On the absence of anomalous dissipation for the Navier-Stokes equations with Navier boundary conditions: a sufficient condition

1. Problem Statement

The paper addresses the vanishing viscosity limit of the incompressible Navier-Stokes equations (NSE) in a bounded three-dimensional domain $\Omega$ with Navier boundary conditions. The central problem is determining under what conditions the solutions of the NSE converge to a solution of the Euler equations as viscosity $\nu \to 0$, and specifically, whether anomalous energy dissipation occurs in this limit.

• Anomalous Dissipation: In turbulent flows, energy may dissipate even as viscosity vanishes (the Kolmogorov zeroth law). Mathematically, this is defined as the non-vanishing of the total viscous dissipation (both bulk and boundary) in the limit $\nu \to 0$:
  $$\liminf_{\nu \to 0} \left( 2\nu \int_0^T \|D(u_\nu)\|_{L^2}^2 dt + 2\nu \sum_{i,j} \int_0^T \int_{\partial\Omega} \lambda_{ij} (u_\nu)_i (u_\nu)_j dx dt \right) > 0$$
• Context: While the Kato criterion (for no-slip Dirichlet conditions) links the vanishing of dissipation to the convergence of solutions, the case for Navier boundary conditions (which allow slip at the boundary) is less understood, particularly regarding the regularity requirements needed to rule out anomalous dissipation without assuming the existence of strong Euler solutions.

2. Mathematical Framework and Definitions

The Equations

• Navier-Stokes: $\partial_t u_\nu + \nabla \cdot (u_\nu \otimes u_\nu) - \nu \Delta u_\nu + \nabla p_\nu = 0$, with $\nabla \cdot u_\nu = 0$.
• Navier Boundary Conditions:
  1. Impermeability: $u_\nu \cdot n = 0$ on $\partial\Omega$.
  2. Generalized Slip Condition: $(D(u_\nu) \cdot n)_\tau + \lambda(\nu, x) \cdot u_\nu = 0$ on $\partial\Omega$.
     Here, $D(v)$ is the symmetric strain rate tensor, and $\lambda$ is a non-negative semidefinite tensor field. This generalizes standard Robin conditions and includes vorticity-based conditions.

Weak Solutions

• Leray-Hopf Weak Solutions (NSE): Defined via a weak formulation (Definition 1) that incorporates the boundary condition directly into the integral identity, satisfying an energy inequality.
• Weak Solutions (Euler): The authors utilize a specific definition (Definition 4) for weak solutions to the Euler equations where the velocity $u$ is Hölder continuous ($C^{0,\alpha}$). Crucially, this definition relies on a non-standard pressure boundary condition derived in their previous work [4]:
  $$\partial_n(p + (u \cdot n)^2)|_{\partial\Omega} = (u \otimes u) : \nabla n$$
  This differs from the classical condition for strong solutions and is necessary for weak solutions with low regularity.

3. Methodology

The authors employ a combination of compactness arguments, regularity theory, and energy estimates:

1. Regularity Propagation: They rely on recent results (from [4]) showing that if the velocity field of a weak Euler solution is Hölder continuous ($C^{0,\alpha}$), the corresponding pressure is also $C^{0,\alpha}$. This allows them to handle the pressure without assuming higher regularity a priori.
2. Compactness (Aubin-Lions): They assume a sequence of NSE solutions $\{u_{\nu_k}\}$ is uniformly bounded in $L^\infty((0,T); C^{0,\alpha}(\Omega))$. Using the Aubin-Lions lemma, they extract a subsequence converging strongly in $C([0,T]; L^2(\Omega))$ to a limit $u$.
3. Weak Formulation Equivalence: They prove (Proposition 5) that for $C^{0,\alpha}$ velocities, the standard weak formulation of the Euler equations (involving pressure) is equivalent to a formulation using only divergence-free test functions. This allows them to pass to the limit in the NSE weak formulation without explicitly tracking the pressure term during the convergence step.
4. Energy Conservation: They invoke the Onsager conjecture results (specifically for bounded domains) which state that if the velocity is Hölder continuous with exponent $\alpha > 1/3$, the weak Euler solution conserves energy.

4. Key Contributions and Main Results

Theorem 6 (Main Result)

The paper establishes a sufficient condition for the absence of anomalous dissipation and the convergence of NSE solutions to Euler solutions.

Assumptions:

• Domain $\Omega$ is $C^4$.
• Initial data $u_{\nu_k, 0}$ converges strongly in $C^{0,\alpha}(\Omega)$ to $u_0$.
• Uniform Bounds: The sequence of solutions satisfies:
  $$\sup_k \|u_{\nu_k}\|_{L^\infty((0,T); C^{0,\alpha}(\Omega))} + \sup_k \nu_k^{1/2} \|u_{\nu_k}\|_{L^2((0,T); H^1(\Omega))} < \infty$$

Conclusions:

1. Convergence: Any weak-$*$ limit $u$ of the sequence is a weak solution to the Euler equations (in the sense of Definition 4) with initial data $u_0$.
2. Energy Conservation: If $\alpha > 1/3$, the limit solution $u$ conserves energy: $\|u(t)\|_{L^2}^2 = \|u_0\|_{L^2}^2$.
3. Absence of Anomalous Dissipation: Under the condition $\alpha > 1/3$, the total viscous dissipation (bulk + boundary) vanishes as $\nu \to 0$:
   $$\lim_{\nu_k \to 0} \left( 2\nu_k \int_0^T \|D(u_{\nu_k})\|_{L^2}^2 dt + \text{Boundary Terms} \right) = 0$$
4. Strong Convergence: There exists a subsequence converging strongly in $C([0,T]; L^2(\Omega))$.

Remark 8 (Weakened Assumptions)

The authors note that the uniform global Hölder bound can be relaxed. It suffices to have:

• Global $C^{0,\beta}$ boundedness for some $\beta > 0$.
• Local $C^{0,\alpha}$ boundedness ($\alpha > 1/3$) on any compact subset strictly inside the domain ($\tilde{\Omega} \subset \subset \Omega$).

5. Significance and Novelty

• No Assumption on Pressure Regularity: Unlike previous works that often required assumptions on the pressure near the boundary or the existence of a strong Euler solution, this result relies solely on the Hölder regularity of the velocity field. The pressure regularity is derived as a consequence.
• No Strong Euler Solution Required: The proof does not assume the existence of a smooth (strong) solution to the Euler equations to start with. It proves convergence to a weak solution that happens to conserve energy due to the regularity of the limit.
• Navier vs. No-Slip: The result highlights a fundamental difference between Navier and no-slip (Dirichlet) boundary conditions. For no-slip, the formation of a Prandtl boundary layer often prevents the uniform Hölder bounds required here. For Navier conditions, the boundary layer is weaker (or non-existent in the velocity profile), making the uniform $C^{0,\alpha}$ bound a plausible physical condition to rule out anomalous dissipation.
• Global Existence (Appendix): The paper provides a sketch of the proof for the global existence of Leray-Hopf weak solutions for the generalized Navier boundary conditions, filling a gap in the literature for this specific generality of $\lambda$.

Conclusion

This paper provides a rigorous sufficient condition ensuring that for Navier boundary conditions, if the velocity field remains uniformly Hölder continuous (with exponent $> 1/3$) as viscosity vanishes, then the limit is an energy-conserving Euler solution, and no anomalous dissipation occurs. This advances the understanding of the inviscid limit by decoupling the problem from assumptions on pressure regularity or the existence of strong Euler solutions.