Stochastic Partial Differential Equations: An Introduction

Liu, Wei; Röckner, Michael

doi:10.1007/978-3-319-22354-4

Cited by 414 publications

(416 citation statements)

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“…x , which was used in many previous works, see for example [LR15] and the references therein, since the non-linear diffusion coefficients behave nicely in L 1…”

Section: Comments On the Literaturementioning

confidence: 99%

Entropy solutions for stochastic porous media equations

Dareiotis

Gerencsér

Gess

2019

Journal of Differential Equations

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“…x , which was used in many previous works, see for example [LR15] and the references therein, since the non-linear diffusion coefficients behave nicely in L 1…”

Section: Comments On the Literaturementioning

confidence: 99%

Entropy solutions for stochastic porous media equations

Dareiotis

Gerencsér

Gess

2019

Journal of Differential Equations

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“…To the best of our knowledge, the case of the operator A has no linear part hasn't been studied yet, such as the porous medium operator and the p-Laplace operator.Hence, the main purpose of this paper is to prove the strong averaging principle for slowfast stochastic partial differential equations within the (generalized) variational framework, i.e., locally monotone and strongly monotone coefficients for the slow and fast equations respectively. Our result covers a large class of examples (see [20, Sections 4 and 5]), especially for the case that the slow equation is a quasilinear stochastic partial differential equation, such as the stochastic porous medium equation or the stochastic p-Laplace equation. Our result is also applicable to the stochastic Burgers type equation and stochastic two dimensional Navier-Stokes equation, whose coefficients only satisfy the local monotonicity conditions.…”

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confidence: 91%

Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients

Liu

Röckner

Sun

et al. 2020

Journal of Differential Equations

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This paper is devoted to proving the strong averaging principle for slow-fast stochastic partial differential equations with locally monotone coefficients, where the slow component is a stochastic partial differential equations with locally monotone coefficients and the fast component is a stochastic partial differential equations (SPDEs) with strongly monotone coefficients. The result is applicable to a large class of examples, such as the stochastic porous medium equation, the stochastic p-Laplace equation, the stochastic Burgers type equation and the stochastic 2D Navier-Stokes equation, which are the nonlinear stochastic partial differential equations. The main techniques are based on time discretization and the variational approach to SPDEs. 1 2 WEI LIU, MICHAEL RÖCKNER, XIAOBIN SUN, AND YINGCHAO XIEwhere ε > 0 is a small parameter describing the ratio of the time scale between the slow component X ε t and the fast component Y ε t , and the coefficientsThe averaging principle has a long and rich history in multiscale models, which has wide applications in material sciences, chemistry, fluid dynamics, biology, ecology and climate dynamics, see, e.g., [1,10,17,21] and the references therein. Usually, a multiscale model can be described through coupled equations, which correspond to the "slow" and "fast" component, respectively. The averaging principle is essential to describe the asymptotic behavior of the slow component, i.e., the slow component will convergence to the so-called averaged equation. Bogoliubov and Mitropolsky [2] first studied the averaging principle for deterministic systems, which then was extended to stochastic differential equations by Khasminskii [18].Since the averaging principle for a general class of stochastic reaction-diffusion systems with two time-scales were investigated by Cerrai and Freidlin in [6], the averaging principle for slow-fast stochastic partial differential equations has initiated further studies in the past decade, including other types of SPDEs, various ways of convergence and rates of convergence. For instance, Bréhier obtained the strong and weak orders in averaging for stochastic evolution equation of parabolic type with slow and fast time scales in [3]. Fu, Wan and Liu proved the strong averaging principle for stochastic hyperbolic-parabolic equations with slow and fast time-scales in [13]. Cerrai and Lunardi studied the averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations in [7]. For some further results on this topic, we refer to [4,12,22,23] and the references therein.However, the references we mentioned above always assume that the coefficients satisfy Lipschitz conditions, and there are few results on the average principle for SPDEs with nonlinear terms. For example, stochastic reaction-diffusion equations with polynomial coefficients [5], stochastic Burgers equation [9], stochastic two dimensional Navier-Stokes equations [19], stochastic Kuramoto-Sivashinsky equation [14], stochastic Schrödinger equation [15] an...

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“…For a general introduction to infinite dimensional Hilbert space valued stochastic calculus, see [8], [10] or [9]. Taking into account, (3.2), (3.3) and that f, g…”

Section: Preliminariesmentioning

confidence: 99%