Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data

Dodson, Benjamin; Lührmann, Jonas; Mendelson, Dana

doi:10.48550/arxiv.1703.09655

Cited by 10 publications

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“…Finally, we control the nonlinear component w n . To handle the low-high interaction term ∇P 1 w ¨∇F ω n , we place w n in the space of frequency-localized functions Y ν N , which is defined in (20).…”

Section: Theorem 13 (Main Theorem)mentioning

confidence: 99%

“…In the following, we recall a refined Strichartz estimate from [29]. This estimate has already been used in the context of the Wiener randomization in [20].…”

Section: Lemma 22 (Strichartz Estimate)mentioning

confidence: 99%

“…After pioneering work by Bourgain [5,6] on nonlinear Schrödinger equations and more recent work by Burq-Tzvetkov [11,12] on nonlinear wave equations, the study of dispersive PDE with random initial data has seen an enormous growth of interest. We mention only some additional references in the context of nonlinear wave equations [8,10,13,18,20,21,32,33,34,35]. We also refer the reader to the survey paper [3] for a summary and further relevant references.…”

Section: Introductionmentioning

confidence: 99%

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Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Bringmann

2020

International Mathematics Research Notices

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We study the defocusing energy-critical nonlinear wave equation in four dimensions. Our main result proves the stability of the scattering mechanism under random pertubations of the initial data. The random pertubation is defined through a microlocal randomization, which is based on a unit-scale decomposition in physical and frequency space. In contrast to the previous literature, we do not require the spherical symmetry of the pertubation. The main novelty lies in a wave packet decomposition of the random linear evolution. Through this decomposition, we can adaptively estimate the interaction between the rough and regular components of the solution. Our argument relies on techniques from restriction theory, such as Bourgain's bush argument and Wolff's induction on scales.Contents MSC2010 : 35L05, 42B20, 42B37

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Section: Theorem 13 (Main Theorem)mentioning

confidence: 99%

“…In the following, we recall a refined Strichartz estimate from [29]. This estimate has already been used in the context of the Wiener randomization in [20].…”

Section: Lemma 22 (Strichartz Estimate)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Bringmann

2020

International Mathematics Research Notices

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“…4. In [1,2,3,27,28,29,17,25,18], the randomization considered there is based on the uniform decomposition of the frequency space,…”

Section: Randomization Proceduresmentioning

confidence: 99%

“…See [1,2,3,27,28,29,4,14] and references therein. For the scattering results, see [17,25,18,10]. For the long time behavior in the context of fucusing wave equation, see [24].…”

Section: Introductionmentioning

confidence: 99%

Almost sure scattering for the nonlinear Klein-Gordon equations with Sobolev critical power

Chen¹,

Wang²

2020

Preprint

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In this paper, we study the almost sure scattering for the Klein-Gordon equations with Sobolev critical power. We obtain the almost sure scattering with random initial data in H s × H s−1 , 11 12 < s < 1 for d = 4, 15 16 < s < 1 for d = 5. We use the induction on scales and bushes argument in [9] where the model equation is wave equation. For d = 5, we use the mass term of the Klein-Gordon equation to obtain the control of the increment of energy in the process of induction on scales.

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Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

Dodson

Lührmann

Mendelson

2019

Advances in Mathematics

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We consider the Cauchy problem for the defocusing cubic nonlinear Schrödinger equation in four space dimensions and establish almost sure local well-posedness and conditional almost sure scattering for random initial data in H s x (R 4 ) with 1 3 < s < 1. The main ingredient in the proofs is the introduction of a functional framework for the study of the associated forced cubic nonlinear Schrödinger equation, which is inspired by certain function spaces used in the study of the Schrödinger maps problem, and is based on Strichartz spaces as well as variants of local smoothing, inhomogeneous local smoothing, and maximal function spaces. Additionally, we prove an almost sure scattering result for randomized radially symmetric initial data in H s x (R 4 ) with 1 2 < s < 1.

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Almost sure scattering for the 4D energy-critical defocusing nonlinear wave equation with radial data

Cited by 10 publications

References 0 publications

Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Almost Sure Local Well-Posedness for a Derivative Nonlinear Wave Equation

Almost sure scattering for the nonlinear Klein-Gordon equations with Sobolev critical power

Almost sure local well-posedness and scattering for the 4D cubic nonlinear Schrödinger equation

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