Susana Gutierrez - University of Edinburgh

Susana Gutierrez
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Name
Susana Gutierrez
Affiliation
University of Edinburgh
City
Edinburgh
Country
United Kingdom

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Mathematics - Analysis of PDEs (8)
 
Mathematics - Classical Analysis and ODEs (3)

Publications Authored By Susana Gutierrez

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. Read More

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 Schr\"odinger 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 $1\leq k\leq d-1$. Read More

We establish smoothing estimates in the framework of hyperbolic Sobolev spaces for the velocity averaging operator $\rho$ of the solution of the kinetic transport equation. If the velocity domain is either the unit sphere or the unit ball, then, for any exponents $q$ and $r$, we find a characterisation of the exponents $\beta_+$ and $\beta_-$, except possibly for an endpoint case, for which $D_+^{\beta_+}D_-^{\beta_-} \rho$ is bounded from space-velocity $L^2_{x,v}$ to space-time $L^q_tL^r_x$. Here, $D_+$ and $D_-$ are the classical and hyperbolic derivative operators, respectively. Read More

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. Read More

We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present a new approach to proving the non-endpoint cases using multilinear analysis. Read More

We prove a logarithmic convexity result for exponentially weighted $L^2$-norms of solutions to electromagnetic Schr\"odinger equation, without needing to assume smallness of the magnetic potential. As a consequence, we can prove a unique continuation result in the style of the Hardy uncertainty principle, which generalizes the analogous theorems which have been recently proved by Escauriaza, Kenig, Ponce and Vega. Read More

In this paper we will study the stability properties of self-similar solutions of 1-d cubic NLS equations with time-dependent coefficients of the form iu_t+u_{xx}+\frac{u}{2} (|u|^2-\frac{A}{t})=0, A\in \R (cubic). The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation iv_t+ v_{xx} +\frac{v}{2t}(|v|^2-A)=0. Read More

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. Read More

2004Apr

We investigate the formation of singularities in a self-similar form of regular solutions of the Localized Induction Approximation (also referred as to the binormal flow). This equation appears as an approximation model for the self-induced motion of a vortex filament in an inviscid incompressible fluid. The solutions behave as 3d-logarithmic spirals at infinity. Read More