Anomalous dissipation and the Burgers–Kato boundary layer in the vanishing-viscosity limit of two-dimensional wall-bounded flows at critical Navier slip

The collision of a vortex dipole with a solid wall is a generic two-dimensional flow event in which all the conditions identified by Kato for anomalous dissipation can be met: vorticity is created at the wall, and dissipation can concentrate in a layer of thickness proportional to the viscosity. Nguyen van yen, Farge and Schneider (2011) simulated this collision with Navier slip conditions whose slip length scales as $\alpha=\mathrm{Re}^{-1}$ and found dissipation localized in such an $O(\mathrm{Re}^{-1})$ wall sublayer — which they named the Burgers–Kato layer — rolling up into a spiral that detaches from the wall, with an integrated dissipation rate that becomes Reynolds-number independent: numerical evidence for anomalous dissipation generated by walls in two dimensions, where interior enstrophy bounds otherwise forbid it. They asked whether the inviscid limit provably fails in this regime, and how general and statistically characterizable the scenario is. Later work widened the proven-convergence regime from the Letter's $\gamma<1/2$ to all $\gamma<1$ (Paddick 2014), leaving only the critical regime $\gamma\ge1$ (slip length at most of order the viscosity, including no-slip) open at collision time scales; only short-time analytic-data results (Trinh T. Nguyen 2020), conditional Kato-type criteria, and a proposed Rayleigh–Tollmien–Schlichting mechanism exist. The open problem is to prove or refute anomalous dissipation and nonconvergence to Euler in the critical regime $\gamma\ge1$ on the collision time scale; a rigorous asymptotic description of the Burgers–Kato layer that produces or precludes it is the expected mechanism through which a resolution proceeds, not a separately required deliverable.

Background

Incompressible fluid motion with small kinematic viscosity $\nu$ is described by the Navier–Stokes equations, which at $\nu=0$ reduce formally to the Euler equations of ideal flow. The vanishing-viscosity (inviscid) limit — whether Navier–Stokes solutions converge, as $\nu\to0$ (equivalently as the Reynolds number $\mathrm{Re}=UL/\nu\to\infty$), to the Euler solution with the same initial data — is a fundamental open problem when the domain has solid boundaries, because the viscous and inviscid problems impose incompatible conditions at the wall. At a solid wall the velocity obeys either the no-slip condition (fluid velocity equal to wall velocity) or the Navier slip condition, which relates the tangential velocity to its wall-normal derivative through a slip length $\alpha$; for a wall with normal in the $x_1$ direction it reads $u_2+\alpha\,\partial_1 u_2=0$, and $\alpha=0$ recovers no-slip. Kato (1984) proved a dichotomy for no-slip flows: the energy dissipation rate $\nu\int|\nabla u^\nu|^2$ tends to zero if and only if $u^\nu$ converges in $L^2$ to the Euler solution, and any nonzero limiting dissipation must already occur inside a wall-attached layer of thickness proportional to $\nu$. A dissipation rate that stays strictly positive as $\nu\to0$ is called anomalous dissipation and is a cornerstone of turbulence theory. In two dimensions (2D) the question is especially sharp: without boundaries the enstrophy $\int|\omega|^2$ (with vorticity $\omega=\nabla\times u$) is nonincreasing along solutions, which forces $\nu\int|\nabla u|^2=\nu\int|\omega|^2\to0$; anomalous dissipation in 2D can therefore only be produced by vorticity created at solid walls.

Nguyen van yen, Farge and Schneider (2011) studied numerically the dipole–wall collision — the impact on a solid wall of a propagating vortex dipole (a self-propelling pair of counter-rotating vortices), a standard generic 2D wall–vorticity interaction — in a periodic channel at $\mathrm{Re}=985$ to $7880$, using volume penalization, a numerical device that enforces solid regions through a large friction term and effectively replaces no-slip by Navier slip with a slip length $\alpha\propto\mathrm{Re}^{-1}$. They observed that dissipation first appears in a vorticity sheet of thickness proportional to $\mathrm{Re}^{-1}$ — far thinner than the classical Prandtl layer of thickness $\mathrm{Re}^{-1/2}$, the near-wall region where viscous effects are traditionally confined — and continues as this sheet rolls into a spiral that detaches from the wall; the dissipation integrated over the near-wall strip and over the detached spiral becomes approximately Re-independent, and is concentrated along Lagrangian trajectories (trajectories of transported fluid particles) that 'crash' into the wall. They named the $O(\mathrm{Re}^{-1})$ sublayer the Burgers–Kato boundary layer, and argued that such energy-dissipating structures may be common in 2D wall-bounded flows and lie outside existing statistical theories of 2D turbulence.

