Gibbsian Dynamics and Ergodicity¶for the Stochastically Forced Navier–Stokes Equation

We study stationary measures for the two-dimensional Navier–Stokes equation with periodic boundary condition and random forcing. We prove uniqueness of the stationary measure under the condition that all “determining modes” are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of the low modes. Received: 21 November 2000 / Accepted: 9 December 2000     

Cauchy–Born Rule and the Stability of Crystalline Solids: Static Problems

We study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature. We prove, under certain sharp stability conditions, that the correct nonlinear elasticity model is given by the classical Cauchy–Born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the Cauchy–Born rule. The analysis is carried out for both simple and complex lattices, and for this purpose, we develop the necessary tools for performing asymptotic analysis on such lattices. Our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids.     

BN‐Embedded Polycyclic Aromatic Hydrocarbon Oligomers: Synthesis,Aromaticity, and Reactivity

BN‐embedded oligomers with different pairs of BN units were synthesized by electrophilic borylation. Up to four pairs of BN units were incorporated in the large polycyclic aromatic hydrocarbons (PAHs). Their geometric, photophysical, electrochemical, and Lewis acidic properties were investigated by X‐ray crystallography, optical spectroscopy, and cyclic voltammetry. The B?N bonds show delocalized double‐bond characteristics and the conjugation can be extended through the trans‐orientated aromatic azaborine units. Calculations reveal the relatively lower aromaticity for the inner azaborine rings in the BN‐embedded PAH oligomers. The frontier orbitals of the longer oligomers are delocalized over the inner aromatic rings. Consequently, the inner moieties of the BN‐embedded PAH oligomers are more active than the outer parts. This is confirmed by a simple oxidation reaction, which has significant effects on the aromaticity and the intramolecular charge‐transfer interactions.     

A class of homogenization problems in the calculus of variations

We study a class of integral functionals for which the integrand f^e(x, u, ?u) is an oscillatory function of both x and u. Our method is based on the concept of Γ-convergenee. Technical difficulties arise because f^e(x, u, ?u) is not convex or equi-continuous in u with respect to e. Two somewhat different approaches, based respectively on abstract convergence theorems and the study of affine functions, are exploited together to overcome these technical difficulties. As an application, we give another proof of a homogenization result of P. L. Lions, G. Papanicolaou, and S. R. S. Varadhan for Hamilton-Jacobi equations.     

Homogenization and convergence of the vortex method for 2-D euler equations with oscillatory vorticity fields

The 2-D incompressible Euler equations with oscillatory vorticity fields are studied. A homogenization result for 2-D Euler equations in velocity-vorticity formulation is obtained and weak continuity of the equations is proved. Convergence of the vortex method is analyzed in the case when the continuous vorticity is not well resolved by the computational particles. Numerical results are given. Comparisons are made with the corresponding finite difference approximation.     

Large time behavior and homogenization of solutions of two-dimensional conservation laws

We study the large time behavior of solutions of scalar conservation laws in one and two space dimensions with periodic initial data. Under a very weak nonlinearity condition, we prove that the solutions converge to constants as time goes to infinity. Even in one space dimension our results improve the earlier ones since we only require the fluxes to be nonlinear in a neighborhood of the mean value of the initial data. We then use these results to study the homogenization problem for scalar conservation laws with oscillatory initial data.     

第Ⅰ类两态叠加多模叠加态光场的N次方X压缩

利用多模辐射场广义非线性等幂次高次差压缩的一般理论,对第Ⅰ类两态叠加多模叠加态光场的等幂次N次方X压缩特性进行了详细研究,发现了其存在等幂次N次方X压缩效应的条件,还发现第Ⅰ、第Ⅱ两类两态叠加多模叠加态光场之间存在相似压缩和压缩简并现象,从而再次证明第Ⅰ类两态叠加多模叠加态光场是一种典型的多模非经典光场.