Linear and NonLinear Waves
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Invariant measures for the dnls equation
(Mathematics of Wave Phenomena, 20201002)We describe invariant measures associated to the integrals of motion of the periodic derivative nonlinear Schr\"odinger equation (DNLS) constructed in \cite{MR3518561, Genovese2018}. The construction works for small $L^2$ ... 
Magnetic domaintwin boundary interactions in NiMnGa
(Acta Materialia, 202004)The stress required for the propagation of twin boundaries in a sample with fine twins increases monotonically with ongoing deformation. In contrast, for samples with a single twin boundary, the stress exhibits a plateau ... 
Sensitivity of twin boundary movement to sample orientation and magnetic field direction in NiMnGa
(Acta Materialia, 2019)When applying a magnetic field parallel or perpendicular to the long edge of a parallelepiped Ni MnGa stick, twin boundaries move instantaneously or gradullay through the sample. We evaluate the sample shape dependence ... 
A sharp lorentzinvariant strichartz norm expansion for the cubic wave equation in \mathbb{R}^{1+3}
(The Quarterly Journal of Mathematics, 2020)We provide an asymptotic formula for the maximal Stri chartz norm of small solutions to the cubic wave equation in Minkowski space. The leading coefficient is given by Foschi’s sharp constant for the linear Strichartz ... 
Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory
(SIAM Journal on Numerical Analysis, 2019)Abstract. We develop a unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations ∂tu − Lσ,μ[φ(u)] = f(x,t) in RN × (0,T), where Lσ,μ is a general ... 
Bayesian approach to inverse scattering with topological priors
(Inverse Problems, 2020)We propose a Bayesian inference framework to estimate uncertainties in inverse scattering problems. Given the observed data, the forward model and their uncertainties, we find the posterior distribution over a finite ... 
The Schrödinger equation and Uncertainty Principles
(202009)The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First we consider the elliptic Schrödinger ... 
On the smallness condition in linear inviscid damping: monotonicity and resonance chains
(Nonlinearity, 2020)We consider the effects of mixing by smooth bilipschitz shear flows in the linearized Euler equations on $\mathbb{T}_{L}\times\mathbb{R}$. Here, we construct a model which is closely related to a small high frequency ... 
A geometric and physical study of Riemann's nondifferentiable function
(20200708)Riemann's nondifferentiable function is a classic example of a continuous but almost nowhere differentiable function, whose analytic regularity has been widely studied since it was proposed in the second half of the 19th ... 
Pseudospectral Methods for the Fractional Laplacian on R
(20200702)In this thesis, first, we propose a novel pseudospectral method to approximate accu rately and efficiently the fractional Laplacian without using truncation. More pre cisely, given a bounded regular function defined over ... 
Vortex Filament Equation for some Regular Polygonal Curves
(20200615)One of the most interesting phenomena in fluid literature is the occurrence and evolution of vortex filaments. Some of their examples in the real world are smoke rings, whirlpools, and tornadoes. For an ideal fluid, there ... 
Exact Constructions in the (Nonlinear) Planar Theory of Elasticity: From Elastic Crystals to Nematic Elastomers
(Archive for Rational Mechanics and Analysis, 202007)In this article we deduce necessary and sufficient conditions for the presence of “Contitype”, highly symmetric, exactly stressfree constructions in the geometrically nonlinear, planar nwell problem, generalising results ... 
Geometric differentiability of Riemann's nondifferentiable function
(Advances in Mathematics, 202006)Riemann’s nondifferentiable 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 ... 
A Hardytype inequality and some spectral characterizations for the DiracCoulomb operator
(Revista Matemática Complutense, 20190702)We prove a sharp Hardytype inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrixvalued potentials V of Coulomb type: we characterise ... 
Evolution of Polygonal Lines by the Binormal Flow
(Springer Nature Switzerland AG 2020, 20200205)The aim of this paper is threefold. First we display solutions of the cubic nonlinear Schr ̈odinger equation on R in link with initial data a sum of Dirac masses. Secondly we show a Talbot effect for the same equation. ... 
Some lower bounds for solutions of Schrodinger evolutions
(SIAM J. MATH. ANAL., 20190821)We present some lower bounds for regular solutions of Schr odinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, ... 
Uniqueness Properties of Solutions to the BenjaminOno equation and related models
(20190131)We prove that if u1, u2 are 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 ... 
Absence of eigenvalues of twodimensional magnetic Schr ̈odinger operators
(20171017)By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding twodimensional Schr ̈odinger 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 W1,8 around a carefully chosen, two term ansatz. Such knowledge ... 
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 ...