On the rigorous side, the inviscid limit with Navier slip is well understood when the slip length is independent of $\nu$ (complete boundary-layer expansions by Iftimie–Sueur; uniform regularity by Masmoudi–Rousset). For viscosity-dependent slip $\alpha=\nu^\gamma$ (the Letter's $\alpha=\mathrm{Re}^{-\gamma}$), Wang–Wang–Xin (2010) proved $L^2$ convergence for $\gamma<1/2$ — the published form of a result the Letter cited as a private communication — and Paddick (2014) extended convergence to the whole range $\gamma<1$ for the physical sign of the friction coefficient, with rate $O(\nu^{(1-\gamma)/2})$; this energy method degenerates exactly at $\gamma=1$, making slip length of order $\nu$ the critical regime, which is precisely the regime the Letter simulates. Paddick also proved that at $\gamma=1/2$ the Prandtl-type asymptotic (Wentzel–Kramers–Brillouin, WKB) expansion becomes nonlinearly unstable in stronger norms ($L^\infty$, $\dot H^s$, $s>1$) even though $L^2$ convergence holds; Quarisa–Rodrigo (2022) extended this instability analysis. At the critical $\gamma=1$, Trinh T. Nguyen (2020) established the inviscid limit in the half-space for analytic data on short times uniform in $\nu$, and Wang–Yin–Zhu (2017) gave Kato-type criteria that guarantee convergence for small slip $\alpha\le O(\nu)$ provided the boundary-strip dissipation vanishes. For genuine no-slip, Maekawa (2014) rigorously justified the classical Prandtl expansion for dipole-type data (initial vorticity away from the wall), but only up to short times before boundary-layer breakdown. Conditional frameworks delimit the possible outcomes without settling the wall-collision regime: localized versions, on domains with boundary, of Onsager's energy-conservation criterion (the conjecture that sufficiently Hölder-regular Euler solutions conserve energy, while rougher weak solutions may dissipate it) and vanishing-dissipation criteria for Leray–Hopf solutions, i.e. weak solutions satisfying the energy inequality (Drivas–Nguyen 2018); the possible emergence of weak, possibly dissipative Euler solutions when second-order structure-function exponents, the scaling exponents of second moments of velocity increments, remain positive uniformly in $\nu$ (Drivas–Nguyen 2019); and Kato-type criteria at regularity near Onsager's critical exponent (Chen–Liang–Wang 2022). In a 2018 follow-up with no-slip walls, the Letter's authors proposed that the dissipation is driven by a Rayleigh–Tollmien–Schlichting instability (an instability of wall-attached shear layers generating growing waves) whose superposition produces a boundary vorticity flux of order $\mathrm{Re}$, but they state that a rigorous description, and the flow after layer detachment, remain open.

Problem Statement

Consider the two-dimensional incompressible Navier–Stokes equations in the periodic-channel geometry of Nguyen van yen–Farge–Schneider (2011), referred to below as the Letter, with the Navier slip boundary condition $u_2+\alpha\,\partial_1 u_2=0$ at the wall (tangential velocity $u_2$ and its wall-normal derivative $\partial_1 u_2$, the wall normal being the $x_1$ direction), where the slip length scales with the Reynolds number $\mathrm{Re}=UL/\nu$ — $L$ the channel half-width and $U$ the initial rms velocity, as in the Letter — as $\alpha=\mathrm{Re}^{-\gamma}$ (the true no-slip condition $\alpha=0$ is included as the extreme case), and with initial data of dipole type (e.g. the Letter's explicit dipole, Eq. (5)) whose inviscid evolution drives $O(1)$ vorticity into the wall (the dipole–wall collision configuration studied by the Letter).

