Delayed blow-up and enhanced diffusion by transport noise for systems of reaction-diffusion equations (Q6606157)

The author considers a stochastic system of reaction-diffusion equations with linear Stratonovich transport noise of the form\N\[\N\mathrm{d} v_i-\nu_i \Delta v_i\,\mathrm{d} t = f_i(x,v)\, \mathrm{d} t + \sqrt{\nu} \langle \nabla v_i, \circ \mathrm{d} W\rangle.\N\]\NThe SPDE is set on the torus \(\mathbb{T}^d\), and \(W=W(t,x)\) is a random velocity field which is taken Brownian in time, coloured and divergence free in space. Stratonovich transport noise is natural, as it preserves properties of solutions like their positivity, which are desirable when \(v^i\) are interpreted as concentrations of chemicals; the same would not be true e.g. for additive noise.\N\NIn the deterministic case (\(\nu=0\)), even in the presence of dissipation of mass, explicit examples of finite-time blow-up of strong solutions are known; and there are numerous cases where it is currently open whether blow-up occurs. This paper shows that, in a regularisation by noise fashion, the presence of \(W\) can improve the situation significantly.\N\NLoosely speaking, the main result of the paper asserts that, under suitable assumption on the nonlinearities \(f_i\), for any initial datum \(v_0\in L^q\) and any fixed \(\varepsilon\ll 1\) and \(T\gg 1\), one can find an explicit choice of \(W\) such that\N\[\N\mathbb{P}(\tau \geq T)> 1-\varepsilon, \quad \text{ where \(\tau\) is the maximal lifetime of \(v\)}.\N\]\NMoreover, with high probability \(v\) is close to a solution \(v^{det}\) of the deterministic PDE with enhanced diffusion\N\[\N\partial_t v^{det}-(\nu+\nu_i) \Delta v^{\det} = f_i(x,v^{det})\N\]\Nin the sense that\N\[\N\mathbb{P}(\tau \geq T, \, \| v-v^{det}\|_{L^r([0,T];L^q)} \geq \varepsilon) >1-\varepsilon.\N\]\NMoreover, such strong solutions \(v\) preserve positivity and become instantaneously regular in the space variable at positive times, thus being classical solutions. Finally, under some additional assumptions on the nonlinearities \(f_i\) guaranteeing exponential decay of the mass, one can strengthen the result to global existence of strong solutions \(v\) with high probability:\N\[\N\mathbb{P}(\tau =+\infty)> 1-\varepsilon, \quad \text{ where \(\tau\) is the maximal lifetime of \(v\)}.\N\]\N\NThe proofs are based on the scaling limit approach developed by \textit{F. Flandoli} and \textit{D. Luo} [Probab. Theory Relat. Fields 180, No. 1--2, 309--363 (2021; Zbl 1469.60205)], \textit{F. Flandoli} et al. [Commun. Partial Differ. Equations 46, No. 9, 1757--1788 (2021; Zbl 1477.60096)]. This argument must be combined with the nature of the PDE in consideration, and the main novelty of the paper is the use of parabolic maximal regularity tools coming from the \(L^p(L^q)\) theory of SPDEs, first pioneered by Krylov and then developed among others by \textit{A. Agresti} and \textit{M. Veraar} [Nonlinearity 35, No. 8, 4100--4210 (2022; Zbl 1496.60068)], \textit{A. Agresti} and \textit{M. Veraar} [J. Evol. Equ. 22, No. 2, Paper No. 56, 96 p. (2022; Zbl 1491.60093)].\N\NThe interpretation offered in the introduction of the scaling limit argument as an homogenization result, with \(v^{det}\) playing the role of an effective equation, is also remarkable.