Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$
J. Colliander Tristan Roy
Communications on Pure & Applied Analysis 2011, 10(2): 397-414 doi: 10.3934/cpaa.2011.10.397
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
Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$
Tristan Roy
Discrete & Continuous Dynamical Systems - A 2009, 24(4): 1307-1323 doi: 10.3934/dcds.2009.24.1307
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.

keywords: Global Well-posedness Defocusing Cubic Wave Equation Adapted Linear-Nonlinear Decomposition.
Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm
M. Keel Tristan Roy Terence Tao
Discrete & Continuous Dynamical Systems - A 2011, 30(3): 573-621 doi: 10.3934/dcds.2011.30.573
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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