Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866917659304525824 |
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| author | Ampatzoglou, Ioakeim Miller, Joseph K. Pavlović, Nataša Tasković, Maja |
| author_facet | Ampatzoglou, Ioakeim Miller, Joseph K. Pavlović, Nataša Tasković, Maja |
| contents | Inspired by ideas stemming from the analysis of the Boltzmann equation, in this paper we expand well-posedness theory of the spatially inhomogeneous 4-wave kinetic equation, and also analyze an infinite hierarchy of PDE associated with this nonlinear equation. More precisely, we show global in time well-posedness of the spatially inhomogeneous 4-wave kinetic equation for polynomially decaying initial data. For the associated infinite hierarchy, we construct global in time solutions using the solutions of the wave kinetic equation and the Hewitt-Savage theorem. Uniqueness of these solutions is proved by using a combinatorial board game argument tailored to this context, which allows us to control the factorial growth of the Dyson series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03984 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces Ampatzoglou, Ioakeim Miller, Joseph K. Pavlović, Nataša Tasković, Maja Analysis of PDEs Inspired by ideas stemming from the analysis of the Boltzmann equation, in this paper we expand well-posedness theory of the spatially inhomogeneous 4-wave kinetic equation, and also analyze an infinite hierarchy of PDE associated with this nonlinear equation. More precisely, we show global in time well-posedness of the spatially inhomogeneous 4-wave kinetic equation for polynomially decaying initial data. For the associated infinite hierarchy, we construct global in time solutions using the solutions of the wave kinetic equation and the Hewitt-Savage theorem. Uniqueness of these solutions is proved by using a combinatorial board game argument tailored to this context, which allows us to control the factorial growth of the Dyson series. |
| title | Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.03984 |