Analysis of Partial Differential Equations (APDE)
http://hdl.handle.net/20.500.11824/1
2021-10-18T23:24:29ZVortex Filament Equation for a regular polygon in the hyperbolic plane
http://hdl.handle.net/20.500.11824/1350
Vortex Filament Equation for a regular polygon in the hyperbolic plane
de la Hoz, F.; Kumar, S.; Vega, L.
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 both of its ends grow exponentially, which makes the problem more challenging from a numerical point of view. However, with fixed boundary conditions, a finite difference scheme and a fourth-order Runge--Kutta method in time, we show that the numerical solution is in complete agreement with the one obtained from algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with that in the Euclidean case.
2020-07-09T00:00:00ZRiemann's non-differentiable function and the binormal curvature flow
http://hdl.handle.net/20.500.11824/1349
Riemann's non-differentiable function and the binormal curvature flow
Banica, V.; Vega, L.
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 has a non-obvious nonlinear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids.
2020-07-14T00:00:00ZA Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator
http://hdl.handle.net/20.500.11824/1348
A Hardy-type inequality and some spectral characterizations for the Dirac–Coulomb operator
Cassano, B.; Pizzichillo, F.; Vega, L.
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 its eigenvalues in terms of the Birman–Schwinger principle and we bound its discrete spectrum from below, showing that the ground-state energy is reached if and only if V verifies some rigidity conditions. In the particular case of an electrostatic potential, these imply that V is the Coulomb potential.
2020-01-01T00:00:00ZUniqueness properties of solutions to the Benjamin-Ono equation and related models
http://hdl.handle.net/20.500.11824/1347
Uniqueness properties of solutions to the Benjamin-Ono equation and related models
Kenig, C. E.; Ponce, G.; Vega, L.
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 type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. We relate our uniqueness results with those for a water wave problem. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.
2020-03-15T00:00:00Z