Quasilinear Parabolic Stochastic Evolution Equations Via Maximal Lp-Regularity

Abstract

We study the Cauchy problem for an abstract quasilinear stochastic parabolic evolution equation on a Banach space driven by a cylindrical Brownian motion. We prove existence and uniqueness of a local strong solution up to a maximal stopping time, that is characterised by a blow-up alternative. The key idea is an iterative application of the theory about maximal L p -regularity for semilinear stochastic evolution equations by Van Neerven, Veraar and Weis. We apply our local well-posedness result to a convection-diffusion equation on a bounded domain with Dirichlet, Neumann or mixed boudary conditions and to a generalized Navier-Stokes equation describing non-Newtonian fluids. In the first example, we can even show that the solution exists globally.

L -estimates for the square root of elliptic systems with mixed boundary conditions

“…Rather than that, we need to prepare our results for the aforementioned applications to nonautonomous parabolic problems, where a family of operators L t with uniform ellipticity parameters in t acts in the spatial variables, and one needs the same uniformity on estimates for L 1/2 t . This was asked for in [17,35,35,42]. It is only implicit in [6] and sometimes all but impossible to track.…”

Stability properties of stochastic maximal L-regularity

“…In a future paper we will use the theory of the current paper to study quasilinear stochastic evolution equations. In particular we plan to obtain a version of [17,18] with weights in time. Because of the weights in time one can treat rough initial data.…”

Pathwise mild solutions for quasilinear stochastic partial differential equations

“…The issue (a) of blow-up certainly also occurs already for many classical SPDEs (see e.g. [19,25,47]) but also plays a key role for quasilinear SPDE problems [36]. For completeness, we provide two very simple quasilinear counter-examples involving (a) and (b) to demonstrate that we cannot expect global-in-time existence for (1.1) in general; see Section 5.…”

“…There are also several works exploiting the additional assumption of monotone coefficients [43] particularly in the case of the stochastic porous medium equation [9,10], where A(u) = ∆(a(u)) for a maximal monotone map a and F ≡ 0. Other approaches to quasilinear SPDEs are based upon a gradient structure [29], approximation methods [38,39], kinetic solutions [20,30], or directly looking at strong (in the PDE sense) solutions [36].One may ask, why one might want to prove the existence of pathwise mild solutions obtained by a suitable variations-of-constants/Duhamel formula [34] instead of working with weak solutions obtained in a formulation via test functions [26]? One reason is that a mild formulation is often more natural to work with in the context of (random) dynamical systems for SPDEs [18,34].…”