arXiv subjects

David Villringer

Publications and source records attributed to David Villringer.

8 recordsLinked to original sources

arXiv2026

Smooth autonomous fast dynamo action on the three-torus

We construct a nonempty $C^k$-open family, for some $k\in\mathbb{N}$, of smooth, autonomous, divergence-free velocity fields on $\mathbb{T}^3$ that generate fast dynamos. This resolves the Fast Dynamo Conjecture of Zeldovich and Sakharov, as recorded in Arnold's book of problems. We first construct a family of smooth, time-periodic fast dynamos via a stretch-fold-shear mechanism that applies to a large class of velocity fields. We then realize the associated period dynamics within a smooth autonomous flow using a global Poincar\'e section and a suitably prescribed return-time function. In both settings, the proof first isolates unstable distributional eigenmodes for the ideal induction operator on an anisotropic Banach space of distributions and then shows that this spectral instability persists under the singular perturbation $\varepsilon \Delta$. The resulting estimates are uniform as $\varepsilon\to0$ and stable under $C^k$ perturbations of the velocity field, yielding an open set of smooth fast dynamo vector fields.

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arXiv2026

A fast dynamo on the three-torus

We study the kinematic dynamo equation on the three-torus and provide a rigorous proof of fast dynamo action for a time-periodic, divergence-free, Lipschitz velocity field. Our construction is based on a stretch-fold-shear mechanism generating a uniformly hyperbolic flow. To analyze the associated dynamics, we develop anisotropic Banach spaces adapted to the underlying hyperbolic structure, allowing us to recover a discrete spectral picture for the ideal dynamo operator. In the strong-chaos regime, we show that this operator admits an eigenvalue with modulus strictly larger than one. We then prove that this instability persists under the singular perturbation induced by diffusion, yielding exponential growth of the magnetic field uniformly in the vanishing resistivity limit.

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arXiv2025

A limsup fast dynamo on $\mathbb{T}^3$

We construct a time-dependent, incompressible, and uniformly-in-time Lipschitz continuous velocity field on $\mathbb{T}^3$ that produces exponential growth of the magnetic energy along a subsequence of times, for every positive value of the magnetic diffusivity. Because this growth is not uniform in time but occurs only along a diverging sequence of times, we refer to the resulting mechanism as a limsup fast dynamo. Our construction is based on suitably rescaled Arnold-Beltrami-Childress (ABC) flows, each supported on long time intervals. The analysis employs perturbation theory to establish continuity of the exponential growth rate with respect to both the initial data and the diffusivity parameter. This proves the weak form of the fast dynamo conjecture formulated by Childress and Gilbert on $\mathbb{T}^3$, but the considerably more challenging version proposed by Arnold on $\mathbb{T}^3$ remains an open problem.

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arXiv2025

Nonlinear instability for the 3D MHD equations around the Taylor-Couette flow

We study the 3D magnetohydrodynamics (MHD) equations in an annular cylinder, perturbed around the explicit steady state given by the 3D Taylor-Couette velocity field and zero magnetic field. Combining a recent linear instability result for the magnetic field with the framework of Friedlander, Pavlovi\'c and Shvydkoy [Comm. Math. Phys. 264 (2006), no. 2, 335-347], we prove nonlinear instability of the solution around this steady state in $L^p$, for any $p>1$. In particular, our results are, to the best of our knowledge, the first rigorous instability results for 3D MHD without forcing, in which the instability is produced as a result of the (exponential) growth of the magnetic field. Furthermore, we offer a mathematical proof of the physically conjectured transfer of energy from the velocity field to the magnetic field in the MHD system.

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arXiv2025

Spectral instability in the smooth Ponomarenko dynamo

We consider the kinematic dynamo equations for a passive vector in $\mathcal{M} \times \mathbb{T} \subseteq \mathbb{R}^2 \times \mathbb{T}$ describing the evolution of a magnetic field with resistivity $\varepsilon>0$, that is transported by a given velocity field. For a broad class of $C^3$ velocity fields with helical geometry, we establish the existence of solutions that exhibit exponential growth over time. We construct an unstable eigenmode via detailed resolvent estimates of the corresponding linear operator, which we carry out by introducing suitable Green's functions that accurately approximate the local behaviour of the true system. This approach yields an explicit asymptotic expression for the growing mode, providing a sharp description of the instability mechanism. We first derive the results with $\mathcal{M}=\mathbb{R}^2$ for a large class of velocity fields that includes finite energy examples. We then consider the case of domains with boundary, where $\mathcal{M}\times\mathbb{T}$ denotes a periodic cylinder, annular cylinder, or the exterior of a cylinder, with the boundary conditions of perfectly conducting walls. Our results offer a rigorous and sharp mathematical justification for the physically conjectured process by which helical flows can sustain magnetic field generation in the Ponomarenko dynamo, with growth rate of order $\varepsilon^{1/3}$.

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arXiv2025

A Stochastic RAGE Theorem and Enhanced Dissipation for Transport Noise

We prove a stochastic version of the classical RAGE theorem that applies to the two-point motion generated by noisy transport equations. As a consequence, we identify a necessary and sufficient condition for the corresponding diffusive equation to be dissipation enhancing. This involves the identification of a non-trivial, finite dimensional subspace that is invariant for the family of self-adjoint operator characterizing the structure of the transport noise. We discuss several examples and prove a sharp enhanced dissipation rate for stochastic shear flows.

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arXiv2025

Alpha-unstable flows and the fast dynamo problem

We construct a time-independent, incompressible, and Lipschitz-continuous velocity field in $\mathbb{R}^3$ that generates a fast kinematic dynamo - an instability characterized by exponential growth of magnetic energy, independent of diffusivity. Specifically, we show that the associated vector transport-diffusion equation admits solutions that grow exponentially fast, uniformly in the vanishing diffusivity limit $\varepsilon\to 0$. Our construction is based on a periodic velocity field $U$ on $\mathbb{T}^3$, such as an Arnold-Beltrami-Childress flow, which satisfies a generic spectral instability property called alpha-instability, established via perturbation theory. This provides a rigorous mathematical framework for the alpha-effect, a mechanism conjectured in the late 1960s to drive large-scale magnetic field generation. By rescaling with respect to $\varepsilon$ and employing a Bloch-type theorem, we extend the solution to the whole space. Finally, through a gluing procedure that spatially localizes the instability, we construct a globally defined velocity field $u$ in $\mathbb{R}^3$ that drives the dynamo instability.

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arXiv2024

Enhanced Dissipation via the Malliavin Calculus

In this work we investigate the phenomenon of enhanced dissipation using techniques from the Malliavin Calculus. In particular, we construct a concise, elementary argument, that allows us to recover the well-known enhanced dissipation timescale for shear flows, first obtained by Bedrossian and Coti Zelati in 2017, as well as the precise hypoelliptic regularisation in x.

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