Hydrodynamic regimes were characterized in terms of overall gas holdup, gas liquid mass transfer coefficient, drift-flux and liquid circulation velocity.
We study the limit of the solution of linear and semilinear second order PDEs of parabolic type, with rapidly oscillating periodic coefficients, singular drift, and singular coefficient of the zeroth order term.
At first, we evaluate the influence of all the dynamic and kinetic parameters (the diffusion coefficient, the drift velocity, and the transition rates between the diffusive and the active transport regimes) on simulated kICS correlation functions.
We further analysed the 30 persistent units by deriving a coefficient of drift of the centroids of the clusters in 3D principal component space.
For the sake of notational convenience, in (17) we denote by b the coefficients of drift and diffusion terms, respectively, for (phi =p_{0},bar {x},q); denote by f the generator for (psi =hat {x}_{0},p,k).
Most of the existing results on the robust stability require that the drift coefficient f and diffusion coefficient g of the stochastic system are either linear or nonlinear with linear growth condition.
We also put forward the following standard local Lipschitz condition on the drift coefficient f and the diffusion coefficient g.
Motivated by system (8) in Section 1, we propose the following nonlinear growth condition on the drift coefficient f and the diffusion coefficient g naturally.
We are given a stochastic differential equation (called forward) mathrm{d} X_{t}=S t,X_{t});mathrm{d} t+ sigma (t,X_{t});mathrm{d} W_{t}, X_{0}=x_{0leq0leq tleq T, where S t,x) is the drift coefficient, σ t,x 2 is the diffusion coefficient, and W t,0≤t≤T is a standard Wiener process.
In this paper we consider the general form of one-dimensional SDE with begin{aligned} begin{gathered} dX t,omega)=fbigl t,X t,omega) bigr),dt+gbigl t,X t,omega bigr),dW t,omega),quad t_{0}leq tleq T, X(t_{0},omega)=X_{0} omega), end{gathered} end{aligned} (1) where f is the drift coefficient, while g is the diffusion coefficient and (W t,omega)) is the Wiener process.
Theorem 3.7 Assume that the drift coefficient satisfies (3.1) and the diffusion coefficient satisfies (3.10) if the following holds for some positive constant λ: 〈 x, f ( x ) 〉 + 1 2 | g ( x ) | 2 D + | F ( x ) | 2 − 〈 x, g ( x ) 〉 2 ( D + | F ( x ) | 2 ) 2 ≤ − λ + P 3 ( | x | ) ( D + | x | 2 ) 2, where F ( x ) = x − θ Δ f ( x ) and D is some positive constant larger than a 1 θ Δ t.
