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### Open Access Journals

CPAA

We prove global well-posedness for the $L^2$-critical cubic
defocusing nonlinear Schrödinger equation on $R^2$ with
data $u_0 \in H^s(R^2)$ for $ s > \frac{1}{3}$. The
proof combines

*a priori*Morawetz estimates obtained in [4] and the improved almost conservation law obtained in [6]. There are two technical difficulties. The first one is to estimate the variation of the improved almost conservation law on intervals given in terms of Strichartz spaces rather than in terms of $X^{s,b}$ spaces. The second one is to control the error of the*a priori*Morawetz estimates on an arbitrary large time interval, which is performed by a bootstrap via a double layer in time decomposition.
keywords:
bootstraps
,
interaction Morawetz
,
scattering
,
global well-posedness.
,
Nonlinear Schrödinger

DCDS

We prove global well-posedness for the defocusing cubic wave
equation

∂_{tt} $u - \Delta u = -u^{3} $

$u(0,x) = u_{0}(x) $

$\partial_{t} u(0,x) = u_{1}(x)$

with data $( u_{0}, u_{1} ) \in H^{s} \times H^{s-1}$, $1 > s > \frac{13}{18} $≈ 0.722. The main task is to estimate the variation of an almost conserved quantity on an arbitrary long time interval. We divide it into subintervals. On each of these subintervals we write the solution as the sum of its linear part adapted to the subinterval and its corresponding nonlinear part. Some terms resulting from this decomposition have a controlled global variation and other terms have a slow local variation.

DCDS

We show that the Maxwell-Klein-Gordon equations in three dimensions
are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s >
\sqrt{3}/2 \approx 0.866$. This extends previous work of
Klainerman-Machedon [24] on finite energy data $s \geq
1$, and Eardley-Moncrief [11] for still smoother data. We
use the method of almost conservation laws, sometimes called the
"I-method", to construct an almost conserved quantity based on the
Hamiltonian, but at the regularity of $H^s_x$ rather than $H^1_x$.
One then uses Strichartz, null form, and commutator estimates to
control the development of this quantity. The main technical
difficulty (compared with other applications of the method of almost
conservation laws) is at low frequencies, because of the poor
control on the $L^2_x$ norm. In an appendix, we demonstrate the
equations' relative lack of smoothing - a property that presents
serious difficulties for studying rough solutions using other known
methods.

keywords:
$X^{s
,
Global well-posedness
,
Maxwell-Klein-Gordon equation
,
b}$ spaces.
,
Coulomb gauge
,
I-method

CPAA

Global dynamics above the ground state for the energy-critical Schrödinger equation with radial data

Consider the focusing energy critical Schrödinger equation in three space dimensions
with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters.
In analogy with [19], the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows
to classify the solutions is the

*one-pass*lemma. The main difference between [19] and this paper is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the*one-pass*lemma by comparing the contribution in the variational region and in the hyperbolic region.
keywords:
blow-up
,
solitons
,
Sobolev critical exponent.
,
scattering theory
,
nonlinear Schrödinger equation

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