arXiv subjects

Daniel Eceizabarrena

Publications and source records attributed to Daniel Eceizabarrena.

14 recordsLinked to original sources

arXiv2026

Almost sure pointwise convergence for the 2D periodic quintic NLS

We prove probabilistic pointwise convergence to the initial datum for the 2D periodic quintic NLS for data in $H^s(\mathbb T^2)$ with $s > 0$. This is an improvement with respect to the deterministic setting, in which convergence is known to fail if $s < 1/3$. The proof is based on a nonlinear maximal characterization for convergence, Bourgain's linear-nonlinear decomposition and the corresponding nonlinear smoothing for which we employ random tensor estimates.

math.AP↗

arXiv2025

Uniform periodic counterexamples to Carleson's convergence problem with polynomial symbols

In Carleson's convergence problem for dispersive equations $i\, \partial_t u + P(D)u=0$ in the periodic setting $\mathbb T^d$, we prove that the Sobolev exponent $d/(2(d+1))$ is necessary for any non-singular polynomial symbol $P$, including the natural powers of the Laplacian $Δ^k$. This is in contrast with the results known in the Euclidean case, in which for symbols $P(ξ) = |ξ|^a$ with $a > 1$ the exponent $d/(2(d+1))$ is sufficient, but we do not know if it is necessary.

math.AP↗

arXiv2025

Bourgain's counterexample in the sequential convergence problem for the Schrödinger equation

We study the problem of pointwise convegence for the Schrödinger operator on $\mathbb R^n$ along time sequences. We show that the sharp counterexample to the sequential Schrödinger maximal estimate given recently by Li, Wang and Yan based in the construction by Lucà and Rogers can also be achieved with the construction of Bourgain, and we extend it to the fractal setting.

math.AP↗

arXiv2025

Multifractality and intermittency in the limit evolution of polygonal vortex filaments

With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality and intermittency of the family of generalized Riemann's non-differentiable functions \begin{equation} R_{x_0}(t) = \sum_{n \neq 0} \frac{e^{2πi ( n^2 t + n x_0 ) } }{n^2}, \qquad x_0 \in [0,1]. \end{equation} These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow. When $x_0$ is rational, we show that $R_{x_0}$ is multifractal and intermittent by completely determining the spectrum of singularities of $R_{x_0}$ and computing the $L^p$ norms of its Fourier high-pass filters, which are analogues of structure functions. We prove that $R_{x_0}$ has a multifractal behavior also when $x_0$ is irrational. The proofs rely on a careful design of Diophantine sets that depend on $x_0$, which we study by crucially using the Duffin-Schaeffer theorem and the Mass Transference Principle.

math.AP↗

arXiv2024

Multifractality and polygonal vortex filaments

In this proceedings article we survey the results in [5] and their motivation, as presented at the 50th Journées EDP 2024. With the aim of quantifying turbulent behaviors of vortex filaments, we study the multifractality of a family of generalized Riemann's non-differentiable functions. These functions represent, in a certain limit, the trajectory of regular polygonal vortex filaments that evolve according to the binormal flow, the classical model for vortex filaments dynamics. We explain how we determined their spectrum of singularities through a careful design of Diophantine sets, which we study by using the Duffin-Schaeffer theorem and the Mass Transference Principle.

math.AP↗

arXiv2022

Convergence over fractals for the periodic Schrödinger equation

We consider a fractal refinement of Carleson's problem for pointwise convergence of solutions to the periodic Schrödinger equation to their initial datum. For $α\in (0,d]$ and \[ s < \frac{d}{2(d+1)} (d + 1 - α), \] we find a function in $H^s(\mathbb{T}^d)$ whose corresponding solution diverges in the limit $t \to 0$ on a set with strictly positive $α$-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that \[ s > \frac{d}{2(d+2)}\left( d+2-α\right) \] is sufficient for the solution corresponding to every datum in $H^s(\mathbb T^d)$ to converge to such datum $α$-almost everywhere.

