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arXiv · 2301.06897

Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems

Antonio Agresti·Mark Veraar
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Abstract

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 It\^o's formula and stochastic Gronwall inequalities. Key novelties include the introduction of new $L^{\zeta}$-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.

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Antonio Agresti, Mark Veraar. 2023-01-17. Reaction-diffusion equations with transport noise and critical superlinear diffusion: Global well-posedness of weakly dissipative systems. https://arxiv.org/abs/2301.06897

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