Susana Gutiérrez

Formation of Singularities and Self-Similar Vortex Motion Under the Localized Induction Approximation†

Abstract The dynamical behavior of an isolated vortex filament in three dimensions within the localized induction approximation (LIA) is investigated. It is shown the existence of a uniparametric family of smooth solutions of LIA that generates a corner in finite time. An explicit formula of the solution is provided in terms of the curvature of the initial regular configuration. The behavior of the uniparametric family of solutions with respect to the free parameter is also considered. †Dedicated to the memory of Björn E. J. Dahlberg.

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 80s 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. (2004), Hwang et al. (2005a b) using direct and strong dispersive effects. Here, we use the weak dispersion estimates of Castella and Perthame (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.

SELF-SIMILAR SOLUTIONS OF THE LOCALIZED INDUCTION APPROXIMATION: SINGULARITY FORMATION

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.The proofs of the results are strongly based on the existing connection between the binormal flow and certain Schrodinger equations.

A note on restricted weak-type estimates for Bochner-Riesz operators with negative index in ℝⁿ, ≥2

It is shown that the Bochner-Riesz operator on Rn of negative order α is of restricted weak type in the critical points (p0, q0) and (q′ 0, p′0), where 1/q0 = 3/4 + α/2, q0 = p′0/3 for −3/2 < α < 0 in the two-dimensional case and 1/q0 = (n + 1 + 2α)/2n, q0 = (n − 1)p′0/(n + 1), for −(n + 1)/2 < α < −1/2 if n ≥ 3.

On the Strichartz Estimates for the Kinetic Transport Equation

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

On the regularity of averages over spheres for kinetic transport equations in hyperbolic Sobolev spaces

We study the smoothing effect of averaging over spheres for solutions of kinetic transport equations in hyperbolic Sobolev spaces.

Transversal multilinear Radon-like transforms: local and global estimates.

Abstract. We prove local “L-improving” estimates for a class of multilinear Radon-like transforms satisfying a strong transversality hypothesis. As a consequence, we obtain sharp multilinear convolution estimates for measures supported on fully transversal submanifolds of Euclidean space of arbitrary dimension. Motivated by potential applications in diffraction tomography, we also prove global estimates for the same class of Radon-like transforms under a natural homogeneity assumption.

Averages over Spheres for Kinetic Transport Equations with Velocity Derivatives in the Right-Hand Side

We prove estimates in hyperbolic Sobolev spaces

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

H^{s,\delta}(R^{1+d})

Non trivial L q solutions to the Ginzburg-Landau equation

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