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522
Global wellposedness, scattering, and blowup for the energycritical, focusing, nonlinear Schrödinger equation in the radial case
, 2006
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Strichartz inequalities and the nonlinear Schrodinger equation on compact manifolds
"... Abstract. We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any Riemannian compact manifold. As a consequence we infer low regularity local wellposedness results in any dimension, as well as global existence results for the Cauchy problem of nonline ..."
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Cited by 140 (28 self)
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Abstract. We prove Strichartz estimates with fractional loss of derivatives for the Schrödinger equation on any Riemannian compact manifold. As a consequence we infer low regularity local wellposedness results in any dimension, as well as global existence results for the Cauchy problem of nonlinear Schrödinger equations on surfaces in the case of defocusing polynomial nonlinearities, and on threemanifolds in the case of cubic defocusing nonlinearities. We also discuss the optimality of these Strichartz estimates on spheres. 1. Introduction and statement of the results
Time decay for solutions of Schrödinger equations with rough and timedependent potentials.
, 2001
"... In this paper we establish dispersive estimates for solutions to the linear SchrSdinger equation in three dimension (0.1) 1.0tO  A0 + Vb = 0, O(s) = f where V(t, x) is a timedependent potential that satisfies the conditions suPllV(t,.)llL(R) + sup f f IV(*'x)l drdy < Co. ..."
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Cited by 118 (14 self)
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In this paper we establish dispersive estimates for solutions to the linear SchrSdinger equation in three dimension (0.1) 1.0tO  A0 + Vb = 0, O(s) = f where V(t, x) is a timedependent potential that satisfies the conditions suPllV(t,.)llL(R) + sup f f IV(*'x)l drdy < Co.
The cubic nonlinear Schrödinger equation in two dimensions with radial data
"... Abstract. We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut + ∆u = ±u  2 u for large spherically symmetric L 2 x(R 2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the grou ..."
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Cited by 91 (14 self)
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Abstract. We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut + ∆u = ±u  2 u for large spherically symmetric L 2 x(R 2) initial data; in the focusing case we require, of course, that the mass is strictly less than that of the ground state. As a consequence, we deduce that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. We also establish some partial results towards the analogous claims in other dimensions and without the assumption of spherical symmetry. Contents
Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation
 Math. Research Letters
"... Abstract. We prove an “almost conservation law ” to obtain globalintime wellposedness for the cubic, defocussing nonlinear Schrödinger equation in Hs (Rn) when n = 2, 3 and s> 4 5, , respectively. 7 6 1. ..."
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Cited by 91 (29 self)
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Abstract. We prove an “almost conservation law ” to obtain globalintime wellposedness for the cubic, defocussing nonlinear Schrödinger equation in Hs (Rn) when n = 2, 3 and s> 4 5, , respectively. 7 6 1.
Strichartz estimates for a Schrödinger operator with nonsmooth coefficients
 Comm. Partial Differential Equations
"... Abstract. We prove Strichartz type estimates for the Schrödinger equation corresponding to a second order elliptic operator with variable coefficients. We assume that the coefficients are a C2 compactly supported perturbation of the identity, satisfying a nontrapping condition. 1. ..."
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Cited by 81 (6 self)
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Abstract. We prove Strichartz type estimates for the Schrödinger equation corresponding to a second order elliptic operator with variable coefficients. We assume that the coefficients are a C2 compactly supported perturbation of the identity, satisfying a nontrapping condition. 1.
Global regularity of wave maps II. Small energy in two dimensions
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... We show that wave maps from Minkowski space R 1+n to a sphere S m−1 are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space ˙ H n/2, in all dimensions n ≥ 2. This generalizes the results in the prequel [37] of this paper, which addressed the highdimensio ..."
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Cited by 72 (15 self)
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We show that wave maps from Minkowski space R 1+n to a sphere S m−1 are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space ˙ H n/2, in all dimensions n ≥ 2. This generalizes the results in the prequel [37] of this paper, which addressed the highdimensional case n ≥ 5. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating nonconcentration of energy.
Global wellposedness of the BenjaminOno equation in H¹(R)
, 2004
"... We show that the BenjaminOno equation is globally wellposed in H s (R) for s ≥ 1. This is despite the presence of the derivative in the nonlinearity, which causes the solution map to not be uniformly continuous in H s for any s [15]. The main new ingredient is to perform a global gauge transforma ..."
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Cited by 71 (3 self)
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We show that the BenjaminOno equation is globally wellposed in H s (R) for s ≥ 1. This is despite the presence of the derivative in the nonlinearity, which causes the solution map to not be uniformly continuous in H s for any s [15]. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative.
GLOBAL EXISTENCE AND SCATTERING FOR ROUGH SOLUTIONS OF A NONLINEAR SCHRÖDINGER EQUATION ON R³
, 2003
"... We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in Hs (R3) for s> 4. The main new estimate in the argument is a Morawetztype inequality for the solution φ. 5 This estimate bounds ‖φ(x, t)‖L4 x,t (R3×R), whereas the wellknown Morawetztype estima ..."
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Cited by 71 (16 self)
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We prove global existence and scattering for the defocusing, cubic nonlinear Schrödinger equation in Hs (R3) for s> 4. The main new estimate in the argument is a Morawetztype inequality for the solution φ. 5 This estimate bounds ‖φ(x, t)‖L4 x,t (R3×R), whereas the wellknown Morawetztype estimate of LinStrauss controls
Global wellposedness and scattering for the defocusing energycritical nonlinear Schrödinger equation
 in R 1+4
"... Abstract. We obtain global wellposedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energyspace solutions to the defocusing energycritical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequencylocal ..."
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Cited by 70 (15 self)
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Abstract. We obtain global wellposedness, scattering, uniform regularity, and global L6 t,x spacetime bounds for energyspace solutions to the defocusing energycritical nonlinear Schrödinger equation in R×R 4. Our arguments closely follow those in [11], though our derivation of the frequencylocalized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the L6 t,xnorm