### Recent Submissions

• #### Sharp local smoothing estimates for Fourier integral operators ﻿

(2019)
The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which ...
• #### ENERGY CONSERVATION FOR 2D EULER WITH VORTICITY IN L(log L)α* ﻿

(2022-01-01)
In these notes we discuss the conservation of the energy for weak solutions of the twodimensional incompressible Euler equations. Weak solutions with vorticity in (Formula presented) with p > 3/2 are always conservative, ...
• #### On the Hausdorff dimension of Riemann's non-differentiable function ﻿

(2021-01-01)
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 ...
• #### A pseudospectral method for the one-dimensional fractional Laplacian on R ﻿

(2021-01-15)
In this paper, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over R, we map ...

(2021-12-01)
We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz s-energy on the sphere Sd. Our results are based on bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished ...
• #### Self-adjointness of two-dimensional Dirac operators on corner domains ﻿

(2021-01-01)
We investigate the self-adjointness of the two-dimensional Dirac operator D, with quantum-dot and Lorentz-scalar i-shell boundary conditions, on piecewise C2 domains (with finitely many corners). For both models, we prove ...
• #### Dirac Operators and Shell Interactions: A Survey ﻿

(2021-01-01)
In this survey we gather recent results on Dirac operators coupled with δ-shell interactions. We start by discussing recent advances regarding the question of self-adjointness for these operators. Afterwards we switch to ...
• #### On the regularity of solutions to the k-generalized korteweg-de vries equation ﻿

(2018-01-01)
This work is concerned with special regularity properties of solutions to the k-generalized Korteweg-de Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial ...
• #### The Frisch–Parisi formalism for fluctuations of the Schrödinger equation ﻿

(2022)
We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of \$\int\lvert x \rvert^{2\delta}\lvert ...
• #### On the Schrödinger map for regular helical polygons in the hyperbolic space ﻿

(2022-01-01)
The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the ...
• #### Pointwise Convergence of the Schr\"odinger Flow ﻿

(2021-01)
In this paper we address the question of the pointwise almost everywhere limit of nonlinear Schr\"odinger flows to the initial data, in both the continuous and the periodic settings. Then we show how, in some cases, certain ...
• #### Quasi-invariance of low regularity Gaussian measures under the gauge map of the periodic derivative NLS ﻿

(2022-01-01)
The periodic DNLS gauge is an anticipative map with singular generator which revealed crucial in the study of the periodic derivative NLS. We prove quasi-invariance of the Gaussian measure on L2(T) with covariance [1+(−Δ)s]−1 ...
• #### Numerical approximation of the fractional Laplacian on R using orthogonal families ﻿

(2020-12-01)
In this paper, using well-known complex variable techniques, we compute explicitly, in terms of the F12 Gaussian hypergeometric function, the one-dimensional fractional Laplacian of the complex Higgins functions, the complex ...
• #### Vortex Filament Equation for a regular polygon in the hyperbolic plane ﻿

(2020-07-09)
The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and ...
• #### Riemann's non-differentiable function and the binormal curvature flow ﻿

(2020-07-14)
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object ...
• #### A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator ﻿

(2020-01-01)
We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials V of Coulomb type: we characterise ...
• #### Uniqueness properties of solutions to the Benjamin-Ono equation and related models ﻿

(2020-03-15)
We prove that if u1,u2 are real solutions of the Benjamin-Ono equation defined in (x,t)∈R×[0,T] which agree in an open set Ω⊂R×[0,T], then u1≡u2. We extend this uniqueness result to a general class of equations of Benjamin-Ono ...
• #### Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation ﻿

(2020-05-01)
We give the asymptotics of the Fourier transform of self-similar solutions for the modified Korteweg-de Vries equation. In the defocussing case, the self-similar profiles are solutions to the Painlevé II equation; although ...
• #### Evolution of Polygonal Lines by the Binormal Flow ﻿

(2020-06-01)
The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schrödinger equation on R in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. Finally ...
• #### On the energy of critical solutions of the binormal flow ﻿

(2020-07-02)
The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic ...