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DCDS

The initial value problem for the cubic defocusing nonlinear
Schrödinger equation $i \partial_t u + \Delta u = |u|^2 u$ on the
plane is shown to be globally well-posed for initial data in $H^s (
\R^2)$ provided $s>1/2$. The same result holds true for the
analogous focusing problem provided the mass of the initial data is
smaller than the mass of the ground state. The proof relies upon an
almost conserved quantity constructed using multilinear correction
terms. The main new difficulty is to control the contribution of
resonant interactions to these correction terms. The resonant
interactions are significant due to the multidimensional setting of
the problem and some orthogonality issues which arise.

DCDS

We study the long-time behaviour of the focusing cubic NLS on $\mathbf R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein [27], [26]. In the sequel to this paper we generalize this result to other nonlinear Schrödinger equations. Our arguments are based on the "$I$-method" from earlier papers [9]-[15] which pushes down from the energy norm, as well as an "upside-down $I$-method" which pushes up from the $L^2$ norm.

CPAA

We continue the study (initiated in [18]) of the orbital stability
of the ground state cylinder for focussing non-linear Schrödinger equations
in the $H^s(\R^n)$ norm for $1-\varepsilon < s < 1$, for small $\varepsilon$. In the $L^2$-subcritical case we
obtain a polynomial bound for the time required to move away from the
ground state cylinder. If one is only in the $H^1$-subcritical case
then we cannot show this, but for defocussing equations we obtain global well-posedness and
polynomial growth of $H^s$ norms for $s$ sufficiently close to 1.

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

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