Almost-sure scattering for the radial energy-critical nonlinear wave equation in three dimensions

Bringmann, Bjoern

doi:10.2140/apde.2020.13.1011

Cited by 21 publications

(14 citation statements)

References 34 publications

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“…The first almost sure scattering result for an energy-critical dispersive equation (besides the small data theory) was given in [21] for the defocusing energy-critical wave equation in R 4 with initial data which is radially symmetric before the randomization. See also [20, Appendix A] for an improvement and [8] for the case d = 3. Building upon ideas from [21], almost sure scattering for the solutions of the defocusing energy-critical NLS in R 4 with randomized radially symmetric initial data from H s (R 4 ) was proven in [25] for 5 6 < s < 1 and then improved in [20] to 1 2 < s < 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In fact, all the above references cited in the context of almost sure local and global wellposedness of the energy-critical wave equation and NLS on the full-space except [7] and [8] employ a Wiener randomization, i.e. a randomization based on a unit-scale decomposition of frequency space (see Subsection 1.2 below for details).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Spitz¹

2021

Preprint

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We study the cubic defocusing nonlinear Schrödinger equation on R 4 with supercritical initial data. For randomized initial data in H s (R 4 ), we prove almost sure local wellposedness for 1 7 < s < 1 and almost sure scattering for 5 7 < s < 1. The randomization is based on a unit-scale decomposition in frequency space, a decomposition in the angular variable, and -for the almost sure scattering result -an additional unit-scale decomposition in physical space. We employ new probabilistic estimates for the linear Schrödinger flow with randomized data, where we effectively combine the advantages of the different decompositions.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Spitz¹

2021

Preprint

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“…We note that in [10] a different randomization than in [22,21] for radially symmetric data was used. In fact, most of the aforementioned results (except [9,40,38]) relied on the so-called Wiener randomization, which is based on a unit-scale decomposition of frequency space (see Subsection 1.2 below for details).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the almost sure scattering for the energy-critical cubic wave equation with supercritical data

Spitz¹

2022

Preprint

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In this article we study the defocusing energy-critical nonlinear wave equation on R 4 with scaling supercritical data. We prove almost sure scattering for randomized initial data in H s (R 4 ) × H s−1 (R 4 ) with 5 6 < s < 1. The proof relies on new probabilistic estimates for the linear flow of the wave equation with randomized data, where the randomization is based on a unitscale decomposition in frequency space, a decomposition in the angular variable, and a unit-scale decomposition of physical space. In particular, we show that the solution to the linear wave equation with randomized data almost surely belongs to L 1 t L ∞ x .

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“…One line of research is to study the question if in a supercritical setting, after randomizing the initial data, one still obtains local wellposedness, global wellposedness, or scattering almost surely. We refer to [4,6,21,33,34,22,11,12] and the references therein for exemplary results in this direction for the Schrödinger and the wave equation.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Randomized final-state problem for the Zakharov system in dimension three

Spitz¹

2020

Preprint

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We consider the final-state problem for the Zakharov system in the energy space in three space dimensions. For (u + , v + ) ∈ H 1 × L 2 without any size restriction, symmetry assumption or additional angular regularity, we perform a physical-space randomization on u + and an angular randomization on v + yielding random final states (u ω + , v ω + ). We obtain that for almost every ω, there is a unique solution of the Zakharov system scattering to the final state (u ω + , v ω + ). The key ingredient in the proof is the use of time-weighted norms and generalized Strichartz estimates which are accessible due to the randomization.

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Almost-sure scattering for the radial energy-critical nonlinear wave equation in three dimensions

Cited by 21 publications

References 34 publications

Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

Almost sure local wellposedness and scattering for the energy-critical cubic nonlinear Schrödinger equation with supercritical data

On the almost sure scattering for the energy-critical cubic wave equation with supercritical data

Randomized final-state problem for the Zakharov system in dimension three

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