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

Martin Spitz

doi:10.3934/cpaa.2022134

Article Contents

2022, Volume 21, Issue 12: 4041-4070. Doi: 10.3934/cpaa.2022134

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

Martin Spitz

Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, 33501 Bielefeld, Germany

Received: August 2022

Revised: August 2022

Early access: September 2022

Published: December 2022

The author is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) SFB 1283/2 2021 317210226.

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Abstract

In this article we study the defocusing energy-critical nonlinear wave equation on $ {\mathbb{R}}^4 $ with scaling supercritical data. We prove almost sure scattering for randomized initial data in $ H^s( {\mathbb{R}}^4) \times H^{s-1}( {\mathbb{R}}^4) $ with $ \frac{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 unit-scale 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^\infty_x $.

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Mathematics Subject Classification: Primary: 35L05; Secondary: 35R60, 35A01, 35B40.

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