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Turbulent solutions of the binormal flow and the 1D cubic Schrödinger equation

Valeria Banica, Luis Vega

TL;DR

This work surveys a program that connects the binormal flow (BF), a geometric model for vortex filaments, with the 1D cubic NLS, revealing turbulent dynamics through singularity formation, Fourier growth, and multifractal behavior. It develops a robust well-posedness theory for critical Fourier-Lebesgue spaces and supercritical Sobolev spaces, enabling construction of BF evolutions from NLS solutions with Dirac-mass-type data and establishing unique continuation across singular times via Hasimoto correspondence. The authors demonstrate self-similar and multi-corner BF solutions, quantify Fourier-energy growth and Talbot-type reconstructions at rational times, and uncover intermittent and multifractal trajectories described by Riemann-type limit functions. They also extend these analyses to periodic settings, proving wave-operator-type results and asymptotic completeness for a periodic NLS with time-dependent nonlinearity, which translates into criteria for generating and continuing BF singularities. Overall, the work provides a rigorous bridge between dispersive PDE turbulence and vortex-filament dynamics, with implications for understanding energy cascades and reconnection phenomena in fluids and superfluids.

Abstract

In the last three decades there has been an intense activity on the exploration of turbulent phenomena of dispersive equations, as for instance the growth of Sobolev norms since the work of Bourgain in the 90s. In general the 1D cubic Schrödinger equation has been left aside because of its complete integrability. In a series of papers of the last six years that we survey here for the special issue of the ICMP 2024 ([12],[13],[14],[15],[16],[7],[8]), we considered, together with the 1D cubic Schrödinger equation, the binormal flow, which is a geometric flow explicitly related to it. We displayed rigorously a large range of complex behavior as creation of singularities and unique continuation, Fourier growth, Talbot effects, intermittency and multifractality, justifying in particular some previous numerical observations. To do so we constructed a class of well-posedness for the 1D cubic Schrödinger equation included in the critical Fourier-Lebesgue space $\mathcal FL^\infty$ and in supercritical Sobolev spaces with respect to scaling. Last but not least we recall that the binormal flow is a classical model for the dynamics of a vortex filament in a 3D fluid or superfluid, and that vortex motions are a key element of turbulence.

Turbulent solutions of the binormal flow and the 1D cubic Schrödinger equation

TL;DR

This work surveys a program that connects the binormal flow (BF), a geometric model for vortex filaments, with the 1D cubic NLS, revealing turbulent dynamics through singularity formation, Fourier growth, and multifractal behavior. It develops a robust well-posedness theory for critical Fourier-Lebesgue spaces and supercritical Sobolev spaces, enabling construction of BF evolutions from NLS solutions with Dirac-mass-type data and establishing unique continuation across singular times via Hasimoto correspondence. The authors demonstrate self-similar and multi-corner BF solutions, quantify Fourier-energy growth and Talbot-type reconstructions at rational times, and uncover intermittent and multifractal trajectories described by Riemann-type limit functions. They also extend these analyses to periodic settings, proving wave-operator-type results and asymptotic completeness for a periodic NLS with time-dependent nonlinearity, which translates into criteria for generating and continuing BF singularities. Overall, the work provides a rigorous bridge between dispersive PDE turbulence and vortex-filament dynamics, with implications for understanding energy cascades and reconnection phenomena in fluids and superfluids.

Abstract

In the last three decades there has been an intense activity on the exploration of turbulent phenomena of dispersive equations, as for instance the growth of Sobolev norms since the work of Bourgain in the 90s. In general the 1D cubic Schrödinger equation has been left aside because of its complete integrability. In a series of papers of the last six years that we survey here for the special issue of the ICMP 2024 ([12],[13],[14],[15],[16],[7],[8]), we considered, together with the 1D cubic Schrödinger equation, the binormal flow, which is a geometric flow explicitly related to it. We displayed rigorously a large range of complex behavior as creation of singularities and unique continuation, Fourier growth, Talbot effects, intermittency and multifractality, justifying in particular some previous numerical observations. To do so we constructed a class of well-posedness for the 1D cubic Schrödinger equation included in the critical Fourier-Lebesgue space and in supercritical Sobolev spaces with respect to scaling. Last but not least we recall that the binormal flow is a classical model for the dynamics of a vortex filament in a 3D fluid or superfluid, and that vortex motions are a key element of turbulence.

Paper Structure

This paper contains 20 sections, 12 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Vortex filaments dynamics and the binormal flow
  3. Links between the binormal flow and Schrödinger equations
  4. Self-similar type solutions generating one singularity
  5. Turbulent features of solutions generating several singularities
  6. Existence of solutions generating several singularities and unique continuation
  7. Sketch of the proof of Theorem \ref{['thNLScrit']}
  8. Sketch of the proof of Theorem \ref{['thevpoly']}
  9. Fourier growth
  10. Sketch of the proof of Theorem \ref{['thgrowth']}
  11. Talbot effects
  12. Sketch of the proof of Theorem \ref{['thTalbotlin']}
  13. Intermittency and multifractality
  14. Sketch of the proof of Theorem \ref{['thmultifrac']}
  15. Sketch of the proof of Theorem \ref{['thBENV']} for $\alpha\in(\frac{1}{2},\frac{3}{4}]$.
  16. ...and 5 more sections

Key Result

Theorem 2.1

(Construction of critical NLS solutions, BanicaVega2022) Let $s>\frac{1}{2},0<\gamma<1$, $\{\alpha_k\}\in l^{2,s}$, i.e. $\|\{\alpha_k\}\|_{l^{2,s}}^2:=\sum_k|\alpha_k|^2(1+k^2)^s<\infty$. There exists $T>0$ and a unique solution of CubicNLS on $(0,T)$ of the form with and Moreover, if $s\geq 1$ then the solution can be extended to $(0,\infty)$.

Theorems & Definitions (14)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 5.1
  • Theorem 5.2
  • Theorem 6.1
  • Theorem 6.2
  • ...and 4 more