Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence*

Abstract

In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an L p -setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers’ equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.

The stochastic primitive equations with non-isothermal turbulent pressure

“…The proof of Theorem 3.4 follows as in [AHHS21] where we checked the applicability of the abstract results of [AV22b,AV22c]. A Sketch of the proof of Theorem 3.4 will be given in Subsection 4.1.…”

“…Proof of Theorem 3.4. The proof of Theorem 3.4 follows as in [AHHS21, Section 6.4] by using the results of [AV22b,AV22c] (see [AHHS21, Section 5.1] for a similar situation). We begin by reformulating (3.1)-(3.2) as a stochastic evolution equation on the Banach space…”

“…This allows the authors of [BS21] to overcome some of the difficulties arose in [DGHT11,DGHTZ12]. In [AHHS21], by combining energy estimates and the functional analytic setting of [AV22b,AV22c] we were able to overcome such drawbacks in presence of gradient noise and transport type noises.…”

“…With this perspective, the local existence and blow-up criterion of Theorem 3.4 follow easily from the theory of critical spaces for stochastic evolution equations developed by the M.C. Veraar and the first author in [AV22b,AV22c].…”

“…Once obtained local existence and blow-up criteria from the abstract setting of [AV22b,AV22c], we turn our attention to global well-posedness, that is the main point of the present manuscript.…”