Schauder-type estimates for higher-order parabolic SPDEs

Abstract

In this paper we consider the Cauchy problem for 2m-order stochastic partial differential equations of parabolic type in a class of stochastic Hölder spaces. The Hölder estimates of solutions and their spatial derivatives up to order 2m are obtained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when m is odd or even: the condition for odd m coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even m it depends on the integrability index p. The sharpness of the new-found coercivity condition is demonstrated by an example.Keywords higher-order stochastic partial differential equations · coercivity condition · Hölder spaces · Schauder estimatesLet (Ω, F , (F t ) t≥0 , P) be a complete filtered probability space and {w k · } a sequence of independent standard Wiener processes adapted to the filtration F t . Consider the Cauchy problem for the following 2m-order stochastic partial differential equations (SPDEs) of non-divergence form:where the coefficients, the free terms, and the unknown function are all random fields defined on R n × [0, ∞) × Ω and adapted to F t . Typical examples of Equation (1.1) include the Zakai equation (see [38,33] for example), linearised stochastic Cahn-Hilliard equations (see [6,3] for example), and so on. General solvability theory for higher-order SPDEs of type (1.1) was first investigated in [26] under the framework of Hilbert spaces. This paper concerns the existence, uniqueness and regularity of solutions of (1.1) in some Hölder-type spaces that will be defined later. Regularity theory for linear equations often plays an important role in the study of nonlinear stochastic equations, see [36,4,7] and references therein. The weak solution of Equation (1.1), which satisfies the equation in the (analytic) distribution sense, and its regularity in the framework of Sobolev spaces have been investigated by many researchers. Results for the secondorder case (namely m = 1) are numerous and fruitful; for instance, a complete L p -theory (p ≥ 2) of secondorder parabolic SPDEs has been developed, see for example [30, 25, 26,

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

“…Although we will only use stochastic maximal regularity in the L p (L q ) scale, it is important to note that it can also be considered in different scales such as the Besov scale and Hölder scale. For details on this we refer to [Brz95,BH09,DL98,LV20] for the Besov scale and to [DL19,DLZ20,WD20]. The Hölder case can be seen as a stochastic analogue of Schauder theory (see [GT83,Kry96a]).…”

“…However a C α -theory (see e.g. [DL19,WD20] for the linear C α -theory) requires regularity assumptions on the noise, the coefficients, and on u 0 which can be either too restrictive for physical applications, or does not fit the scaling property of the SPDE considered. Moreover, L p (L q )-theory provides blow-up criteria that can be combined with energy estimates to prove global existence.…”

Higher order moments for SPDE with monotone nonlinearities

“…• semilinear second-order parabolic equations, namely the stochastic Burgers' equation (section 4.3) and the stochastic Navier-Stokes equations in two dimensions (section 4.4), • systems of SPDEs (section 4.5) and higher-order SPDEs (section 4.6) as treated in [12,23],…”

“…The above equation was considered in [23], and below we will show that the p-dependent wellposedness results can be obtained within our abstract framework. Additionally, our coefficients to be space dependent.…”