Asymptotic behaviour of time‐dependent Ginzburg–Landau equations of superconductivity

In this paper, we establish the global fast dynamics for the time‐dependent Ginzburg–Landau equations of superconductivity. We show the squeezing property and the existence of finite‐dimensional exponential attractors for the system. In addition we prove the existence of the global attractor in L² × L² for the Ginzburg–Landau equations in two spatial dimensions. Copyright © 1999 John Wiley & Sons, Ltd.     

Approximate Inertial Manifolds to the Newton-Boussinesq Equations

Approximate inertial manifolds are related to the study of long time behaviour of dissipative partial differential equa tions. In this paper, we construct two approximate inertial manifolds for the two dimensional Newton-Boussinesq Equations. The orders of approximations of these manifolds to the global attractor are derived.     

Hydroxyl Radical-Mediated Efficient Photoelectrocatalytic NO Oxidation with Simultaneous Nitrate Storage Using A Flow Photoanode Reactor

The selective conversion of dilute NO pollutant into low-toxic product and simultaneous storage of metabolic nitrogen for crop plants remains a great challenge from the perspective of waste management and sustainable chemistry. This study demonstrates that this bottleneck can be well tackled by refining the reactive oxygen species (ROS) on Ni-modified NH₂-UiO-66(Zr) (Ni@NU) using nickel foam (NF) as a three-dimensional (3D) substrate through a flow photoanode reactor via the gas-phase photoelectrocatalysis. By rationally refining the ROS to ⋅OH, Ni@NU/NF can rapidly eliminate 82 % of NO without releasing remarkable NO₂ under a low bias voltage (0.3 V) and visible light irradiation. The abundant mesoporous pores on Ni@NU/NF are conducive to the diffusion and storage of the formed nitrate, which enables the progressive conversion NO into nitrate with selectivity over 99 % for long-term use. Through calculation, 90 % of NO could be recovered as the nitrate species, indicating that this state-of-the-art strategy can capture, enrich and recycle the pollutant N source from the atmosphere. This study offers a new perspective of NO pollutant treatment and sustainable nitrogen exploitation, which may possess great potential to the development of highly efficient air purification systems for industrial and indoor NO_x control.     

Random attractors for stochastic reaction-diffusion equations on unbounded domains

The existence of a pullback attractor is established for a stochastic reaction-diffusion equation on all n-dimensional space. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears as spatially distributed temporal white noise. The reaction-diffusion equation is recast as a random dynamical system and asymptotic compactness for this is demonstrated by using uniform a priori estimates for far-field values of solutions.     

Asymptotic behavior of the Newton–Boussinesq equation in a two-dimensional channel

We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor.     

GEVREY CLASS REGULARITY AND APPROXIMATE INERTIAL MANIFOLDS FOR THE NEWTON-BOUSSINESQ EQUATIONS

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Pullback attractors for non-autonomous reaction-diffusion equations on ?n

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space Rn when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L2(Rn) and H1(Rn), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.     

Exponential Attractors for the Generalized Ginzburg-Landau Equation

Abstract In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation * The author is supported by the Postdoctoral Foundation of China     

APPROXIMATION TO THE GLOBAL ATTRACTOR FOR A NONLINEAR SCHRODINGER EQUATION

Abstract. In the present paper, we deal with the long-time behavior of dissipative partial differenttial equations, and we construct the approximate inertial mardfolds for the nonlbaear Stringer equation with a zero order dlssipation. The order of approximation of these manlfolde to the global attractor is derived.