Analysis of Partial Differential Equations (APDE)
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El efecto de Talbot: de la óptica a la ecuación de Schrödinger
(TEMat, 201707)El objetivo de este artículo es dar a conocer un bello efecto óptico que se denomina efecto de Talbot. Primero, describiremos el fenómeno y comentaremos su descubrimiento a mediados del siglo XIX. A continuación, analizaremos ... 
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part II: Schemes and experiments
(SIAM Journal on Numerical Analysis, 2018)\noindent We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u\mathfrak{L}[\varphi(u)]=f(x,t) \qquad\text{in}\qquad ... 
Uniqueness of degreeone Ginzburg–Landau vortex in the unit ball in dimensions N ≥ 7
(Comptes Rendus Mathematique, 20180901)For ε>0, we consider the GinzburgLandau functional for RNvalued maps defined in the unit ball BN⊂RN with the vortex boundary data x on ∂BN. In dimensions N≥7, we prove that for every ε>0, there exists a unique global ... 
Shear flow dynamics in the BerisEdwards model of nematic liquid crystals
(Proceedings of the Royal Society AMathematical, Physical and Engineering Sciences, 20180214)We consider the BerisEdwards model describing nematic liquid crystal dynamics and restrict to a shear flow and spatially homogeneous situation. We analyze the dynamics focusing on the effect of the flow. We show that in ... 
Dynamics and flow effects in the BerisEdwards system modeling nematic liquid crystals
(Archive for Rational Mechanics and Analysis, 20180810)We consider the BerisEdwards system modelling incompressible liquid crystal flows of nematic type. This couples a NavierStokes system for the fluid velocity with a parabolic reactionconvectiondiffusion equation for the ... 
Some remark on the existence of infinitely many nonphysical solutions to the incompressible NavierStokes equations
(Journal of Mathematical Analysis and Applications, 201810)We prove that there exist infinitely many distributional solutions with infinite kinetic energy to both the incompressible NavierStokes equations in $ \mathbb{R}^2 $ and Burgers equation in $\mathbb{R} $ with vanishing ... 
Proof of an extension of E. Sawyer's conjecture about weighted mixed weaktype estimates
(Mathematische Annalen, 201809)We show that if $v\in A_\infty$ and $u\in A_1$, then there is a constant $c$ depending on the $A_1$ constant of $u$ and the $A_{\infty}$ constant of $v$ such that $$\Big\\frac{ T(fv)} {v}\Big\_{L^{1,\infty}(uv)}\le c\, ... 
On the global wellposedness of a class of 2D solutions for the Rosensweig system of ferrofluids
(Journal of Differential Equations, 201809)We study a class of 2D solutions of a BlochTorrey regularization of the Rosensweig system in the whole space, which arise when the initial data and the external magnetic field are 2D. We prove that such solutions are ... 
Weighted norm inequalities for rough singular integral operators
(Journal of Geometric Analysis, 20180817)In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $T_\Omega$ with $\Omega\in L^\infty(\mathbb{S}^{n1})$ and the BochnerRiesz multiplier at the critical index ... 
Defects in Nematic Shells: a Gammaconvergence discretetocontinuum approach
(Archive for Rational Mechanics and Analysis, 201807)In this paper we rigorously investigate the emergence of defects on Nematic Shells with a genus different from one. This phenomenon is related to a nontrivial interplay between the topology of the shell and the alignment ... 
On the regularity of solutions to the kgeneralized kortewegde vries equation
(Proceedings of the American Mathematical Society, 201807)This work is concerned with special regularity properties of solutions to the kgeneralized Kortewegde Vries equation. In [Comm. Partial Differential Equations 40 (2015), 1336–1364] it was established that if the initial ... 
Absence of eigenvalues of twodimensional magnetic Schroedinger operators
(20180101)By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding twodimensional Schroedinger operator possesses no point ... 
Asymptotics in Fourier space of selfsimilar solutions to the modified Kortewegde Vries equation
(20180706)We give the asymptotics of the Fourier transform of selfsimilar solutions to the modified Kortewegde Vries equation, through a fixed point argument in weighted $W^{1,\infty}$ around a carefully chosen, two term ansatz. ... 
On the improvement of the Hardy inequality due to singular magnetic fields
(20180712)We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the AharonovBohm field in all dimensions and establish a sharp Hardytype ... 
Regularity of fractional maximal functions through Fourier multipliers
(2018)We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions $n \geq 2$. We also show that the spherical fractional maximal function ... 
Variable coefficient Wolfftype inequalities and sharp local smoothing estimates for wave equations on manifolds
(2018)The sharp Wolfftype decoupling estimates of BourgainDemeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian ... 
Sparse bounds for pseudodifferential operators
(Journal d'Analyse Mathématique, 2018)We prove sparse bounds for pseudodifferential operators associated to H\"ormander symbol classes. Our sparse bounds are sharp up to the endpoint and rely on a single scale analysis. As a consequence, we deduce a range of ... 
Spectral stability of Schrödinger operators with subordinated complex potentials
(Journal of Spectral Theory, 20180628)We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the nonnegative semiaxis for all potentials satisfying a formsubordinate smallness condition. By developing ... 
On the Relationship between the OneCorner Problem and the $M$Corner Problem for the Vortex Filament Equation
(Journal of Nonlinear Science, 20180628)In this paper, we give evidence that the evolution of the vortex filament equation (VFE) for a regular Mcorner polygon as initial datum can be explained at infinitesimal times as the superposition of M onecorner initial ... 
Sharp exponential localization for eigenfunctions of the Dirac Operator
(2018)We determine the fastest possible rate of exponential decay at infinity for eigenfunctions of the Dirac operator $\mathcal D_n + \mathbb V$, being $\mathcal D_n$ the massless Dirac operator in dimensions $n=2,3$ and ...