201 research outputs found
Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
We consider the defocusing, -critical Hartree equation for the
radial data in all dimensions . We show the global well-posedness
and scattering results in the energy space. The new ingredient in this paper is
that we first take advantage of the term in the localized Morawetz
identity to rule out the possibility of energy concentration, instead of the
classical Morawetz estimate dependent of the nonlinearity.Comment: 23 pages, 1 figur
Dynamics for the focusing, energy-critical nonlinear Hartree equation
In \cite{LiMZ:e-critical Har, MiaoXZ:09:e-critical radial Har}, the dynamics
of the solutions for the focusing energy-critical Hartree equation have been
classified when , where is the ground state. In this paper, we
continue the study on the dynamics of the radial solutions with the threshold
energy. Our arguments closely follow those in
\cite{DuyMerle:NLS:ThresholdSolution, DuyMerle:NLW:ThresholdSolution,
DuyRouden:NLS:ThresholdSolution, LiZh:NLS, LiZh:NLW}. The new ingredient is
that we show that the positive solution of the nonlocal elliptic equation in
is regular and unique by the moving plane method in
its global form, which plays an important role in the spectral theory of the
linearized operator and the dynamics behavior of the threshold solution.Comment: 53 page
The low regularity global solutions for the critical generalized KdV equation
We prove that the Cauchy problem of the mass-critical generalized KdV
equation is globally well-posed in Sobolev spaces for . Of
course, we require that the mass is strictly less than that of the ground state
in the focusing case. The main approach is the "I-method" together with the
multilinear correction analysis. Moreover, we use some "partially refined"
argument to lower the upper control of the multiplier in the resonant
interactions. The result improves the previous works of Fonseca, Linares, Ponce
(2003) and Farah (2009).Comment: 27pages, the mistake in the previous version is corrected; using
I-method with the resonant decomposition gives an improvement over our
previous result
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