On the convergence rates of discrete solutions to the Wave Kinetic Equation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929324157829120 |
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| author | Dolce, Michele Grande, Ricardo |
| author_facet | Dolce, Michele Grande, Ricardo |
| contents | In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation (WKE). This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and Velázquez showed that, starting with initial data given by countably many Dirac masses, solutions remain a linear combination of countably many Dirac masses at all times. Moreover, there is convergence to a single Dirac mass at long times. The first goal of this paper is to give quantitative rates for the speed of said convergence. In order to study the optimality of the bounds we obtain, we introduce and analyze a toy model accounting only for the leading order quadratic interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14400 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the convergence rates of discrete solutions to the Wave Kinetic Equation Dolce, Michele Grande, Ricardo Analysis of PDEs Mathematical Physics 35B40, 45G05, 35Q55 In this paper, we consider the long-term behavior of some special solutions to the Wave Kinetic Equation (WKE). This equation provides a mesoscopic description of wave systems interacting nonlinearly via the cubic NLS equation. Escobedo and Velázquez showed that, starting with initial data given by countably many Dirac masses, solutions remain a linear combination of countably many Dirac masses at all times. Moreover, there is convergence to a single Dirac mass at long times. The first goal of this paper is to give quantitative rates for the speed of said convergence. In order to study the optimality of the bounds we obtain, we introduce and analyze a toy model accounting only for the leading order quadratic interactions. |
| title | On the convergence rates of discrete solutions to the Wave Kinetic Equation |
| topic | Analysis of PDEs Mathematical Physics 35B40, 45G05, 35Q55 |
| url | https://arxiv.org/abs/2404.14400 |