Vanishing viscosity limits for axisymmetric no-swirl flows in an infinite cylinder: anomalous dissipation, rough data, and energy conservation

Ken Abe (J. Math. Pures Appl. 137 (2020) 1-32) constructed global weak solutions of the incompressible Euler equations in the infinite cylinder $\Pi=\{x\in\mathbb{R}^3: |(x_1,x_2)|<1\}$ for axisymmetric initial data without swirl whose relative vorticity $\omega_0^\theta/r$ lies in $L^q$, $q\in[3/2,3)$, by passing to the zero-viscosity limit of Navier-Stokes solutions with the Neumann (free-slip) boundary condition; uniqueness of the limit and the $L^2$ energy equality were obtained only under $\omega_0^\theta/r\in L^\infty$, and the energy dissipation was shown to vanish for $q\in[3/2,2]$. The paper's own closing discussion leaves open: (i) whether the energy dissipation has a strictly positive limit (anomalous dissipation) for rougher data, namely $\omega_0^\theta/r\in L^q$ with $q\in[1,3/2)$ — in particular the endpoint range $q\in[1,6/5]$ where the vorticity decay estimate no longer controls the dissipation — together with the prerequisite, open in the paper as well, of global well-posedness of the Navier-Stokes problem for $q\in[1,3/2)$; and (ii) the energy equality under weaker assumptions, for instance via Onsager-type criteria in the unbounded cylinder, and for all constructed weak solutions without extra regularity. The whole-space ($\mathbb{R}^3$) analogues of the energy question were settled by Nobili-Seis (2022) and Brkic-Wiedemann (2024), but those results do not cover the cylinder with boundary, where the problem remains open.

Background

An incompressible fluid of viscosity $\nu>0$ is governed by the Navier-Stokes equations $\partial_t u-\nu\Delta u+u\cdot\nabla u+\nabla p=0$, $\operatorname{div}u=0$; the inviscid limit $\nu\to0$ formally yields the Euler equations. The inviscid limit in domains with boundary is notoriously hard because thin boundary layers form, and Kolmogorov's turbulence theory predicts that the energy dissipation $\nu\int|\nabla u^\nu|^2$ may tend to a strictly positive constant (anomalous dissipation) as $\nu\to0$, which for a convergent subsequence would produce an Euler weak solution that strictly loses energy. A vector field is axisymmetric if it is invariant under rotations about the $x_3$-axis; in cylindrical coordinates $(r,\theta,z)$ it decomposes as $u=u^r e_r+u^\theta e_\theta+u^z e_z$, and $u^\theta$ is the swirl. For axisymmetric flows without swirl ($u^\theta=0$), the vorticity is purely azimuthal, $\omega=\omega^\theta e_\theta$, and the quantity $\xi=\omega^\theta/r$ (relative vorticity) is transported by Euler flows, $\partial_t\xi+u\cdot\nabla\xi=0$, while $\omega^\theta$ itself undergoes vortex stretching (amplification by stretching of the vortex lines); this makes the class a 3D analogue of 2D vorticity dynamics and a natural setting in which to test energy balance questions beyond smooth data. Abe considered the infinite cylinder $\Pi=\{x: |x_h|<1\}$, $x_h=(x_1,x_2)$, with the Neumann (free-slip) boundary condition $\nabla\times u\times n=0$, $u\cdot n=0$ on $\partial\Pi$; under it, $\omega^\theta/r$ satisfies a drift-diffusion equation with homogeneous Dirichlet boundary condition, and the Biot-Savart law (the recovery of the velocity from the vorticity) takes the form $u=\nabla\times(-\Delta_D)^{-1}(\nabla\times u)$ with the Dirichlet Laplacian (the Laplacian with zero boundary condition) in the cross-section. Here $L^p_\sigma(\Pi)$ denotes the $L^p$-closure of smooth compactly supported divergence-free fields with vanishing normal trace. Abe proved: (1) global well-posedness of the Navier-Stokes problem in $L^p$ for all $p\in[3,\infty)$ for axisymmetric no-swirl data, via a new a priori estimate for $\omega^\theta/r$; (2) existence of global Euler weak solutions for $\omega_0^\theta/r\in L^q$, $q\in[3/2,3)$, obtained as vanishing-viscosity limits (Theorem 1.2(i)), which are Hölder continuous of exponent $1-3/s$ if $\omega_0^\theta/r\in L^s$, $s>3$, and unique if $\omega_0^\theta/r\in L^\infty$ (Theorem 1.2(ii)-(iii)); (3) for $q\in[3/2,2]$ the dissipation vanishes like $O(\nu^{5/2-3/q})$, and under the additional assumption $\omega_0^\theta/r\in L^\infty$ the viscous solutions converge in $L^2$ to the limit, which then satisfies the energy equality $\int|u|^2=\int|u_0|^2$ for all $t\ge 0$ (Theorem 1.3). Onsager's conjecture asserts that Hölder-continuous Euler weak solutions with exponent $>1/3$ conserve energy; this is known in the whole space (Constantin-E-Titi) and in bounded domains (Bardos-Titi), but its applicability to the unbounded cylinder is unverified, and the constructed solutions are Hölder with exponent $1-3/s>1/3$ only when $s>9/2$. For rough data, in $\mathbb{R}^3$ the Cauchy problem is well posed for $\omega_0^\theta/r\in L^1$ (Gallay-Šverák), global weak Euler solutions exist for $\omega_0^\theta/r\in L^1\cap L^p$, $p>1$ (Jiu-Liu-Niu), and single vortex rings (vorticity concentrated on a circle) are handled by Feng-Šverák, with the full vanishing-viscosity asymptotics of such circular filaments recently established by Gallay-Šverák (2024); the cylinder counterparts with boundary are not covered. For the vanishing viscosity limit in $\mathbb{R}^3$, Nobili-Seis proved that limits with $\omega^\theta/r\in L^p$, $p>1$, are renormalized solutions — in the sense of DiPerna-Lions, the transport equation for $\omega^\theta/r$ also holds after composition with any smooth nonlinearity — and conserve kinetic energy for $p>3/2$ (nonnegative vorticity, finite second moments), Brkic-Wiedemann upgraded the convergence of the relative vorticity to strong $L^p$ convergence, and Brkic-Wiedemann (2025) showed by convex integration (a construction scheme producing non-unique "wild" weak solutions) that admissible swirl-free axisymmetric Euler weak solutions in $\mathbb{R}^3$ are in general non-unique. None of these whole-space results extends to the cylinder with boundary, which is the setting of the present problem.

