Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\R^2$
J. Colliander M. Keel Gigliola Staffilani H. Takaoka T. Tao
Discrete & Continuous Dynamical Systems - A 2008, 21(3): 665-686 doi: 10.3934/dcds.2008.21.665
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.
keywords: Strichartz estimates well-posedness Nonlinear Schrödinger equation resonant decomposition.
Polynomial upper bounds for the orbital instability of the 1D cubic NLS below the energy norm
J. Colliander M. Keel G. Staffilani H. Takaoka T. Tao
Discrete & Continuous Dynamical Systems - A 2003, 9(1): 31-54 doi: 10.3934/dcds.2003.9.31
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.
keywords: orbital stability. upper bound on sobolev norms Schrödinger equation
Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm
J. Colliander M. Keel G. Staffilani H. Takaoka T. Tao
Communications on Pure & Applied Analysis 2003, 2(1): 33-50 doi: 10.3934/cpaa.2003.2.33
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.
keywords: Schrödinger equation upper bounds on Sobolev norms orbital stability.
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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