arXiv subjects

Susana Gutiérrez

Publications and source records attributed to Susana Gutiérrez.

5 recordsLinked to original sources

arXiv2020

Self-similar shrinkers of the one-dimensional Landau-Lifshitz-Gilbert equation

The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere $\mathbb{S}^2$, at an exponential rate. In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

math.AP↗

arXiv2019

The Cauchy problem for the Landau-Lifshitz-Gilbert equation in BMO and self-similar solutions

We prove a global well-posedness result for the Landau-Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any dimension. In the one-dimensional case, we characterize the self-similar solutions associated with an initial data given by some ($\mathbb{S}^2$-valued) step function and establish their stability. We also show the existence of multiple solutions if the damping is strong enough. Our arguments rely on the study of a dissipative quasilinear Schrödinger obtained via the stereographic projection and techniques introduced by Koch and Tataru.

math.AP↗

arXiv2017

A sharp $k$-plane Strichartz inequality for the Schrödinger equation

We prove that $$ \|X(|u|^2)\|_{L^3_{t,\ell}}\leq C\|f\|_{L^2(\mathbb{R}^2)}^2, $$ where $u(x,t)$ is the solution to the linear time-dependent Schrödinger equation on $\mathbb{R}^2$ with initial datum $f$, and $X$ is the (spatial) X-ray transform on $\mathbb{R}^2$. In particular, we identify the best constant $C$ and show that a datum $f$ is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions $d$, where the X-ray transform is replaced by the $k$-plane transform for any $1\leq k\leq d-1$. In the process we obtain sharp $L^2(μ)$ bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures $μ$ supported on natural "co-$k$-planarity" sets.

math.CA↗

arXiv2014

Self-similar solutions of the one-dimensional Landau-Lifshitz-Gilbert equation

We consider the one-dimensional Landau-Lifshitz-Gilbert (LLG) equation, a model describing the dynamics for the spin in ferromagnetic materials. Our main aim is the analytical study of the bi-parametric family of self-similar solutions of this model. In the presence of damping, our construction provides a family of global solutions of the LLG equation which are associated to a discontinuous initial data of infinite (total) energy, and which are smooth and have finite energy for all positive times. Special emphasis will be given to the behaviour of this family of solutions with respect to the Gilbert damping parameter. We would like to emphasize that our analysis also includes the study of self-similar solutions of the Schrödinger map and the heat flow for harmonic maps into the 2-sphere as special cases. In particular, the results presented here recover some of the previously known results in the setting of the 1d-Schrödinger map equation.

math.AP↗

arXiv2007

Global existence for a kinetic model of chemotaxis via dispersion and Strichartz estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of 'run and tumble'. The existence of solutions has been obtained in several papers [Chalub et al. Monatsh. Math. 142, 123--141 (2004), Hwang et al. SIAM J. Math. Anal. 36, 1177--1199 (2005)] using direct and strong dispersive effects. Here, we use the weak dispersion estimates of [Castella et al. C. R. Acad. Sci. Paris 322, 535--540 (1996)] to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.

math.AP↗