Linear and Non-Linear Waves
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Static and Dynamical, Fractional Uncertainty Principles
(2021-03)We study the process of dispersion of low-regularity solutions to the Schrödinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get ... -
Invariant measures for the dnls equation
(2020-10-02)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 domain-twin boundary interactions in Ni-Mn-Ga
(2020-04)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 Ni-Mn-Ga
(2019)When applying a magnetic field parallel or perpendicular to the long edge of a parallelepiped Ni- Mn-Ga stick, twin boundaries move instantaneously or gradullay through the sample. We evaluate the sample shape dependence ... -
A sharp lorentz-invariant strichartz norm expansion for the cubic wave equation in \mathbb{R}^{1+3}
(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
(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
(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
(2020-09)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
(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 non-differentiable function
(2020-07-08)Riemann's non-differentiable 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
(2020-07-02)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
(2020-06-15)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 (Non-linear) Planar Theory of Elasticity: From Elastic Crystals to Nematic Elastomers
(2020-07)In this article we deduce necessary and sufficient conditions for the presence of “Conti-type”, highly symmetric, exactly stress-free constructions in the geometrically non-linear, planar n-well problem, generalising results ... -
Geometric differentiability of Riemann's non-differentiable function
(2020-06)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 ... -
A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator
(2019-07-02)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 ... -
Evolution of Polygonal Lines by the Binormal Flow
(2020-02-05)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
(2019-08-21)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 Benjamin-Ono equation and related models
(2019-01-31)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 two-dimensional magnetic Schr ̈odinger operators
(2017-10-17)By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schr ̈odinger operator possesses no point ... -
Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation
(2018-07-06)We give the asymptotics of the Fourier transform of self-similar solutions to the modified Korteweg-de Vries equation, through a fixed point argument in weighted W1,8 around a carefully chosen, two term ansatz. Such knowledge ...
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