Random attractors for locally monotone stochastic partial differential equations

Abstract

The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to SPDE with compact embeddings in the Gelfand triple and singular coercivity. For ergodic, monotone, contractive random dynamical systems it is proven that the attractor consists of a single random point. In case of real, linear multiplicative noise finite time extinction is obtained. Applications include stochastic generalized fast diffusion equations and stochastic generalized singular p-Laplace equations perturbed by Lévy noise with jump measure having finite first and second moments.2000 Mathematics Subject Classification AMS: Primary: 37L55, 60H15. Secondary: 37L30.

Ergodicity for Singular-Degenerate Stochastic Porous Media Equations

“…A different approach to the long-time behaviour of solutions to SPDEs is to analyze the existence and the structure of random attractors of random dynamical systems, as e. g. in [10,15,16,24,27,28]. A property which has turned out to be very useful in this context is order preservation of trajectories which are driven by the same noise, see, e. g., [1,12,23,25].…”

Small time asymptotics for SPDEs with locally monotone coefficients

“…On the other hand, some very interesting quasilinear SPDEs have been studied a lot recently, such as stochastic porous media equation and stochastic p-Laplace equation, see e.g. [9,26,27,28,29,38,39,42,43,46,49,58]) and references therein. We would like to investigate whether small time asymptotics (LDP) results also hold for those SPDE models or not?…”

“…Recently, this framework has been substantially extended by the second named author and Röckner in [40,41,42,43] for more general class of SPDE with coefficients satisfying the generalized coercivity and local monotonicity conditions. In recent years, various properties for SPDEs with monotone or locally monotone coefficients has been intensively investigated in the literature, such as small noise LDP [39,45,49,59], random attractors [26,27,28,29], Harnack inequality and applications [38], Wong-Zakai approximation and support theorem [46], ultra-exponential convergence [58], and existence of optimal controls [16].…”

“…Therefore, the assertion follows by Lemma 2.1 and Theorem 2. This model was pioneered by Ladyzhenskaya [35] and further analyzed by various authors (see [29,30] and the references therein). Compared to the power law fluids considered above, there is an additional fourth order term div(−2µ 1 ∆e(u)) present in the equation.…”

“…In fact, for general d ≥ 2, using the Gagliardo-Nirenberg inequality, it is possible to find a "maximal" range (1, p d ] of p to which the locally monotone variational framework could apply (cf. [29]).…”