Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in ${\mathbb{R}}^{1+n}$

Abstract

Using the same induction on energy argument in both frequency space and spatial space simultaneously as in \cite{CKSTT07}, \cite{RyV05} and \cite{Vi05}, we obtain global well-posedness and scattering of energy solutions of defocusing energy-critical nonlinear Hartree equation in $\mathbb{R}\times {\mathbb{R}}^n$($n\ge 5$), which removes the radial assumption on the data in \cite{MiXZ07a}. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition \ref{freqdelocaimplystbound} and Corollary \ref{frequencylocalization}) of the minimal energy blow up solutions, which can not be obtained from the classical long time perturbation and bilinear estimate and that we obtain the spatial concentration of minimal energy blow up solution after proving that $L_x^{\frac{2n}{n-2}}$-norm of minimal energy blow up solutions is bounded from below, the $L_x^{\frac{2n}{n-2}}$-norm is larger than the potential energy.

41pages

References in corpus (5)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data
• Global well-posedness and scattering for the mass-critical Hartree equation with radial data
• Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case
• Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$-subcritical Hartree equation in ${\mathbb{R}}^d$

Cited by in corpus (4)

• Bounds on the growth of high Sobolev norms of solutions to 2D Hartree Equations
• Energy Scattering for a Klein-Gordon Equation with a Cubic Convolution
• Global well-posedness for the defocusing Hartree equation with radial data in ${\mathbb{R}}^4$
• Global well-posedness of the energy-critical stochastic Hartree nonlinear wave equation