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A Nonlinear Plancherel Theorem with Applications to Global Well-Posedness for the Defocusing Davey-Stewartson Equation and to the Inverse Boundary Value Problem of Calderón
We prove a Plancherel theorem for a nonlinear Fourier transform in two
dimensions arising in the Inverse Scattering method for the defocusing
Davey-Stewartson II equation. We then use it to prove global well-posedness and
scattering in for defocusing DSII. This Plancherel theorem also implies
global uniqueness in the inverse boundary value problem of Calder\'on in
dimension , for conductivities \sigma>0 with .
The proof of the nonlinear Plancherel theorem includes new estimates on
classical fractional integrals, as well as a new result on -boundedness of
pseudo-differential operators with non-smooth symbols, valid in all dimensions
Explicit approximate controllability of the Schr\"odinger equation with a polarizability term
We consider a controlled Schr\"odinger equation with a dipolar and a
polarizability term, used when the dipolar approximation is not valid. The
control is the amplitude of the external electric field, it acts non linearly
on the state. We extend in this infinite dimensional framework previous
techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in
finite dimension. We consider a highly oscillating control and prove the
semi-global weak stabilization of the averaged system using a Lyapunov
function introduced by Nersesyan. Then it is proved that the solutions of the
Schr\"odinger equation and of the averaged equation stay close on every finite
time horizon provided that the control is oscillating enough. Combining these
two results, we get approximate controllability to the ground state for the
polarizability system
Persistence of solutions to the Derivative Nonlinear Schr\"odinger equation in weighted Sobolev spaces
In this paper we show the persistence property for solutions of the
derivative nonlinear Schr\"odinger equation with initial data in weighted
Sobolev spaces , .Comment: 13 page
On applications of modulation spaces to dispersive equations (Harmonic Analysis and Nonlinear Partial Differential Equations)
"Harmonic Analysis and Nonlinear Partial Differential Equations". July 4~6, 2016. edited by Hideo Kubo and Hideo Takaoka. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed
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