Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise

Abstract

Motivated by applications to a manifold of semilinear and quasilinear stochastic partial differential equations (SPDEs) we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Lévy processes. We illustrate the main result of our paper by showing how it can be applied to various types of SPDEs such as stochastic reaction-diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature. investigated by many authors, see e.g. Kallianpur and Xiong [27], Albeverio et al [2], Mueller et al [43, 44], Applebaum and Wu [3], Mytnik [45], Truman and Wu [58], Hausenblas [25, 26], Mandrekar and Rüdiger [39], Röckner and Zhang [52], Dong et al [16,17,18], Marinelli and Röckner [40], Bo et al [5], Brzeźniak et al [9,10,12] and the recent monograph by Peszat and Zabczyk [47]. The last reference can also be used for more detailed expositions and references.In this paper, we aim to establish a framework in which one can treat a large number of SPDEs driven by Lévy type noises including stochastic reaction-diffusion equations, stochastic Burgers type equations, stochastic 2D Navier-Stokes equations and stochastic equations of non-Newtonian fluids etc. The line of investigation proposed in this paper began with the celebrated works by Pardoux [46] and Krylov and Rozovskii [29], and later it was further developed by many authors, see e.g. Gyöngy and Krylov [22], Gyöngy [24]. Ren et al [49], Röckner and Wang [51] and Zhang [59]. Roughly speaking, for stochastic equations in finite dimensional spaces, the existence and uniqueness result was obtained under the local monotonicity assumption for the coefficients, see [29] for SDEs driven by Brownian motion and [22] for SDEs driven by (possibly discontinuous) locally square integrable martingales. However, concerning the existence and uniqueness of strong solutions to SPDEs in infinite dimensional spaces driven by Wiener processes or local martingales, all results were established for the globally monotone coefficients SPDE (cf. [29,24,49,59]).Recently, the classical variational framework has been extended by the second named author and Röckner in [36] for SPDE driven by Wiener process in Hilbert space with locally monotone coefficients. In [36] the authors showed that the local monotonicity method first used by Menaldi and Sritharan [41] for stochastic 2D Navier-Stokes equations (and later used by Sritharan and Sundar [55], Chueshov and Millet [13] for various stochastic equations of hydrodynamics) can be generalized to such an extent that the extended variational framework is applicable to all the equations investigated in [29,48,41,55,13].On the other hand, there are not many papers studying non-Lipschitz SPDEs driven by Lévy type noises with small jumps. The first and third named author proved in [11] the existence and uniqueness of solutions to stochastic nonlinear beam equations driven by Lévy t...

Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives

“…For the deterministic case, one might refer to the monographs [4,9,30,55,64] and the references therein. But also the stochastic case (SPDE) has attracted more and more attention in recent years, we refer to some classical references [7,29,47] and recent works [6,12,25,32,33,34,35,51,65] (see also the references therein).…”

Approximations of stochastic Navier–Stokes equations

“…Under the assumption (H.2) and h ∈ H, it is known that for ε ≤ ε 0 , equation (3.1) admits a unique solution (see e.g. [4]).…”

Maximal Inequalities and Exponential Estimates for Stochastic Convolutions Driven by Lévy-type Processes in Banach Spaces with Application to Stochastic Quasi-Geostrophic Equations

“…Remark 4.10. One might also study (4.1) by following the method from previous works [4,5]. However, our approach here is different.…”