Almost sure scattering for the energy critical nonlinear wave equation

Bringmann, Bjoern

doi:10.1353/ajm.2021.0050

Cited by 15 publications

(18 citation statements)

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“…Furthermore, other randomizations have been applied to the almost sure global well-posedness and scattering of energy critical models. In [11], Bringmann introduced a randomization based on wave packet decomposition to study the non-radial 4D NLW in H s…”

Section: Introductionmentioning

confidence: 99%

Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases

Shen¹,

Soffer²,

Wu³

2021

Preprint

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In this paper, we study the defocusing energy-critical nonlinear Schrödinger equations i∂ t u + ∆u = |u| 4 d−2 u. When d = 3, 4, we prove the almost sure scattering for the equations with nonradial data in H s x for any s ∈ R. In particular, our result does not rely on any spherical symmetry, size or regularity restrictions. Contents 1. Introduction 1.1. Definition of randomization 1.2. Main result 1.3. Sketch of the proof. 1.4. Organization of the paper 2. Preliminary 2.1. Notation 2.2. Useful lemmas 2.3. Probabilistic theory 3. Almost sure Strichartz estimates 3.1. Strichartz estimates 3.2. Linear estimates in 3D case 3.3. Linear estimates in 4D case 4. Global well-posedness and scattering in 3D case 4.1. Reduction to the deterministic problem 4.2. Local theory 4.3. Modified Interaction Morawetz 4.4. Almost conservation law 4.5. Perturbations 4.6. Proof of Proposition 4.1 5. Global well-posedness and scattering in 4D case 5.1. Reduction to the deterministic problem 5.

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Section: Introductionmentioning

confidence: 99%

Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases

Shen¹,

Soffer²,

Wu³

2021

Preprint

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“…The first almost sure scattering result without the radial symmetry assumption was proven in [10] for the energy-critical NLW in dimension d = 4, i.e. (1.1), for 11 12 < s < 1.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…(1.1), for 11 12 < s < 1. 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%

“…Moreover, we obtain this L 1 t L ∞ x -bound almost surely for linear solutions of the wave equation with randomized initial data from H s (R 4 ) × H s−1 (R 4 ) with 5 6 < s < 1, where the original data need not be radially symmetric. Consequently, we improve the regularity threshold for almost sure scattering of (1.1) from s > 11 12 in [10] to s > 5 6 . The proof of Theorem 1.2 is thus based on new probabilistic estimates for the linear flow of the wave equation with randomized initial data, the main novelty being the almost sure L 1 t L ∞ x -estimate of e ±it|∇| f ω .…”

Section: Unit Scale Decomposition In Frequency Spacementioning

confidence: 98%

“…. We use the overall strategy developed in [22] and successfully applied in [25,21,10,40] to obtain almost sure scattering results for the energy-critical cubic nonlinear wave and Schrödinger equation. The strategy consists in developing a suitable local wellposedness and perturbation theory for the forced equation (1.19), which is then combined with the existing deterministic theory for (1.1) to derive a scattering result for (1.19) conditioned on an a priori bound of the energy E(v) of v. Note that the energy of v is not conserved as v is not a solution of the energy-critical NLW (1.1).…”

Section: Unit Scale Decomposition In Frequency Spacementioning

confidence: 99%

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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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Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

Krieger

Lührmann

Staffilani

2023

Arch Rational Mech Anal

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Almost sure scattering for the energy critical nonlinear wave equation

Cited by 15 publications

References 0 publications

Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases

Almost sure scattering for the nonradial energy-critical NLS with arbitrary regularity in 3D and 4D cases

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

Probabilistic Small Data Global Well-Posedness of the Energy-Critical Maxwell–Klein–Gordon Equation

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