Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems
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| Format: | Preprint |
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2023
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| author | Agresti, Antonio Veraar, Mark |
| author_facet | Agresti, Antonio Veraar, Mark |
| contents | In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the $d$-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g. the Allen-Cahn equation), and dissipative systems (e.g. equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for $d\in\{1, 2, 3, 4\}$ and the Brusselator for $d\in \{1, 2, 3\}$. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new $L^ζ$-coercivity/dissipativity conditions and the development of an $L^p(L^q)$-framework for systems of reaction-diffusion equations, which are needed when treating dimensions $d\in \{2, 3\}$ in the case of cubic or higher order nonlinearities. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2301_06897 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems Agresti, Antonio Veraar, Mark Analysis of PDEs Functional Analysis Probability Primary: 35K57, Secondary: 60H15, 35R60, 35A01, 35B44, 35B65, 35K90, 42B37, 58D25 In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the $d$-dimensional torus. We show new global well-posedness results for a large class of scalar equations (e.g. the Allen-Cahn equation), and dissipative systems (e.g. equations in coagulation dynamics). Moreover, we prove global well-posedness for two weakly dissipative systems: Lotka-Volterra equations for $d\in\{1, 2, 3, 4\}$ and the Brusselator for $d\in \{1, 2, 3\}$. Many of the results are also new without transport noise. The proofs are based on maximal regularity techniques, positivity results, and sharp blow-up criteria developed in our recent works, combined with energy estimates based on Itô's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new $L^ζ$-coercivity/dissipativity conditions and the development of an $L^p(L^q)$-framework for systems of reaction-diffusion equations, which are needed when treating dimensions $d\in \{2, 3\}$ in the case of cubic or higher order nonlinearities. |
| title | Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems |
| topic | Analysis of PDEs Functional Analysis Probability Primary: 35K57, Secondary: 60H15, 35R60, 35A01, 35B44, 35B65, 35K90, 42B37, 58D25 |
| url | https://arxiv.org/abs/2301.06897 |