Reaction-Diffusion Equations with Transport Noise and Critical Superlinear Diffusion: Global Well-Posedness of Weakly Dissipative Systems

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4870-4927, August 2024.
Abstract. In this paper, we investigate the global well-posedness of reaction-diffusion systems with transport noise on the [math]-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 [math] and the Brusselator for [math]. 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 [math]-coercivity/dissipativity conditions and the development of an [math]-framework for systems of reaction-diffusion equations, which are needed when treating dimensions [math] in the case of cubic or higher order nonlinearities.