1,720,970 research outputs found
The modified energy technique and applications
No full textThe main aim of this article is to overview a series of results obtained by the author in collaboration with F. Planchon and N. Tzvetkov. The main point is to show how the introduction of suitable modified energies, well adapted to the problem of interest, can be useful in order to get deterministic informations on the long-time behaviour of solutions, as well as informations on the transport of Gaussian measures along the corresponding deterministic flow
Max-Min characterization of the mountain pass energy level for a class of variational problems
No full textTransport of Gaussian measures by the flow of the nonlinear Schrödinger equation
Full text linkWe prove a new smoothing type property for solutions of the 1d quintic Schrödinger 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 we only get local in time quasi-invariance because of a blow-up obstruction. Our results extend as well to generic odd power nonlinearities
Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space
No full textWe consider the nonlinear Schrödinger equation with multiplicative spatial white noise and an arbitrary polynomial nonlinearity on the two-dimensional full space domain. We prove global well-posedness by using a gauge-transform introduced by Hairer and Labbé (Electron Commun Probab 20(43):11, 2015) and constructing the solution as a limit of solutions to a family of approximating equations. This paper extends a previous result by Debussche and Martin (Nonlinearity 32(4):1147–1174, 2019) with a sub-quadratic nonlinearity
Standing waves for a class of nonlinear Schr"odinger equations with potentials in .
No full textWe prove the existence of standing waves
to the following family of nonlinear Sch\"odinger equations:
provided that is small,
when ,
when and
is assumed to have a sublevel with positive and finite measure
Orbital stability of ground states for a Sobolev critical Schrödinger equation
Full text linkWe study the existence of ground state standing waves, of prescribed mass, for the nonlinear Schrödinger equation with mixed power nonlinearities i∂tv+Δv+μv|v|q−2+v|v|2javax.xml.bind.JAXBElement@500a707a−2=0,(t,x)∈R×RN, where N≥3, v:R×RN→C, μ>0, 2<2+4/N and 2⁎=2N/(N−2) is the critical Sobolev exponent. We show that all ground states correspond to local minima of the associated Energy functional. Next, despite the fact that the nonlinearity is Sobolev critical, we show that the set of ground states is orbitally stable. Our results settle a question raised by N. Soave [35]
On a Minimization Problem Involving the Critical Sobolev Exponent.
No full textWe study the following minimization problem:
in any dimension and under suitable assumptions on .
\noindent Mainly we assume that
belongs to the Lorentz space
and the set
has positive Lebesgue measure. Notice that this last condition is
satisfied when the set has a nontrivial interior part
(in fact this is the typical assumption imposed in the literature on
the set )
Global Dynamics of the 2d NLS with White Noise Potential and Generic Polynomial Nonlinearity
Full text linkUsing an approach introduced by Hairer–Labbé we construct a unique global dynamics for the NLS on T2 with a white noise potential and an arbitrary polynomial nonlinearity. We build the solutions as a limit of classical solutions (up to a phase shift) of the same equation with smoothed potentials. This is an improvement on previous contributions of us and Debussche–Weber dealing with quartic nonlinearities and cubic nonlinearities respectively. The main new ingredient are space–time estimates for the approximate nonlinear solutions exploiting the time averaging effect for dispersive equations (the previous works were based only on fixed time spatial estimates)
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