I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow : which preserve the length parametrization. Above is a curve in , the arclength parameter, and denote the temporal variable. This flow appeared for the first time in the work of Da Rios (1906) as a crude approximation to the evolution of a vortex filament under Euler equation, and it is intimately related to the focusing cubic nonlinear Schrödinger...
In the talk we shall present some recent results obtained with F. Merle about compactness of blow up solutions of the critical nonlinear Schrödinger equation for initial data in . They are based on and are complementary to some previous work of J. Bourgain about the concentration of the solution when it approaches to the blow up time.
We prove unique continuation for solutions of the inequality , a connected set contained in and is in the Morrey spaces , with and . These spaces include for (see [H], [BKRS]). If , the extra assumption of being small enough is needed.
We prove bilinear virial identities for the nonlinear Schrödinger equation, which are extensions of the Morawetz interaction inequalities. We recover and extend known bilinear improvements to Strichartz inequalities and provide applications to various nonlinear problems, most notably on domains with boundaries.
We extend in a nonlinear context previous results obtained in [8], [9], [10]. In particular we present a precised version of Morawetz type estimates and a uniqueness criterion for solutions to subcritical NLS.
Let us consider in a domain Ω of Rn solutions of the differential inequality
|Δu(x)| ≤ V(x)|u(x)|, x ∈ Ω,
where V is a non smooth, positive potential.
We are interested in global unique continuation properties. That means that u must be identically zero on Ω if it vanishes on an open subset of Ω.
The purpose of this note is twofold. First it is a corrigenda of our paper [RV1]. And secondly we make some remarks concerning the interpolation properties of Morrey spaces.
We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions form a family of evolving regular curves in that develop a singularity in finite time, indexed by a parameter . We consider curves that are small regular perturbations of for a fixed time . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...
We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy’s version of the uncertainty principle. We also obtain corresponding results for heat evolutions.
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