Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

We consider the defocusing, $\dot{H}^1$-critical Hartree equation for the radial data in all dimensions $(n\geq 5)$. 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 $\displaystyle - \int_{I}\int_{|x|\leq A|I|^{1/2}}|u|^{2}\Delta \Big(\frac{1}{|x|}\Big)dxdt$ 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 $E(u_0)<E(W)$, where $W$ 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 $L^{\frac{2d}{d-2}}(\R^d)$ 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 $H^s(\R)$ for $s>6/13$. 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