Reaction-diffusion equations with transport noise and critical superlinear diffusion: local well-posedness and positivity

Antonio Agresti; Mark Veraar

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arXiv · 2209.14759

Reaction-diffusion equations with transport noise and critical superlinear diffusion: local well-posedness and positivity

Antonio Agresti·Mark Veraar

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Abstract

In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions -- even in case of non-smooth initial data. Crucial tools are $L^p(L^q)$-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work (Agresti and Veraar: Global existence ... ), the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.

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Antonio Agresti, Mark Veraar. 2022-09-29. Reaction-diffusion equations with transport noise and critical superlinear diffusion: local well-posedness and positivity. https://arxiv.org/abs/2209.14759

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