Abstract
The dynamics and invariant manifolds for a nonlocal stochastic Swift-Hohenberg equation with multiplicative noise are investigated. Using a stochastic transformation process, a nonlocal stochastic Swift-Hohenberg equation is studied with either a positive kernel or a non-negative kernel. Then the dynamics, existence, and uniqueness of a global random attractor for the nonlocal stochastic Swift-Hohenberg equation is shown. Moreover, the existence of a local random invariant manifold of the corresponding random dynamical system for the nonlocal stochastic Swift-Hohenberg equation with multiplicative noise is obtained using the technique of a cut-off function and the Lyapunov-Perron method.
Highlights
1 Introduction It is well known that fluid convection due to density gradients arises in geophysical fluid flows in the atmosphere, oceans, and the earth’s mantle
The global random attractor and existence of invariant manifolds are investigated for the two-dimensional nonlocal stochastic Swift-Hohenberg equation with multiplicative noise defined on a bounded planar domain D in Rd (d = )
3 Nonlocal stochastic Swift-Hohenberg equation with positive kernel we mainly show the global random attractor, for the nonlocal stochastic Swift-Hohenberg equation with positive kernel
Summary
It is well known that fluid convection due to density gradients arises in geophysical fluid flows in the atmosphere, oceans, and the earth’s mantle. When the distance from the change of stability is sufficiently small, or the Rayleigh number is near thermal equilibrium, the influence of small noise or molecular noise is detected in various convection experiments [ – ]. It is difficult to stabilize the control parameters (e.g. the temperature in Rayleigh-Bénard convection) to the precision of the noise strength, which is extremely small in the case of thermal fluctuations. When the effects of thermal fluctuations (i.e. additive noise) on the onset of convective motion are considered in the Bénard system, a local stochastic Swift-Hohenberg equation with additive noise [ ] is proposed: ut = –( + ∂xx) u + μu – u + σ ξ. A local stochastic Swift-Hohenberg equation with multiplicative noise[ ] arises, ut = –( + ∂xx) u + μu – u + σ u ◦ ξ
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