math.AP↗

arXiv2022

Pointwise convergence over fractals for dispersive equations with homogeneous symbol

We study the fractal pointwise convergence for the equation $i\hbar\partial_tu + P(D)u = 0$, where the symbol $P$ is real, homogeneous and non-singular. We prove that for initial data $f\in H^s(\mathbb{R}^n)$ with $s>(n-α+1)/2$ the solution $u$ converges to $f$ $\mathcal{H}^α$-a.e, where $\mathcal{H}^α$ is the $α$-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of $P$. On the other hand, for a family of polynomials $P$ and given $α$, we exploit a Talbot-like effect to construct initial data whose solutions $u$ diverge in sets of Hausdorff dimension $α$. To compute the dimension of the sets of divergence, we adopt the Mass Transference Principle from Diophantine approximation. We also construct counterexamples for quadratic symbols like the saddle to show that our positive results are sometimes best possible.

math.AP↗

arXiv2022

An analytical study of flatness and intermittency through Riemann's non-differentiable functions

In the study of turbulence, intermittency is a measure of how much Kolmogorov's theory of 1941 deviates from experiments. It is quantified with the flatness of the velocity of the fluid, usually based on structure functions in the physical space. However, it can also be defined with Fourier high-pass filters. Experimental and numerical simulations suggest that the two approaches do not always give the same results. Our purpose is to compare them from the analytical point of view of functions. We do that by studying generalizations of Riemann's non-differentiable function, yielding computations that are related to some classical problems in Fourier analysis. The conclusion is that the result strongly depends on regularity. To visualize this, we establish an analogy between these generalizations and the influence of viscosity in turbulent flows. This article is motivated by the mathematical works on the multifractal formalism and the discovery of Riemann's non-differentiable function as a trajectory of polygonal vortex filaments.

math-ph↗

arXiv2021

Counterexamples for the fractal Schrödinger convergence problem with an intermediate space trick

We construct counterexamples for the fractal Schrödinger convergence problem by combining a fractal extension of Bourgain's counterexample and the intermediate space trick of Du--Kim--Wang--Zhang. We confirm that the same regularity as Du's counterexamples for weighted $L^2$ restriction estimates is achieved for the convergence problem. To do so, we need to construct the set of divergence explicitly and compute its Hausdorff dimension, for which we use the Mass Transference Principle, a technique originated from Diophantine approximation.

math.AP↗

arXiv2021

Intermittency of Riemann's non-differentiable function through the fourth-order flatness

Riemann's non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch-Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure functions in the physical space and with high-pass filters in the Fourier space.

math.CA↗

arXiv2021

On the Hausdorff dimension of Riemann's non-differentiable function

Recent findings show that the classical Riemann's non-differentiable function has a physical and geometric nature as the irregular trajectory of a polygonal vortex filament driven by the binormal flow. In this article, we give an upper estimate of its Hausdorff dimension. We also adapt this result to the multifractal setting. To prove these results, we recalculate the asymptotic behavior of Riemann's function around rationals from a novel perspective, underlining its connections with the Talbot effect and Gauss sums, with the hope that it is useful to give a lower bound of its dimension and to answer further geometric questions.

math.CA↗

arXiv2021

The Talbot effect as the fundamental solution to the free Schrödinger equation

The Talbot effect is usually modeled using the Helmholtz equation, but its main experimental features are captured by the solution to the free Schrödinger equation with the Dirac comb as initial datum. This simplified description is a consequence of the paraxial approximation in geometric optics. However, it is a heuristic approximation that is not mathematically well justified, so K. I. Oskolkov raised the problem of "mathematizing" it. We show that it holds exactly in the sense of distributions.

math.AP↗

arXiv2020

Geometric differentiability of Riemann's non-differentiable function

Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity.

math.CA↗

arXiv2019

Some geometric properties of Riemann's non-differentiable function

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.

math.CA↗