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Harmonic Analysis and Differential Equations: New Challenges

Periodic Reporting for period 2 - HADE (Harmonic Analysis and Differential Equations: New Challenges)

Reporting period: 2017-06-01 to 2018-11-30

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 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. 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.

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 paves 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.

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 se
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 the progress has been substantial, both with respect to Frisch-Parisi conjecture and with respect to the pseudo-randomness of the trajectories of regular polygons. The existence of particular solutions given by polygonal helices have opened an unexpected path that we are willing to explore.

In part (2) the results concerning the Dirac operator are very satisfactory 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. Recently, we have already written a paper on the Aharanov-Bohm potential that is one of the objectives of the ERC proposal.

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. And finally, we gave the first relevant step forward in the direction of Nazarov’s uncertainty principle.
The result with Arrizabalaga and Mas, that has appeared in Comm. Math. Phys. opens a new field of research that connects two different groups of researchers. One of them is the group in Mathematical Physics that studies the so-called delta interactions hamiltonians as understood in the classical monography by S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden. And the other, is the group in Harmonic Analysis working on the boundedness properties of the restriction of Cauchy operator (and its generalizations) to very rough sets of codimension one. I think that the variational problem proposed in our paper, that links the classical “positive” Newtonian potential and the classical “nonpositive” (Clifford) Cauchy operator is just the tip of an iceberg that we have just started to explore. Even the case of two dimensions seems extremely rich as recent work by Le Trust and Ourmiere-Bonafos indicates.

The paper with De La Hoz that will appear in the J. of Nonlinear Science, and that we will carefully explained in item 1.1 is well beyond the state of the art. Among other things it strongly suggests that the Vortex Filament Equation is one of the simplest models that exhibits all the desired properties that a model in turbulence, as understood by Uriel Frisch in his classical book on the topic, should have.

Finally, the recent result with Banica shows the path to follow to rigorously prove that the Vortex Filament Equation is indeed a fundamental toy model to understand some of key aspects of real turbulent flows.