(1) Critical-slip and no-slip regime, $\gamma\ge1$ (slip length at most of order the viscosity): prove or refute the Letter's central expectation that, on the $O(1)$ time scale of the wall collision, the solutions fail to converge in $L^2$ to the Euler solution because the energy dissipation stays bounded away from zero. A resolution must either (a) exhibit, for an explicitly characterized nondegenerate class of dipole-type data (nondegenerate meaning that the inviscid evolution drives $O(1)$ vorticity into the wall), a time interval containing the collision, for the Letter's critical slip length $\alpha=\mathrm{Re}^{-1}$ (or, stated explicitly, for some $\alpha>0$ with $\alpha\le O(\nu)$), with

$$\liminf_{\nu\to0}\ \int_0^T\nu\int|\nabla u^\nu|^2\,dx\,dt \;\ge\; c(T)>0,$$

which by Kato's criterion is equivalent to persistent dissipation in the $O(\nu)$ wall strip and to failure of $L^2$ convergence to the smooth Euler solution; or (b) prove $L^2$ convergence to the Euler solution on time intervals containing the collision in the regime $\alpha\le O(\nu)$ (all $\gamma\ge1$, the critical case $\gamma=1$ included), implying vanishing dissipation and refuting the scenario; convergence for no-slip $\alpha=0$ alone does not refute the Letter's slip scenario.

Expected mechanism — Burgers–Kato layer: a rigorous asymptotic description of the $O(\nu)$ wall sublayer at and after the collision — its formation from the crashing vorticity, its stability or instability, its nonlinear roll-up into the detaching spiral, and its dissipation budget — is the expected route through which a resolution of (1) proceeds, and would account for the structures observed numerically in the near-wall and detached-spiral regions. Such a description strengthens an answer to (1), but is not part of the completion boundary of this problem.

The completion boundary of this problem is item (1): the rigorous fate of the Letter's wall-generated anomalous-dissipation scenario in the critical regime $\gamma\ge1$ on the collision time scale. The Letter also poses two broader strands, recorded as context rather than bounded deliverables of this problem: the generality of the dissipative scenario across other two-dimensional wall-bounded configurations, boundary conditions and initial data, and a statistical theory of two-dimensional wall-bounded turbulence encompassing persistent wall-generated dissipating structures (including their contribution to ensemble-averaged energy balances). Progress on the layer mechanism or on the broader strands strengthens, but is not required for, a resolution.