Problem Statement

The overarching objective is to complete the energy-balance program of the vanishing-viscosity theory that Abe initiated for axisymmetric flows without swirl in the infinite cylinder $\Pi=\{x\in\mathbb{R}^3: |(x_1,x_2)|<1\}$ subject to the Neumann boundary condition $\nabla\times u\times n=0$, $u\cdot n=0$ on $\partial\Pi$: extend the construction of global Euler weak solutions to rougher initial data and settle the energy dissipation and the energy equality of the limits. Concretely, the following components, posed by the source paper, remain to be resolved in this setting. (i) Energy dissipation for rough data: for data $\omega_0^\theta/r\in L^q(\Pi)$ with $q\in[1,3/2)$, determine whether $\lim_{\nu\to0}\nu\int_0^T\!\int_\Pi|\nabla u^\nu|^2$ is strictly positive (anomalous dissipation, producing a vanishing-viscosity limit that strictly decreases the energy), zero, or fails to converge, and decide whether this behaviour is uniform over the whole class or data-dependent — for a data-dependent answer, state the dichotomy and exhibit a datum realising each behaviour that occurs, a single datum settling only that case — including the endpoint range $q\in[1,6/5]$ where the known vorticity decay estimate does not imply convergence of the dissipation; this includes establishing whatever viscous well-posedness in the $L^p$ classes with $p<3$ is needed, itself open for $q\in[1,3/2)$. (ii) Energy conservation: determine whether the energy equality $\int_\Pi|u|^2=\int_\Pi|u_0|^2$ (for all $t\ge 0$) holds under conditions weaker than $\omega_0^\theta/r\in L^\infty$ — e.g. by verifying Onsager-type criteria (Hölder exponent $>1/3$, available for $\omega_0^\theta/r\in L^s$ with $s>9/2$) for the unbounded cylinder, whose current proofs cover only the whole space and bounded domains — and decide whether it holds for all weak solutions constructed in Theorem 1.2(i) without any extra regularity; this second form is the strongest of the two and is the target, so establishing the Onsager-type criterion only for the smoother range $\omega_0^\theta/r\in L^s$, $s>9/2$ is partial progress but does not by itself complete (ii). These components jointly support the single objective of carrying the energy-balance theory in the cylinder to the level of roughness available without boundary in $\mathbb{R}^3$; partial resolutions of individual components constitute progress but not completion.

References

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