Search

This article concerns the dynamical transitions of the stochastic Swift-Hohenberg equation with multiplicative noise on a two-dimensional domain (-L, L) × (-L, L). With α and L regarded as parameters, we show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation near the critical points, and the impact of noise on stochastic bifurcation of the Swift-Hohenberg equation. We find the approximation representation of the manifold and the corresponding reduced systems for stochastic Swift-Hohenberg equation when L2 and √ 2L1 are close together. [ABSTRACT FROM AUTHOR]

The stochastic 3D Navier–Stokes equation with damping driven by a multiplicative noise is considered in this paper. The stability of weak solutions to the stochastic 3D Navier–Stokes equations with damping is proved for any β > 3 with any α > 0 and α ≥ 1 2 as β = 3. The weak solutions converge exponentially in the mean square and almost surely exponentially to the stationary solutions are proved for any β > 3 with any α > 0 and α ≥ 1 2 as β = 3. The stabilization of these equations is obtained for any β > 3 with any α > 0 and α ≥ 1 2 as β = 3. [ABSTRACT FROM AUTHOR]

In this paper, bifurcation of the generalized Swift–Hohenberg equation is considered. We first study the bifurcation of the generalized Swift–Hohenberg equation in one spatial dimension with three kinds of boundary conditions. With the help of Liapunov–Schmidt reduction, the original equation is transformed to the reduced system, and then the bifurcation analysis is carried out. Secondly, bifurcation of the generalized Swift–Hohenberg equation in two spatial dimensions with periodic boundary conditions is also considered, using the perturbation method, asymptotic expressions of the nontrivial solutions bifurcated from the trivial solution are obtained. Moreover, the stability of the bifurcated solutions is discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study a viscous capillary model of plasma as a so called Navier–Stokes–Poisson–Korteweg model. Our purpose is to prove the existence of global weak solutions for large data in a two-dimensional torus. We utilize the effective velocity and some interesting identities to remove the restrictions on the coefficients. It is worth pointing out that we prove the critical case that the value of viscosity coefficient is equivalent to the capillary coefficient ν = κ. In this case, the B-D entropy and method used in Antonelli and Spirito (2017) cannot be applied directly. Moreover, there is no friction term and cold pressure term in the model. In some senses, we improve the previous results. [ABSTRACT FROM AUTHOR]