Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4

Ryckman, E.; Vişan, Monica

doi:10.1353/ajm.2007.0004

Cited by 228 publications

(281 citation statements)

References 20 publications

(33 reference statements)

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“…These results are established in [19]. In the H 1 -critical three dimensional case (n = 3 and σ = 2), it is proved in [15] that solutions with H s regularity (s > 1) remain in H s for all time; the same is true in the four dimensional case (n = 4 and σ = 1), from [29]. On the other hand, if the nonlinearity is H 1 -supercritical (σ > 2 n−2 ), then it is not known in general whether the solution remains smooth for all time or not.…”

Section: Exercise 363])mentioning

confidence: 69%

Loss of regularity for supercritical nonlinear Schrödinger equations

2008

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Abstract. We consider the nonlinear Schrödinger equation with defocusing, smooth, nonlinearity. Below the critical Sobolev regularity, it is known that the Cauchy problem is ill-posed. We show that this is even worse, namely that there is a loss of regularity, in the spirit of the result due to G. Lebeau in the case of the wave equation. As a consequence, the Cauchy problem for energy-supercritical equations is not well-posed in the sense of Hadamard. We reduce the problem to a supercritical WKB analysis. For super-cubic, smooth nonlinearity, this analysis is new, and relies on the introduction of a modulated energy functionalà la Brenier.

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Section: Exercise 363])mentioning

confidence: 69%

Loss of regularity for supercritical nonlinear Schrödinger equations

2008

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“…On the slab I j0 × R n , we treat the first nonlinearity as a perturbation to the energy-critical NLS (5.24) iw t + ∆w = |w| By the global well-posedness results in [6,23,30], there exists a unique global solution w to (5.24) with initial data u(a) at time t = a and moreover,…”

Section: Global Bounds In the Casementioning

confidence: 99%

“…However, as in this case we do not have a good a priori interaction Morawetz inequality, we will rely instead on the small mass assumption. As in subsection 5.4, in this case we also compare (1.1) to the energy-critical problem iw t + ∆w = |w| 4 n−2 w w(0) = u 0 , which by [6,23,30] is globally wellposed and moreover, …”

Section: Global Bounds Formentioning

confidence: 99%

“…However, the proof of global well-posedness for the energy-critical nonlinear Schrödinger equation, (1.5) (i∂ t + ∆)w = |w| 4 n−2 w w(0, x) = w 0 (x) ∈Ḣ 1 x relies heavily on the scale invariance for this equation; see [6,23,30]. Hence, adding an energy-subcritical perturbation to (1.5), which destroys the scale invariance, is of particular interest.…”

Section: Introductionmentioning

confidence: 99%

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The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

Tao

Vişan

Zhang

2007

Communications in Partial Differential Equations

267

270

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Abstract. We undertake a comprehensive study of the nonlinear Schrödinger equationwhere u(t, x) is a complex-valued function in spacetime Rt × R n x , λ 1 and λ 2 are nonzero real constants, and 0 < p 1 < p 2 ≤ 4 n−2. We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H 1 (R n ) and in the pseudoconformal space Σ := {f ∈ H 1 (R n ); xf ∈ L 2 (R n )}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L 2 x -critical, respectivelyḢ 1 x -critical NLS, that is, λ 1 , λ 2 > 0 and. The results at the endpoint p 1 = 4 n are conditional on a conjectured global existence and spacetime estimate for the L 2x -critical nonlinear Schrödinger equation.As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H 1 x for solutions to the nonlinear Schrödinger equation iut + ∆u = |u| p u,

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“…See Ryckman-Vişan [46], Vişan [52], and Kenig-Merle [33] for global-in-time results. In the following, we state a local well-posedness result of (1.12) with random initial data below the critical space.…”

Section: 4mentioning

confidence: 99%

Excursions in Harmonic Analysis, Volume 4

Bălan¹,

Begue²,

Benedetto³

et al. 2015

Applied and Numerical Harmonic Analysis

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Abstract. We consider a randomization of a function on R d that is naturally associated to the Wiener decomposition and, intrinsically, to the modulation spaces. Such randomized functions enjoy better integrability, thus allowing us to improve the Strichartz estimates for the Schrödinger equation. As an example, we also show that the energycritical cubic nonlinear Schrödinger equation on R 4 is almost surely locally well-posed with respect to randomized initial data below the energy space.

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Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4

Cited by 228 publications

References 20 publications

Loss of regularity for supercritical nonlinear Schrödinger equations

Loss of regularity for supercritical nonlinear Schrödinger equations

The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

Excursions in Harmonic Analysis, Volume 4

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