# Harmonic Analysis and Differential Equations: New Challenges

## Periodic Reporting for period 4 - HADE (Harmonic Analysis and Differential Equations: New Challenges)

Reporting period: 2020-06-01 to 2021-11-30

The research we have developed has three main axes:
(1) The Vortex Filament Equation;
(2) Relativistic and Non-relativistic Critical Electromagnetic Hamiltonians;
(3) Uncertainty Principles and Applications.

The Vortex Filament Equation (VFE) is a partial differential equation obtained as a simplified model for the dynamics of an ideal fluid whose vorticity is concentrated on a curve (the filament), as for example the smoke ring or the bath tub vortex. VFE is a nonlinear Schrödinger (NLS) equation as shown using a remarkable transformation found by Hasimoto in 1971: the unknown is a wave function constructed using the curvature and the torsion of the filament.
Our main objective is to show that the complex dynamics of some turbulent flows, as the non-circular jets generated by nozzles with a polygonal shape, can be explained by the Talbot effect. This effect was discovered by Talbot in 1836 in his studies of the patterns that the light creates when it crosses a periodic array of very thin slits. Using Fresnel´s theory of diffraction and the so called paraxial approximation, the Talbot effect is described by a wave function that solves the (linear) Schrödinger equation. The geometric analogy is to take the periodic array of corners of a regular polygon instead of the slits, and to use this polygon as initial datum for VFE.
In a first breakthrough result with F. De La Hoz (Journal of Nonlinear Science, 2018) we give concluding numerical evidence, supported by rigorous analytical results, that at infinitesimal times the corners of the regular polygon do not interact with each other. In other words, the evolution of the tangent vector for very small times is self-similar and is determined just by the angle of one corner. This implies that the right way of understanding the regular polygon is as a superposition of filaments, one for each corner, so that by periodicity the number of filaments becomes infinite.
The interaction of these filaments in later times is qualitatively as the Talbot effect predicts, but due to the non-linear potential important differences appear. The main one is the existence of a transfer of energy. In the case of a filament with just one corner this transfer creates a discontinuity (a jump) in some appropriate norm that measures the interchange energy of the filament. For the regular polygon this discontinuity appears at any rational multiple of the time period and numerically we show that the jumps are infinitely large.
The second breakthrough is with V. Banica. We prove that VFE is well posed for initial data given by a skew polygonal line (Annals of PDE 2020). As a byproduct we obtain that the self-similar solutions of NLS have finite energy, even though when seen as solutions of an initial value problem they are ill posed. This is because of a phase loss that occurs due to the creation of a singularity, in our case a Dirac delta function. Nevertheless, the well-posedness of VFE allows us to continue the solution beyond the singularity time. This answers in the positive to some questions proposed by F. Merle, and J. Bourgain and W. Wang in the 90's.
The third breakthrough has been achieved in collaboration with V. Banica (Annales IHP, 2022) and concerns the proof of the existence of a cascade of energy for filaments with a finite number of corners. This is carefully explained in section 1.1.

Regarding (2), and together with N. Arrizabalaga and A. Mas, we have given a satisfactory answer to the optimality of the spectrum of the so-called delta shell interactions in the relativistic setting. This is one of the milestones of the project that has paved the way to consider the MIT bag model for quark confinement. This model has been used successfully to predict many properties of hadrons, and the equivalent model in two dimensions arises in the study of the graphene. In the last two years of the project we have given a substantial progress concerning the confinement problem, mainly thanks to the hiring of T. Sanz-Perela as postdoc. The outcome is the article arXiv preprint arXiv:2106.08348 submitted to Comm. in Math. Phys and that I consider as one of the main achievements of the project. The paper proposes even more questions that those that are solved. From my point of view, it opens the way to a new and quite promising territory that would need the effort of researchers from other fields. In fact, we are already collaborating with J. Serra (ETH, Zurich) to explore some of the natural paths suggested by that work.

