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

Global well-posedness and scattering for the mass-critical Hartree equation with radial data

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Hartree equation $iu_t+\Delta u=\pm(|x|^{-2}*|u|^2)u$ for large spherically symmetric $L^2_x(\Bbb{R}^d)$ initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state.Comment: 38 pages, 1 figur

Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case

We establish global existence, scattering for radial solutions to the energy-critical focusing Hartree equation with energy and $\dot{H}^1$ norm less than those of the ground state in $\mathbb{R}\times \mathbb{R}^d$, $d\geq 5$.Comment: 35 pages, 2 figure

The dynamics of the NLS with the combined terms in five and higher dimensions

In this paper, we continue the study in \cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schr\"{o}dinger equation with the energy below the threshold $m$ in the energy space $H^1(\R^d)$, iu_t + \Delta u = -|u|^{4/(d-2)}u + |u|^{4/(d-1)}u, \; d\geq 5. \tag{CNLS} The threshold is given by the ground state $W$ for the energy-critical NLS: $iu_t + \Delta u = -|u|^{4/(d-2)}u$. Compared with the argument in \cite{MiaoWZ:NLS:3d Combined}, the new ingredient is that we use the double duhamel formula in \cite{Kiv:Clay Lecture, TaoVZ:NLS:mass compact} to lower the regularity of the critical element in $L^{\infty}_tH^1_x$ to $L^{\infty}\dot H^{-\epsilon}_x$ for some $\epsilon>0$ in five and higher dimensions and obtain the compactness of the critical element in $L^2_x$, which is used to control the spatial center function $x(t)$ of the critical element and furthermore used to defeat the critical element in the reductive argument.Comment: To publish in: Some Topics in Harmonic Analysis and Applications, Pages265-298, Advanced Lectures in Mathematics, ALM34, Higher Education Press, Beijing, and International Press, USA, 201

Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions d≥9d\geq9

We consider the defocusing energy-critical nonlinear Schr\"odinger equation of fourth order $iu_t+\Delta^2 u=-|u|^\frac{8}{d-4}u$. We prove that any finite energy solution is global and scatters both forward and backward in time in dimensions $d\geq9$.Comment: 23 pages, some errors in Proposition 5.1 and section 7 are fixed. Other typos are correcte

Plasma kinetics: Discrete Boltzmann modelling and Richtmyer-Meshkov instability

A discrete Boltzmann model (DBM) for plasma kinetics is proposed. The constructing of DBM mainly considers two aspects. The first is to build a physical model with sufficient physical functions before simulation. The second is to present schemes for extracting more valuable information from massive data after simulation. For the first aspect, the model is equivalent to a magnetohydrodynamic model plus a coarse-grained model for the most relevant TNE behaviors including the entropy production rate. A number of typical benchmark problems including Orszag-Tang (OT) vortex problem are used to verify the physical functions of DBM. For the second aspect, the DBM use non-conserved kinetic moments of (f-feq) to describe non-equilibrium state and behaviours of complex system. The OT vortex problem and the Richtmyer-Meshkov instability (RMI) are practical applications of the second aspect. For RMI with interface inverse and re-shock process, it is found that, in the case without magnetic field, the non-organized momentum flux shows the most pronounced effects near shock front, while the non-organized energy flux shows the most pronounced behaviors near perturbed interface. The influence of magnetic field on TNE effects shows stages: before the interface inverse, the TNE strength is enhanced by reducing the interface inverse speed; while after the interface inverse, the TNE strength is significantly reduced. Both the global average TNE strength and entropy production rate contributed by non-organized energy flux can be used as physical criteria to identify whether or not the magnetic field is sufficient to prevent the interface inverse.Comment: 20 pages, 15 figure