Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912193281261568 |
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| author | Pavlović, Nataša Tasković, Maja Velasco, Luisa |
| author_facet | Pavlović, Nataša Tasković, Maja Velasco, Luisa |
| contents | Six-wave interactions are used for modeling various physical systems, including in optical wave turbulence [16] (where a cascade of photons displays this kind of behavior) and in quantum wave turbulence [31] (for the interaction of Kelvin waves in superfluids). In this paper, we initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation. More precisely, we obtain the existence and uniqueness of non-negative mild solutions to this equation in exponentially weighted $L^\infty_{xv}$ spaces. This is accompanied by an analysis of the long-time behavior of such solutions - we prove that the solutions scatter, that is, they converge to solutions of the transport equation in the limit as $t \to \pm \infty$. Compared with the study of four-wave kinetic equations, the main challenge we face is to address the increased complexity of the geometry of the six-wave interactions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces Pavlović, Nataša Tasković, Maja Velasco, Luisa Analysis of PDEs 35Q82 Six-wave interactions are used for modeling various physical systems, including in optical wave turbulence [16] (where a cascade of photons displays this kind of behavior) and in quantum wave turbulence [31] (for the interaction of Kelvin waves in superfluids). In this paper, we initiate the analysis of the Cauchy problem for the spatially inhomogeneous six-wave kinetic equation. More precisely, we obtain the existence and uniqueness of non-negative mild solutions to this equation in exponentially weighted $L^\infty_{xv}$ spaces. This is accompanied by an analysis of the long-time behavior of such solutions - we prove that the solutions scatter, that is, they converge to solutions of the transport equation in the limit as $t \to \pm \infty$. Compared with the study of four-wave kinetic equations, the main challenge we face is to address the increased complexity of the geometry of the six-wave interactions. |
| title | Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces |
| topic | Analysis of PDEs 35Q82 |
| url | https://arxiv.org/abs/2501.10565 |