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52
Nonlinear Schrödinger equations at critical regularity
 CLAY LECTURE NOTES
, 2009
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THE FOCUSING ENERGYCRITICAL NONLINEAR SCHRÖDINGER EQUATION IN DIMENSIONS FIVE AND HIGHER
"... Abstract. We consider the focusing energycritical nonlinear Schrödinger equation iut + ∆u = 4 −u  d−2 u in dimensions d ≥ 5. We prove that if a maximallifespan solution u: I ×Rd → C obeys supt∈I ‖∇u(t)‖2 < ‖∇W‖2, then it is global and scatters both forward and backward in time. Here W denotes ..."
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Cited by 54 (8 self)
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Abstract. We consider the focusing energycritical nonlinear Schrödinger equation iut + ∆u = 4 −u  d−2 u in dimensions d ≥ 5. We prove that if a maximallifespan solution u: I ×Rd → C obeys supt∈I ‖∇u(t)‖2 < ‖∇W‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters. We also show that any solution that blows up with bounded kinetic energy must concentrate at least the kinetic energy of the ground state. Similar results were obtained by Kenig and Merle for spherically symmetric initial data and dimensions d =3,4,5.
Global wellposedness and scattering for the defocusing L²critical nonlinear Schrödinger equation when d = 1
, 2015
"... In this paper we prove global well posedness and scattering for the defocusing, one dimensional mass critical nonlinear Schrödinger equation. We make use of a long time Strichartz estimate and a frequency localized Morawetz estimate. This continues work begun in [28] and [30] for dimensions d ≥ ..."
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Cited by 34 (7 self)
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In this paper we prove global well posedness and scattering for the defocusing, one dimensional mass critical nonlinear Schrödinger equation. We make use of a long time Strichartz estimate and a frequency localized Morawetz estimate. This continues work begun in [28] and [30] for dimensions d ≥ 3 and d = 2 respectively.
Nondispersive radial solutions to energy supercritical nonlinear wave equations, with applications
, 2009
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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.
Global wellposedness and scattering for the energycritical, defocusing Hartree equation for radial data
 J. Funct. Anal
"... Using the same induction on energy argument in both frequency space and spatial space simultaneously as in [6], [31] and [35], we obtain global wellposedness and scattering of energy solutions of defocusing energycritical nonlinear Hartree equation in R × R n (n ≥ 5), which removes the radial assu ..."
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Cited by 21 (15 self)
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Using the same induction on energy argument in both frequency space and spatial space simultaneously as in [6], [31] and [35], we obtain global wellposedness and scattering of energy solutions of defocusing energycritical nonlinear Hartree equation in R × R n (n ≥ 5), which removes the radial assumption on the data in [25]. The new ingredients are that we use a modified long time perturbation theory to obtain the frequency localization (Proposition 3.1 and Corollary 3.1) of the minimal energy blow up solutions, which can not be obtained from the classical long time perturbation and bilinear estimate and that we obtain the spatial concentration of minimal energy blow up solution after proving that L 2n n−2 xnorm of minimal energy blow up solutions is bounded from below, the L 2n n−2 xnorm is larger than the potential energy.
Global wellposedness and scattering for the mass critical nonlinear Schrödinger . . .
, 2013
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