Analysis of a stochastic 2D–Navier-Stokes model with infinite delay

Some results concerning a stochastic 2D Navier-Stokes system when the external forces contain hereditary characteristics are established. The existence and uniqueness of solutions in the case of unbounded (infinite) delay are first proved by using the classical technique of Galerkin approximations. The local stability analysis of constant solutions (equilibria) is also carried out by exploiting two approaches. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. The asymptotic stability and hence, the uniqueness of equilibrium solution are obtained by constructing Lyapunov functionals. Moreover, some sufficient conditions ensuring the polynomial stability of the equilibrium solution in a particular case of unbounded variable delay will be provided. Exponential stability for other special cases of infinite delay remains as an open problem.Ministerio de Economía y Competitividad (MINECO). EspañaJunta de AndalucíaNational Natural Science Foundation of ChinaScience and Technology Commission of Shanghai Municipalit

A novel directional coupler utilizing a left-handed material

A novel directional coupler with a left-handed material (LHM) layer between two single-mode waveguides of usual material is introduced. The coupling system is analyzed with the supermode theory. It is shown that such a LHM layer of finite length can shorten significantly the coupling length for the two single-mode waveguides. A LHM layer with two slowly tapered ends is used to avoid the reflection loss at the ends.Comment: 3 pages, 2 figure

Dynamics of a non-autonomous incompressible non-Newtonian fluid with delay

We first study the well-posedness of a non-autonomous incompressible non-Newtonian fluid with delay. The existence of global solution is obtained by classical Galerkin approximation and the energy method. Actually, we also prove the uniqueness of solution as well as the continuous dependence on the initial value. Then we analyze the long time behavior of the dynamical system associated to the incompressible non-Newtonian fluid. Finally, we establish the existence of pullback attractors for the non-autonomous dynamical system associated to the problem.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de AndalucíaNational Natural Science Foundation of ChinaScience and Technology Commission of Shanghai MunicipalityShanghai Leading Academic Discipline Projec

Stability results for 2D Navier-Stokes equations with unbounded delay

Some results related to 2D Navier-Stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed. First, the existence and uniqueness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solution is then established by means of the Lax-Milgram theorem and the Schauder fixed point theorem. The local stability analysis of stationary solutions is studied by several different methods: the classical Lyapunov function method, the Razumikhin-Lyapunov technique and by constructing appropriate Lyapunov functionals. Finally, we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay. Exponential stability in this infinite delay setting remains as an open problem.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalJunta de AndalucíaNational Science Foundation of ChinaScience and Technology Commission of Shanghai MunicipalityShanghai Leading Academic Discipline Projec

Interaction between negative and positive index medium waveguides

The coupling between negative and positive index medium waveguides is investigated theoretically in this paper. A coupled mode theory is developed for such a waveguide system and its validity is verified. Interesting phenomena in the coupled waveguides are demonstrated, which occur in the case when the negative index medium waveguide in isolation guides its mode backward. A new type of coupled mode solution that varies exponentially with the coupling length is found in the special case when the propagation constants of two individual waveguides are nearly the same. A coupler operating in this case is insensitive to the coupling length, and its coupling efficiency can reach 100% as long as the coupling length is long enough. However, when the propagation constants of the two individual waveguides differ greatly, the coupled mode solution is still a periodic function of the coupling length, but the coupled power is output backward. In addition, the modes in the composite waveguide system are also studied using the coupled mode theory, and their fundamental properties are revealed.Comment: 7 page

Finite-size effects of a left-handed material slab on the image quality

The characteristics of an imaging system formed by a left-handed material (LHM) slab of finite length are studied, and the influence of the finite length of the slab on the image quality is analyzed. Unusual phenomena such as surface bright spots and negative energy stream at the image side are observed and explained as the cavity effects of surface plasmons excited by the evanescent components of the incident field. For a thin LHM slab, the cavity effects are found rather sensitive to the length of the slab, and the bright spots on the bottom surface of the slab may stretch to the image plane and degrade the image quality.Comment: changes in the content and the title, and also the figure

Imaging Studies of Characteristics for a Slab of a Lossy Left-handed Material

The characteristics of an imaging system formed by a slab of a lossy left-handed material (LHM) are studied. The transfer function of the LHM imaging system is written in an appropriate product form with each term having a clear physical interpretation. A tiny loss of the LHM may suppress the transmission of evanescent waves through the LHM slab and this is explained physically. An analytical expression for the resolution of the imaging system is derived. It is shown that it is impossible to make a subwavelength imaging by using a realistic LHM imaging system unless the LHM slab is much thinner than the wavelength.Comment: 15 pages, 10 figure