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

Investigation of the functions of 53BP1 in DNA demethylation

DNA damage can be caused by various forms of genotoxic stress, including endogenous (reactive oxygen species, abnormal replication intermediates) and exogenous (UV, IR, and reactive chemicals) sources. DNA double-strand break (DSB) is believed to be one of the most serious lesions to cells because it can result in loss or rearrangement of genetic information, leading to cell death or carcinogenesis. The DNA damage response (DDR) involves multiple signal transduction pathways in that several different components act in concert to activate the cellular checkpoint. These components consist of sensors that sense DNA damage, signal transducers that generate and amplify the DNA damage signal, and effectors that induce cell cycle delay, programmed cell death, and DNA repair. Even though several candidate proteins have been implicated in DNA damage response, an official checkpoint-specific damage sensor is still unknown. 53BP1 seems to be one of the key-sensors of DNA DSBs, upstream of ATM. The function of 53BP1 is important for coupling ATM to its downstream targets, including p53 and Gadd45a. The activation of Gadd45a as a stress protein promotes epigenetic gene activation by repair-mediated DNA demethylation, thus linking both processes. DNA methylation is mediated by MBD2 as well as a class of DNMTs, which encompassing DNMT1, DNMT3a and DNMT3b. Recent studies have demonstrated that the DNA methylation mediated by DNMTs is associated with p53 signalling in maintaining genome stability. Since p53 is one of the downstream targets of 53BP1, it will be of interest to investigate the functions of 53BP1 in DNA demethylation and determine the possible link between 53BP1 and these related genes. The data presented here indicate that 53BP1 can induce DNA demethylation of single copy gene as well as repetitive elements in A549 cells. Meanwhile, the transient expression of 53BP1 can enhance DNA demethylation in combination with IR. Furthermore, the tumor suppressor gene RASSF1A was re-expressed following predominantly demethylation of CpG islands in the promoter analyzed by MSP and RT-PCR. Moreover, overexpression of 53BP1 caused a marked decrease in DNMT1 and DNMT3a mRNA expression as well as a significant increase in Gadd45a and MBD2 mRNA expression. To our best knowledge, the present study shows for the first time the involvement of 53BP1 in DNA demethylation process. Understanding the 53BP1-mediated network will certainly have an impact on numerous fields of medicine. However, how 53BP1 regulates different member of DNMTs need to be characterized. Further experiments of the precise mechanisms of 53BP1 in DNA demethylation may clarify this association and develop therapeutic alternatives designed to promote hypomethylation and re-activation of tumor suppressor genes

Análisis de sistemas dinámicos infinito-dimensionales asociados a ecuaciones en derivadas parciales funcionales.

Based on the theory of functional diferential equations, theory of semigroup, theory of random dynamical systems and theory of in nite dimensional dynamical systems, this thesis studies the long time behavior of several kinds of in nite dimensional dynamical systems associated to partial diferential equations containing some kinds of hereditary characteristics (such as variable delay, distributed delay or memory, etc), including existence and upper semicontinuity of pullback/random attractors and the stability analysis of stationary (steady-state) solutions. Three important mathematical-phyiscal models are considered, namely, reaction-di usion equation, 2D-Navier-Stokes equation as well as in-compressible non-Newtonian uids. Chapter 1 is devoted to the dynamics of an integer order stochastic reaction-difusion equation with thermal memory when the nonlinear term is subcritical or critical. Notice that our model contains not only memory but also white noise, which means it is not easy to prove the existence and uniqueness of solutions directly. In order to deal with this problem, we need introduce a new variable to transform our model into a system with two equations, and we use the Ornstein-Uhlenbeck to transfer this system into a deterministic ones only with random parameter. Then a semigroup method together with the Lax-Milgram theorem is applied to prove the existence, uniqueness and continuity of mild solutions. Next, the dynamics of solutions is analyzed by a priori estimates, and the existence of pullback random attractors is established. Besides, we prove that this pullback random attractors cannot explode, a property known as upper semicontinuity. But the dimension of the random attractor is still unknown. On the other hand, it has been proved that sometimes, especially when self-orgnization phenomena, anisotropic di usion, anomalous difusion occurs, a fractional order diferential equation can model this phenomena more precisely than a integer one. Hence, in Chapter 2, we focus on the asymptotical behavior of a fractional stochastic reaction-difusion equation with memory, which is also called fractional integro-diferential equation. First of all, the Ornstein-Uhlenbeck is applied to change the stochastic reaction-difusion equation into a deterministic ones, which makes it more convenient to solve. Then existence and uniqueness of mild solutions is proved by using the Lumer-Phillips theorem. Next, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining some uniform estimates and solutions decomposition. Moreover, the random attractor is shown to have nite Hausdorf dimension, which means the asymptotic behavior of the system is determined by only a nite number of degrees of freedom, though the random attractor is a subset of an in nite-dimensional phase space. But we still wonder whether this random attractor has inertial manifolds, which means this random attractor needs to be exponentially attracting. Besides, the long time behavior of time-fractional reaction-difusion equation and fractional Brownian motion are still unknown. The rst two chapters consider an important partial function diferential equations with in nite distributed delay. However, partial functional diferential equations include more than only distributed delays; for instance, also variable delay terms can be considered. Therefore, in the next chapter, we consider another signi cant partial functional diferential equation but with variable delay. In Chapter 3, we discuss the stability of stationary solutions to 2D Navier-Stokes equations when the external force contains unbounded variable delay. Notice that the classic phase space C which is used to deal with diferential equations with in nite delay does not work well for our unbounded variable delay case. Instead, we choose the phase space of continuous bounded functions with limits at1. Then the existence and uni- queness of solutions is proved by Galerkin approximations and the energy method. The existence of stationary solutions is established by means of the Lax-Milgram theorem and the Schauder xed point theorem. Afterward, the local stability analysis of stationary solutions is carried out by three diferent approaches: the classical Lyapunov function method, the Razumikhin-Lyapunov technique and by constructing appropriate Lyapunov functionals. It worths mentioning that the classical Lyapunov function method requires diferentiability of delay term, which in some extent is restrictive. Fortunately, we could utilize Razumikhin-Lyapunov argument to weak this condition, and only requires continuity of every operators of this equation but allows more general delay. Neverheless, by these methods, the best result we can obtain is the asymptotical stability of stationary solutions by constructing a suitable Lyapunov functionals. Fortunately, we could obtain polynomial stability of the steady-state in a particular case of unbounded variable delay, namely, the proportional delay. However, the exponential stability of stationary solutions to Navier-Stokes equation with unbounded variable delay still seems an open problem. We can also wonder about the stability of stationary solutions to 2D Navier-Stokes equations with unbounded delay when it is perturbed by random noise. Therefore, in Chapter 4, a stochastic 2D Navier-Stokes equation with unbounded delay is analyzed in the phase space of continuous bounded functions with limits at1. Because of the perturbation of random noise, the classical Galerkin approximations alone is not enough to prove the existence and uniqueness of weak solutions. By combing a technical lemma and Faedo-Galerkin approach, the existence and uniqueness of weak solutions is obtained. Next, the local stability analysis of constant solutions (equilibria) is carried out by exploiting two methods. Namely, the Lyapunov function method and by constructing appropriate Lyapunov functionals. Although it is not possible, in general, to establish the exponential convergence of the stationary solutions, the polynomial convergence towards the stationary solutions, in a particular case of unbounded variable delay can be proved. We would like to point out that the Razumikhin argument cannot be applied to analyze directly the stability of stationary solutions to stochastic equations as we did to deterministic equations. Actually, we need more technical, and this will be our forthcoming paper. We also would like to mention that exponential stability of other special cases of in nite delay remains as an open problem for both the deterministic and stochastic cases. Especially, we are interested in the pantograph equation, which is a typical but simple unbounded variable delayed diferential equation.We believe that the study of pantograph equation can help us to improve our knowledge about 2D{Navier-Stokes equations with unbounded delay. Notice that Chapter 3 and Chapter 4 are both concerned with delayed Navier-Stokes equations, which is a very important Newtonian uids, and it is extensively applied in physics, chemistry, medicine, etc. However, there are also many important uids, such as blood, polymer solutions, and biological uids, etc, whose motion cannot be modeled pre- cisely by Newtonian uids but by non-Newtonian uids. Hence, in the next two chapters, we are interested in the long time behavior of an incompressible non-Newtonian uids ith delay. In Chapter 5, we study the dynamics of non-autonomous incompressible non-Newtonian uids with nite delay. The existence of global solution is showed by classical Galerkin approximations and the energy method. Actually, we also prove the uniqueness of solutions as well as the continuous dependence of solutions on the initial value. Then, the existence of pullback attractors for the non-autonomous dynamical system associated to this problem is established under a weaker condition in space C([h; 0];H2) rather than space C([h; 0];L2), and this improves the available results that worked on non-Newtonian uids. However, we still would like to analyze the Hausdor dimension or fractal dimension of the pullback attractor, as well as the existence of inertial manifolds and morsedecomposition. Finally, in Chapter 6, we consider the exponential stability of an incompressible non Newtonian uids with nite delay. The existence and uniqueness of stationary solutions are rst established, and this is not an obvious and straightforward work because of the nonlinearity and the complexity of the term N(u). The exponential stability of steady state solutions is then analyzed by means of four diferent approaches. The rst one is the classical Lyapunov function method, which requires the diferentiability of the delay term. But this may seem a very restrictive condition. Luckily, we could use a Razumikhin type argument to weaken this condition, but allow for more general types of delay. In fact, we could obtain a better stability result by this technique. Then, a method relying on the construction of Lyapunov functionals and another one using a Gronwall-like lemma are also exploited to study the stability, respectively. We would like to emphasize that by using a Gronwall-like lemma, only the measurability of delay term is demanded, but still ensure the exponential stability. Furthermore, we also would like to discuss the dynamics of stochastic non-Newtonian uids with both nite delay and in nite delay. All the problems deserve our attraction, and actually, these are our forthcoming work

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

Master of Science

thesisA wireless, wearable, real-time gait asymmetry detection system-the Lower Extremity Ambulatory Feedback System (LEAFS)-has been validated by comparison to clinical motion capture (force plate and three-dimensional cameras) measurements, and evaluated in training sessions with seven subjects. LEAFS is a low-cost in-shoe gait detection device that provides real-time auditory feedback based on stance time ratio and allows long-term gait asymmetry training to be performed outside of the clinical environment. Stance time ratio, which is also known as Symmetry Ratio (SR), is calculated by dividing the stance time on one limb (typically the more affected limb) by the other, and control subjects have been shown to have SR of 1.02 ± 0.02. The validation test results demonstrate that the SR measured by LEAFS as compared to clinical motion capture results has a mean error of 0.003 ± 0.05 for control subjects and 0.008 ± 0.04 for subjects with unilateral trans-tibial amputations. The LEAFS was used for gait asymmetry training in seven subjects with unilateral trans-tibial amputations; subjects received six 30-minute training sessions over a 3-week training period. The results demonstrate that LEAFS is accurate at measuring mean SR of a trial of steps, and it is reliable and practical to use LEAFS to train the gait of patients with unilateral trans-tibial amputations by bringing their SR towards a normal range