Results 1  10
of
39
The cubic nonlinear Schrödinger equation in two dimensions with radial data
, 2008
"... 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 ..."
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Cited by 90 (14 self)
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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.
Minimalmass blowup solutions of the masscritical NLS
, 2006
"... We consider the minimal mass m0 required for solutions to the masscritical nonlinear Schrödinger (NLS) equation iut + ∆u = µu  4/d u to blow up. If m0 is finite, we show that there exists a minimalmass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, wh ..."
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Cited by 63 (20 self)
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We consider the minimal mass m0 required for solutions to the masscritical nonlinear Schrödinger (NLS) equation iut + ∆u = µu  4/d u to blow up. If m0 is finite, we show that there exists a minimalmass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in L 2 x (Rd) is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, [17], in dimensions 1, 2 and Begout and Vargas, [2], in dimensions d ≥ 3 for the masscritical NLS and by Kenig and Merle, [18], in the energycritical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in L 2 x (Rd) for the defocusing NLS in three and higher dimensions with spherically symmetric data.
Nonlinear Schrödinger equations at critical regularity
 CLAY LECTURE NOTES
, 2009
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The masscritical nonlinear Schrödinger equation with radial data in dimensions three and higher
"... Abstract. We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut+∆u = ±u  4/d u for large spherically symmetric L 2 x (Rd) initial data in dimensions d ≥ 3. In the focusing case we require that the mass is strictly less than that of ..."
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Cited by 52 (10 self)
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Abstract. We establish global wellposedness and scattering for solutions to the masscritical nonlinear Schrödinger equation iut+∆u = ±u  4/d u for large spherically symmetric L 2 x (Rd) initial data in dimensions d ≥ 3. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time. 1.
Resonant decompositions and the Imethod for cubic nonlinear Schrödinger
 on R 2 . Disc. Cont. Dynam. Systems A
"... Abstract. The initial value problem for the cubic defocusing nonlinear Schrödinger equation i∂tu + ∆u = u  2 u on the plane is shown to be globally wellposed for initial data in H s (R 2) provided s> 1/2. The proof relies upon an almost conserved quantity constructed using multilinear correcti ..."
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Cited by 28 (4 self)
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Abstract. The initial value problem for the cubic defocusing nonlinear Schrödinger equation i∂tu + ∆u = u  2 u on the plane is shown to be globally wellposed for initial data in H s (R 2) provided s> 1/2. 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. 1.
Why are solitons stable?
 BULL. AMER. MATH. SOC. (N.S
, 2009
"... The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be st ..."
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Cited by 26 (0 self)
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The theory of linear dispersive equations predicts that waves should spread out and disperse over time. However, it is a remarkable phenomenon, observed both in theory and practice, that once nonlinear effects are taken into account, solitary wave or soliton solutions can be created, which can be stable enough to persist indefinitely. The construction of such solutions can be relatively straightforward, but the fact that they are stable requires some significant amounts of analysis to establish, in part due to symmetries in the equation (such as translation invariance) which create degeneracy in the stability analysis. The theory is particularly difficult in the critical case in which the nonlinearity is at exactly the right power to potentially allow for a selfsimilar blowup. In this article we survey some of the highlights of this theory, from the more classical orbital stability analysis of Weinstein and GrillakisShatahStrauss, to the more recent asymptotic stability and blowup analysis of MartelMerle and MerleRaphael, as well as current developments in using this theory to rigorously demonstrate controlled blowup for several key equations.
A pseudoconformal compactification of the nonlinear Schrödinger equation and applications
, 2009
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THE RADIAL DEFOCUSING ENERGYSUPERCRITICAL NONLINEAR WAVE EQUATION IN ALL SPACE DIMENSIONS
, 2010
"... We consider the defocusing nonlinear wave equation utt − Δu + u  p 4 4 u = 0 with sphericallysymmetric initial data in the regime <p< d−2 d−3 (which is energysupercritical) and dimensions 3 ≤ d ≤ 6; we also consider d ≥ 7, but for a smaller range of p> 4 d−2. The principal result is th ..."
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Cited by 21 (1 self)
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We consider the defocusing nonlinear wave equation utt − Δu + u  p 4 4 u = 0 with sphericallysymmetric initial data in the regime <p< d−2 d−3 (which is energysupercritical) and dimensions 3 ≤ d ≤ 6; we also consider d ≥ 7, but for a smaller range of p> 4 d−2. The principal result is that blowup (or failure to scatter) must be accompanied by blowup of the critical Sobolev norm. An equivalent formulation is that maximallifespan solutions with bounded critical Sobolev norm are global and scatter.
Sharp linear and bilinear restriction estimates for the paraboloid in the cylindrically symmetric case
, 2007
"... In this paper, for cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and linear adjoint restriction estimates. As corollaries of them, for such functions, we first extend the ranges of exponents for the classical linear or bilinear adjoint ..."
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Cited by 20 (2 self)
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In this paper, for cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and linear adjoint restriction estimates. As corollaries of them, for such functions, we first extend the ranges of exponents for the classical linear or bilinear adjoint restriction conjectures, which are sharp up to certain endpoints; Secondly we show that the linear adjoint restriction conjecture for the paraboloid holds for all cylindrically symmetric functions; Lastly, we interpret the restriction estimates in terms of the solutions to the Schrödinger equation. Analogously, we also establish the restriction estimates when the paraboloid is replaced by the lower third of the sphere.
Global regularity of wave maps III. Large energy from R 1+2 to hyperbolic spaces
, 2009
"... We show that wave maps φ from twodimensional Minkowski space R 1+2 to hyperbolic spaces H m are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, as well as the selfsimilar and travelling (or stationary so ..."
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Cited by 20 (4 self)
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We show that wave maps φ from twodimensional Minkowski space R 1+2 to hyperbolic spaces H m are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, as well as the selfsimilar and travelling (or stationary solutions); we will address these claims in the sequels [67], [68], [69], [70] to this paper. Following recent work in critical dispersive equations, the strategy is to reduce matters to the study of an almost periodic maximal Cauchy development in the energy class. We then repeatedly analyse the stressenergy tensor of this development (as in [12], [64]) to extract either a selfsimilar, travelling, or degenerate nontrivial energy class solution to the wave maps equation. We will then rule out such solutions in the sequels to this paper, establishing the desired global regularity result for wave maps.