30 research outputs found
Local well-posedness for the nonlinear Schr\"odinger equation in the intersection of modulation spaces
We introduce a Littlewood-Paley characterization of modulation spaces and use
it to give an alternative proof of the algebra property, somehow implicitly
contained in Sugimoto (2011), of the intersection for , and
. We employ this algebra property to show the local well-posedness of
the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the above
intersection. This improves Theorem 1.1 by B\'enyi and Okoudjou (2009), where
only the case is considered, and closes a gap in the literature. If and or if and then
and the
above intersection is superfluous. For this case we also reobtain a
H\"older-type inequality for modulation spaces.Comment: 14 page
Endpoint Estimates for N-dimensional Hardy Operators and Their Commutators
In this paper, it is proved that the higher dimensional Hardy operator is
bounded from Hardy space to Lebesgue space. The endpoint estimate for the
commutator generated by Hardy operator and (central) BMO function is also
discussed.Comment: 8 page
A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations
42 pagesInternational audienceWe consider the defocusing nonlinear Schr\"odinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in . Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure