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Kink solutions of the binormal flow

Luis Vega — 2003

Journées équations aux dérivées partielles

I shall present some recent work in collaboration with S. Gutierrez on the characterization of all selfsimilar solutions of the binormal flow : X t = X s × X s s which preserve the length parametrization. Above X ( s , t ) is a curve in 3 , s the arclength parameter, and t 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...

Remarks on global existence and compactness for L 2 solutions in the critical nonlinear schrödinger equation in 2D

Luis Vega Gonzalez — 1998

Journées équations aux dérivées partielles

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 L 2 ( 𝐑 2 ) . 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.

Unique continuation for the solutions of the laplacian plus a drift

Alberto RuizLuis Vega — 1991

Annales de l'institut Fourier

We prove unique continuation for solutions of the inequality | Δ u ( x ) | V ( x ) | u ( x ) | , x Ω a connected set contained in R n and V is in the Morrey spaces F α , p , with p ( n - 2 ) / 2 ( 1 - α ) and α < 1 . These spaces include L q for q ( 3 n - 2 ) / 2 (see [H], [BKRS]). If p = ( n - 2 ) / 2 ( 1 - α ) , the extra assumption of V being small enough is needed.

Bilinear virial identities and applications

Fabrice PlanchonLuis Vega — 2009

Annales scientifiques de l'École Normale Supérieure

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.

Scattering for 1D cubic NLS and singular vortex dynamics

Valeria BanicaLuis Vega — 2012

Journal of the European Mathematical Society

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 χ a ( t , x ) form a family of evolving regular curves in 3 that develop a singularity in finite time, indexed by a parameter a > 0 . We consider curves that are small regular perturbations of χ a ( t 0 , x ) for a fixed time t 0 . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...

Hardy's uncertainty principle, convexity and Schrödinger evolutions

Luis EscauriazaCarlos E. KenigG. PonceLuis Vega — 2008

Journal of the European Mathematical Society

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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