author derives parabolic smoothing estimates for the vorticity (Lemma 5.2) that are sensitive to the scaling of the friction coefficient. These estimates, combined with the Biot–Savart law and Morrey's inequality, yield Hölder estimates for the velocity near the boundary. This allows the author to bound the normal drift $u_\nu \cdot \nabla d$ effectively, even when $\alpha_\nu$ grows.
Uniform Vorticity Bounds: By combining the drift estimates with the boundary condition growth rate, the author derives a uniform bound for the boundary contribution $b_\nu$ in $L^\rho(\Omega)$ (Proposition 6.1 and Lemma 6.2). The bound vanishes as $\nu \to 0$ provided the growth exponent $q$ satisfies a specific threshold relative to the integrability $p$ .
Compactness and Stability: The uniform bounds allow the application of the Aubin–Lions lemma to extract a strongly convergent subsequence for the velocity. The stability of the homogeneous problem ensures that the initial contribution $a_\nu$ converges strongly to the limiting Euler vorticity.

Key Contributions and Results

Extension to $p \in (1, \infty)$ : The paper removes the restriction $p > 2$ found in prior works (e.g., Lopes Filho et al. [12]) regarding the integrability of initial vorticity. Strong vorticity convergence is established for any $\omega_0 \in L^p(\Omega)$ with $p > 1$ .
Viscosity-Dependent Friction: The author establishes convergence for friction coefficients $\alpha_\nu$ that grow as $\nu \to 0$ . The main result (Theorem 1.1) states that for $\omega_0 \in L^p(\Omega)$ , there exists a sequence $\nu_k \to 0$ such that $\omega_{\nu_k} \to \text{curl } u$ strongly in $C([0, T]; L^\rho(\Omega))$ for $\rho \in (1, p]$ , provided the growth exponent $q$ satisfies:
$\rho q < \min\left\{ \frac{p-1}{2p}, \frac{1}{4} \right\}$
This condition ensures that the boundary contribution $b_\nu$ vanishes in the limit.
Strong Vorticity Convergence: Unlike previous results that focused on velocity convergence, this work establishes the strong convergence of vorticity. This is a significant strengthening, as it implies the limiting Euler solution is a renormalized solution, thereby conserving kinetic energy (Corollary 1.2).
Counterexample and Sharpness: The paper notes that strong vorticity convergence can fail if $q = 1/2$ (Lemma B.1), aligning with the threshold derived in the main theorem. The counterexample utilizes a boundary obstruction similar to Kelliher's analysis of the no-slip problem, demonstrating that strong vorticity convergence implies strong convergence of tangential velocities at the boundary, which fails in the counterexample regime.

Significance and Claims
The paper claims to resolve the open question raised in [12, Remark 1] regarding the extension of strong convergence results to the range $p \in (1, 2]$ . Furthermore, it generalizes these results to viscosity-dependent friction coefficients, a regime motivated by experimental observations where slip length decreases as viscosity vanishes.

The author emphasizes that the restriction on $q$ arises from the specific analytical methods employed (specifically the interplay between the drift bound and the renormalization identity) and is not expected to be optimal. However, the work provides a rigorous framework for handling the boundary layer effects in the inviscid limit under Navier slip conditions with growing friction, ensuring that the limiting Euler solutions conserve kinetic energy and possess strong vorticity convergence.

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