44 research outputs found
Transport of gaussian measures by the flow of the nonlinear Schr\"odinger equation
We prove a new smoothing type property for solutions of the 1d quintic
Schr\"odinger equation. As a consequence, we prove that a family of natural
gaussian measures are quasi-invariant under the flow of this equation. In the
defocusing case, we prove global in time quasi-invariance while in the focusing
case because of a blow-up obstruction we only get local in time
quasi-invariance. Our results extend as well to generic odd power
nonlinearities.Comment: Presentation improve
Existence and Stability of standing waves for supercritical NLS with a Partial Confinement
We prove the existence of orbitally stable ground states to NLS with a
partial confinement together with qualitative and symmetry properties. This
result is obtained for nonlinearities which are -supercritical, in
particular we cover the physically relevant cubic case. The equation that we
consider is the limit case of the cigar-shaped model in BEC.Comment: Revised version, accepted on Comm. Math. Physic
On the orbital stability for a class of nonautonomous NLS
Following the original approach introduced by T. Cazenave and P.L. Lions in
\cite{CaLi} we prove the existence and the orbital stability of standing waves
for the following class of NLS: \label{intr1} i\partial_t u+ \Delta u - V(x) u
+ Q(x) u|u|^{p-2}=0, \hbox{} (t,x) \in \R\times \R^n, \hbox{} 2<p<2+\frac 4n
and \label{intr2} i\partial_t u - \Delta^2 u - V(x) u + Q(x) u|u|^{p-2}=0,
\hbox{} (t,x) \in \R\times \R^n, \hbox{} 2<p<2+\frac 8n under suitable
assumptions on the potentials and . More precisely we assume
and for
a suitable . The main point is the analysis of the compactness of
minimiziang sequences to suitable constrained minimization problems related to
\eqref{intr1} and \eqref{intr2}
Modified energies for the periodic generalized KdV equation and applications
We construct modified energies for the generalized KdV equation. As a
consequence, we obtain quasi-invariance of the high order Gaussian measures
along with regularity on the corresponding Radon-Nykodim density, as well
as new bounds on the growth of the Sobolev norms of the solutions
The nonlinear Schrödinger equation ground states on product spaces
We study the nature of the nonlinear Schrödinger equation ground states on the product spaces Rn x Mk ,
where Mk is a compact Riemannian manifold. We prove that for small L2 masses the ground states
coincide with the corresponding Rn ground states. We also prove that above a critical mass the ground
states have nontrivial Mk dependence. Finally, we address the Cauchy problem issue, which transforms
the variational analysis into dynamical stability results
The Strauss conjecture on asymptotically flat space-times
By assuming a certain localized energy estimate, we prove the existence
portion of the Strauss conjecture on asymptotically flat manifolds, possibly
exterior to a compact domain, when the spatial dimension is 3 or 4. In
particular, this result applies to the 3 and 4-dimensional Schwarzschild and
Kerr (with small angular momentum) black hole backgrounds, long range
asymptotically Euclidean spaces, and small time-dependent asymptotically flat
perturbations of Minkowski space-time. We also permit lower order perturbations
of the wave operator. The key estimates are a class of weighted Strichartz
estimates, which are used near infinity where the metrics can be viewed as
small perturbations of the Minkowski metric, and the assumed localized energy
estimate, which is used in the remaining compact set.Comment: Final version, to appear in SIAM Journal on Mathematical Analysis. 17
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