Regularization by noise for some strongly non-resonant modulated dispersive PDEs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914998337404928 |
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| author | Robert, Tristan |
| author_facet | Robert, Tristan |
| contents | In this work, we pursue our investigations on the Cauchy problem for a class of dispersive PDEs where a rough time coefficient is present in front of the dispersion. We show that if the PDE satisfies a strong non-resonance condition (Theorem 1.6), eventually up to a completely resonant term (Theorem 1.9), then the modulated PDE is well-posed at any regularity index provided that the noise term in front of the dispersion is irregular enough. This extends earlier pioneering work of Chouk-Gubinelli and Chouk-Gubinelli-Li-Li-Oh to a more general context. We quantify the irregularity of the noise required to reach a given regularity index in terms of the regularity of its occupation measure in the sense of Catellier-Gubinelli. As examples, we discuss the cases of dispersive perturbations of the Burger's equation, including the dispersion-generalized Korteweg-de Vries and Benjamin-Ono equations, the intermediate long wave equation, the Wick-ordered modified dispersion-generalized Korteweg-de Vries equation, and the fifth-order Korteweg-de Vries equation. We also treat the completely non-resonant nonlinear Schrödinger equation and the Wick-ordered fractional cubic nonlinear Schrödinger equation, all with periodic boundary conditions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_23051 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularization by noise for some strongly non-resonant modulated dispersive PDEs Robert, Tristan Analysis of PDEs Probability 35Q53, 35Q55, 60L50 In this work, we pursue our investigations on the Cauchy problem for a class of dispersive PDEs where a rough time coefficient is present in front of the dispersion. We show that if the PDE satisfies a strong non-resonance condition (Theorem 1.6), eventually up to a completely resonant term (Theorem 1.9), then the modulated PDE is well-posed at any regularity index provided that the noise term in front of the dispersion is irregular enough. This extends earlier pioneering work of Chouk-Gubinelli and Chouk-Gubinelli-Li-Li-Oh to a more general context. We quantify the irregularity of the noise required to reach a given regularity index in terms of the regularity of its occupation measure in the sense of Catellier-Gubinelli. As examples, we discuss the cases of dispersive perturbations of the Burger's equation, including the dispersion-generalized Korteweg-de Vries and Benjamin-Ono equations, the intermediate long wave equation, the Wick-ordered modified dispersion-generalized Korteweg-de Vries equation, and the fifth-order Korteweg-de Vries equation. We also treat the completely non-resonant nonlinear Schrödinger equation and the Wick-ordered fractional cubic nonlinear Schrödinger equation, all with periodic boundary conditions. |
| title | Regularization by noise for some strongly non-resonant modulated dispersive PDEs |
| topic | Analysis of PDEs Probability 35Q53, 35Q55, 60L50 |
| url | https://arxiv.org/abs/2410.23051 |