References

1. Nguyen van yen, R.; Farge, M.; Schneider, K. (2011). Energy Dissipating Structures Produced by Walls in Two-Dimensional Flows at Vanishing Viscosity. Physical Review Letters 106, 184502. DOI: 10.1103/PhysRevLett.106.184502. https://doi.org/10.1103/PhysRevLett.106.184502
2. Kato, T. (1984). Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. In: Seminar on Nonlinear Partial Differential Equations (Berkeley, 1983), Mathematical Sciences Research Institute Publications 2, Springer, pp. 85-98. DOI: 10.1007/978-1-4612-1110-5_6. https://doi.org/10.1007/978-1-4612-1110-5_6
3. Wang, X.-P.; Wang, Y.-G.; Xin, Z. (2010). Boundary layers in incompressible Navier-Stokes equations with Navier boundary conditions for the vanishing viscosity limit. Communications in Mathematical Sciences 8(4), 965-998. DOI: 10.4310/CMS.2010.V8.N4.A10. https://doi.org/10.4310/CMS.2010.V8.N4.A10
4. Iftimie, D.; Sueur, F. (2011). Viscous boundary layers for the Navier-Stokes equations with the Navier slip conditions. Archive for Rational Mechanics and Analysis 199(1), 145-175. DOI: 10.1007/s00205-010-0320-z. https://doi.org/10.1007/s00205-010-0320-z
5. Masmoudi, N.; Rousset, F. (2012). Uniform Regularity for the Navier-Stokes Equation with Navier Boundary Condition. Archive for Rational Mechanics and Analysis 203(2), 529-575. DOI: 10.1007/s00205-011-0456-5. https://doi.org/10.1007/s00205-011-0456-5
6. Paddick, M. (2014). Stability and instability of Navier boundary layers. Differential and Integral Equations 27(9-10), 893-930. DOI: 10.57262/die/1404230050. https://doi.org/10.57262/die/1404230050 (preprint arXiv:1103.5009, https://arxiv.org/abs/1103.5009)
7. Maekawa, Y. (2014). On the Inviscid Limit Problem of the Vorticity Equations for Viscous Incompressible Flows in the Half-Plane. Communications on Pure and Applied Mathematics 67(7), 1045-1128. DOI: 10.1002/cpa.21516. https://doi.org/10.1002/cpa.21516
8. Wang, Y.-G.; Yin, J.; Zhu, S. (2017). Vanishing viscosity limit for incompressible Navier-Stokes equations with Navier boundary conditions for small slip length. Journal of Mathematical Physics 58, 101507. DOI: 10.1063/1.5004975. https://doi.org/10.1063/1.5004975
9. Drivas, T. D.; Nguyen, H. Q. (2018). Onsager's Conjecture and Anomalous Dissipation on Domains with Boundary. SIAM Journal on Mathematical Analysis 50(5), 4785-4811. DOI: 10.1137/18M1178864. https://doi.org/10.1137/18M1178864 (preprint arXiv:1803.05416, https://arxiv.org/abs/1803.05416)
10. Nguyen van yen, N.; Waidmann, M.; Klein, R.; Farge, M.; Schneider, K. (2018). Energy dissipation caused by boundary layer instability at vanishing viscosity. Journal of Fluid Mechanics 849, 676-717. DOI: 10.1017/jfm.2018.396. https://doi.org/10.1017/jfm.2018.396 (preprint arXiv:1706.00942, https://arxiv.org/abs/1706.00942)
11. Drivas, T. D.; Nguyen, H. Q. (2019). Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit. Journal of Nonlinear Science 29(2), 709-721. DOI: 10.1007/s00332-018-9500-z. https://doi.org/10.1007/s00332-018-9500-z (preprint arXiv:1808.01014, https://arxiv.org/abs/1808.01014)
12. Nguyen, T. T. (2020). The Inviscid Limit of Navier-Stokes with Critical Navier-Slip Boundary Conditions for Analytic Data. SIAM Journal on Mathematical Analysis 52(5), 4616-4637. DOI: 10.1137/19M1258967. https://doi.org/10.1137/19M1258967 (preprint arXiv:1904.12943, https://arxiv.org/abs/1904.12943)
13. Quarisa, L.; Rodrigo, J. L. (2022). Instability of Boundary Layers with the Navier Boundary Condition. Journal of Mathematical Fluid Mechanics 24(3), Article 91. DOI: 10.1007/s00021-022-00714-2. https://doi.org/10.1007/s00021-022-00714-2
14. Chen, R. M.; Liang, Z.; Wang, D. (2022). A Kato-Type Criterion for Vanishing Viscosity Near Onsager's Critical Regularity. Archive for Rational Mechanics and Analysis 246(2-3), 535-559. DOI: 10.1007/s00205-022-01822-z. https://doi.org/10.1007/s00205-022-01822-z (preprint arXiv:2007.12746, https://arxiv.org/abs/2007.12746)
15. Sutherland, D.; Macaskill, C.; Dritschel, D. G. (2013). The effect of slip length on vortex rebound from a rigid boundary. Physics of Fluids 25(9), 093104. DOI: 10.1063/1.4821774. https://doi.org/10.1063/1.4821774
16. Clercx, H. J. H.; van Heijst, G. J. F. (2017). Dissipation of coherent structures in confined two-dimensional turbulence. Physics of Fluids 29, 111103. DOI: 10.1063/1.4993488. https://doi.org/10.1063/1.4993488
17. Miller, L.; Deremble, B.; Venaille, A. (2024). Gyre turbulence: Anomalous dissipation in a two-dimensional ocean model. Physical Review Fluids 9, L051801. DOI: 10.1103/PhysRevFluids.9.L051801. https://doi.org/10.1103/PhysRevFluids.9.L051801
18. Eyink, G.; Quan, H. (2025). Weak-strong uniqueness and extreme wall events at high Reynolds number. Physical Review Fluids 10, 064610. DOI: 10.1103/PhysRevFluids.10.064610. https://doi.org/10.1103/PhysRevFluids.10.064610 (preprint arXiv:2502.00994, https://arxiv.org/abs/2502.00994)
19. Riquier, A.; Dormy, E. (2026). Do irrotational water waves remain irrotational in the limit of a vanishing viscosity? Proceedings of the Royal Society A 482, 20250772. DOI: 10.1098/rspa.2025.0772. https://doi.org/10.1098/rspa.2025.0772 (preprint arXiv:2411.09291, https://arxiv.org/abs/2411.09291)