Nonlinear parabolic stochastic evolution equations in critical spaces part II

Abstract

This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical $$L^2$$ L 2 -settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.

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

“…The results of this paper fall within this line of research providing new results and highlighting new view points on the works [FL21,FGL21a]. One of the main contribution is the connection with the theory of critical spaces for SPDEs developed in [AV21a,AV22a] which relies on the L p (L q )theory for SPDEs, pioneered by Krylov [Kry94,Kry99] and later by Van Neerven, Veraar and Weis [NVW07,NVW12]. To the best of our knowledge, the current paper is the first regularization by noise result exploiting L p (L q )-estimates.…”

“…Let (V, V 0 ) be as in Step 2, see (6.41)-(6.42). By (6.41), the Burkholder-Davis-Gundy inequality yields, for some constant c 0 (p, q, K, r, δ, θ, η) > 0 and for all t ∈ [0, T ] (see [AV22a,Theorem 4.15] for similar computations)…”

“…Proposition 4.2 does not follow directly from the results of [AV21a,AV22a] as the setting used there does not allow for the non-local (in time) operator v → φ R,r (•, v). However, the methods of [AV21a,AV22a] are still applicable with minor modifications. Below we give some indications how to extend the proofs of [AV21a,AV22a] to the present situation.…”

“…However, the methods of [AV21a,AV22a] are still applicable with minor modifications. Below we give some indications how to extend the proofs of [AV21a,AV22a] to the present situation.…”

“…In the works [FL21,FGL21a] the analogue of Theorem 4.1 is proven by showing global existence and pathwise uniqueness which, in combination with a Yamada-Watanabe type argument, yields existence of global unique solutions. Our approach is different and it based on the construction of maximal solutions and blow-up criteria, following the scheme of [AV21a,AV22a]. This strategy has two basic advantages.…”