Vortex reconnection in the three dimensional Navier–Stokes equations

Enciso, Alberto; Lucà, Renato; Peralta‐Salas, Daniel

doi:10.1016/j.aim.2017.01.025

Cited by 25 publications

(34 citation statements)

References 17 publications

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“…Before presenting the main ideas of the proof of this theorem, it is worth comparing it with our previous result with Lucà on vortex reconnection for the 3D Navier-Stokes equation [15]. From the point of view of what we prove, the main difference is that the Navier-Stokes result shows that one can take a finite number of "observation times" T 0 < T 1 < • • • < T N such that the vortex structures present at the fluid at time T k are not topologically equivalent to those at time T k±1 , which shows in an indirect way that at least one reconnection event must have taken place.…”

Section: Reconnection Of Quantum Vortices For the Gross-pitaevskii Equationmentioning

confidence: 94%

Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection

Enciso

Peralta‐Salas

2021

Commun. Math. Phys.

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We prove the existence of smooth solutions to the Gross–Pitaevskii equation on $$\mathbb {R}^3$$ R 3 that feature arbitrarily complex quantum vortex reconnections. We can track the evolution of the vortices during the whole process. This permits to describe the reconnection events in detail and verify that this scenario exhibits the properties observed in experiments and numerics, such as the $$t^{1/2}$$ t 1 / 2 and change of parity laws. We are mostly interested in solutions tending to 1 at infinity, which have finite Ginzburg–Landau energy and physically correspond to the presence of a background chemical potential, but we also consider the cases of Schwartz initial data and of the Gross–Pitaevskii equation on the torus. In the proof, the Gross–Pitaevskii equation operates in a nearly linear regime, so the result applies to a wide range of nonlinear Schrödinger equations. Indeed, an essential ingredient in the proofs is the development of novel global approximation theorems for the Schrödinger equation on $$\mathbb {R}^n$$ R n . Specifically, we prove a qualitative approximation result that applies for solutions defined on very general spacetime sets and also a quantitative result for solutions on product sets in spacetime $$D\times \mathbb {R}$$ D × R . This hinges on frequency-dependent estimates for the Helmholtz–Yukawa equation that are of independent interest.

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Section: Reconnection Of Quantum Vortices For the Gross-pitaevskii Equationmentioning

confidence: 94%

Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection

Enciso

Peralta‐Salas

2021

Commun. Math. Phys.

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“…We highlight the recent experimental results presented in [11,17], where the authors study how vortex lines (or tubes) of different knotted topologies reconnect in actual fluids using cleverly designed hydrofoils. In contrast with the wealth of heuristic, numerical and experimental results on this subject, the first mathematically rigorous scenario of vortex reconnection was constructed very recently in [7] (see Theorem 2.1).…”

Section: Classical Fluids: the Navier-stokes Equationsmentioning

confidence: 99%

Vortex reconnections in classical and quantum fluids

Enciso

Peralta‐Salas

2021

SeMA

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We review recent rigorous results on the phenomenon of vortex reconnection in classical and quantum fluids. In the context of the Navier–Stokes equations in $$\mathbb {T}^3$$ T 3 we show the existence of global smooth solutions that exhibit creation and destruction of vortex lines of arbitrarily complicated topologies. Concerning quantum fluids, we prove that for any initial and final configurations of quantum vortices, and any way of transforming one into the other, there is an initial condition whose associated solution to the Gross–Pitaevskii equation realizes this specific vortex reconnection scenario. Key to prove these results is an inverse localization principle for Beltrami fields and a global approximation theorem for the linear Schrödinger equation.

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“…(x j dy j − y j dx j ), (19) where we write the complex coordinates (z 1 , z 2 ) of C 2 in terms of their real and imaginary parts: z j = x j + iy j . There are two interesting geometric interpretations of the standard contact structure ξ 0 .…”

Section: Contact Structures and Legendrian Linksmentioning

confidence: 99%

“…However, these links cannot be expected to be stable, since they usually undergo reconnection events as time progresses and the field evolves according to Maxwell's equations, or they disappear altogether. While there are not many rigorous results concerning the time evolution and reconnections of knots in electromagnetic fields, the setting of the Navier-Stokes equations has been analysed in some detail [19]. Necessary and sufficient conditions for the stability of knotted field lines in electromagnetic fields are known [28], but so far only the family of torus links has been constructed and thereby been proven to arise as stable knotted field lines in electromagnetism.…”

Section: Introductionmentioning

confidence: 99%

Stable Knots and Links in Electromagnetic Fields

Bode

2021

Commun. Math. Phys.

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Persistent topological structures in physical systems have become increasingly important over the last years. Electromagnetic fields with knotted field lines play a special role among these, since they can be used to transfer their knottedness to other systems like plasmas and quantum fluids. In null electromagnetic fields the electric and the magnetic field lines evolve like unbreakable elastic filaments in a fluid flow. In particular, their topology is preserved for all time, so that all knotted closed field lines maintain their knot type. We use an approach due to Bateman to prove that for every link L there is such an electromagnetic field that satisfies Maxwell’s equations in free space and that has closed electric and magnetic field lines in the shape of L for all time. The knotted and linked field lines turn out to be projections of real analytic Legendrian links with respect to the standard contact structure on the 3-sphere.

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Vortex reconnection in the three dimensional Navier–Stokes equations

Cited by 25 publications

References 17 publications

Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection

Approximation Theorems for the Schrödinger Equation and Quantum Vortex Reconnection

Vortex reconnections in classical and quantum fluids

Stable Knots and Links in Electromagnetic Fields

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