As for (3) and as a major achievement, we proved Hardy’s uncertainty principle (UP) for heat evolutions with L. Escauriaza, C.E. Kenig, and G. Ponce, connecting UP with the uniqueness of parabolic equations. Moreover, a completely new lower bound for the tails of gaussian probability distributions was found. Showing the strength of our result, with A. Fernández-Bertolin, we have partially extended it to solutions in the discrete setting. Fernández-Bertolin has started a collaboration with E. Malinnikova which is very promising as the article appeared in Bull. AMS, 2021 shows. Similarly, we have obtained in collaboration with C.E. Kenig and G. Ponce some unexpected results concerning non-local operators. The most striking one is the one concerning water waves that appear in J. of Functional Anal. in 2020. The approach is completely different and no Carleman estimates are used. In fact, the non-locality is the key property, and more in particular, the role played by the canonical operators of Harmonic Analysis: the Hilbert and Riesz transforms.
The proposal had three main axes:
(1) Vortex Filament Equation (VFE);
(2) Relativistic and Non-relativistic Critical Electromagnetic Hamiltonians;
(3) Uncertainty Principles and Applications;
and the following two main objectives in each of them:

A.1.1 Prove/disprove the Frisch-Parisi conjecture in the VFE setting.
A.1.2 Use VFE and the Schrödinger Map (SM) equation as a generator of pseudo-random numbers, in particular of random points on the unit sphere.
A.2.1 Prove that the sphere is among all the regular and bounded hypersurfaces in three dimensions and under suitably chosen constraints the one that minimizes the norm of the Dirac-Cauchy operator.
A.2.2 Establish a satisfactory well-posedness theory in the relativistic and non-relativistic settings for critical electro-magnetic Hamiltonians.
A.3.1 Prove a discrete version of Hardy’s uncertainty principle.
A.3.2 Develop a theory about uniqueness in the framework of dispersive equations that includes as part of it the uncertainty principles of Beurling and Nazarov.

Regarding part (1) of the proposal we have achieved all the objectives of A.1.1. Even more, we have found new paths that have a very promising future, mainly thanks to the connection we have established between the Talbot effect and some intermittency phenomena exhibited by the dynamics of vortex filaments with corners. As recently exposed in a colloquium at the U. of Lyon 1 (France) and last fall at ICERM in Providence (USA) (http://www.bcamath.org/en/people/lvega/conferences-and-talks/presentations) we propose the Talbot effect as a mechanism that can describe some of the intermittent behavior shown in non-circular jets. Mathematically, the new evidence is due to the relation of the Talbot effect with some fractional uncertainty principles proved with Ponce-Vanegas and Kumar (Trans. AMS, 2022) that naturally appear for describing the fluctuations of delta-moments of free quantum particles. Intermittency is also linked in this case to Frisch-Parisi formalism and it is very reminiscent to the one satisfied by alpha-Levy processes as proved by S. Jaffard. In fact, we think that this connection explains why A.1.2 has turned out to be very challenging and the reason why the progress in this question has been so slow.

In part (2) the results concerning the Dirac operator are almost optimal as will be explained with detail in item 1.1. Concerning classical electromagnetic hamiltonians we realized that it would be better to start looking at non-selfajoint perturbations and to adapt the multiplier technique to this case. At this respect the collaboration with L. Fanelli, D. Krejcirik, and A Laptev has been very satisfactory.

With respect to part (3) we have got three important results. Firstly, we obtained the last piece of our program on the connection between the uncertainty principles and some uniqueness properties of partial differential equations of evolution type. Secondly, we extended one of the fundamental ingredients in that program to the discrete setting. This extension, done with A. Fernández-Bertolin, has turned out to be very robust. Finally, in the PhD thesis of M. Agirre a relevant result in the direction of Nazarov’s uncertainty principle is given. Nevertheless, this work also shows that new ideas seem to be needed to develop a PDE approach to Nazarov’s Uncertainty Principle, something that remains as a challenge. In the positive side, we obtained with C. E. Kenig and G. Ponce some unexpected results on unique continuation for non-local operators that apply to solutions of the water wave problem. The approach is completely different and does not use Carleman estimates.
It is carefully explained in section "Major Achievements"