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

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

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

Exponential stability of an incompressible non-Newtonian fluid with delay

The existence and uniqueness of stationary solutions to an incompressible non-Newtonian fluid are first established. The exponential stability of steady-state solutions is then analyzed by means of four different approaches. The first is the classical Lyapunov function method, while the second one is based on a Razumikhin type argument. 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. Some comments concerning several open research directions about this model are also included.Ministerio de Economía y Competitividad (MINECO). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Junta de AndalucíaNational Science Foundation of ChinaScience and Technology Commission of Shanghai MunicipalityShanghai Leading Academic Discipline Projec

Long time behavior of stochastic parabolic problems with white noise in materials with thermal memory

The existence and limiting behavior of the solutions of stochastic parabolic problems with thermal memory are investigate in the cases that the nonlinear term satisfies subcritical and critical growth conditions. The existence, uniqueness and continuity of solutions is proved by a semigroup method and the Lax-Milgram theorem, then the dynamics of solutions is analyzed by a priori estimates. In particular, the existence of pullback random attractors for the random dynamical system associated to the problem is established and the upper semi-continuity of the pullback random attractors is verified.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

The asymptotic behaviour of fractional lattice systems with variable delay

The existence and uniqueness of global solutions for a fractional functional differential equation is established. The asymptotic behaviour of a lattice system with a fractional substantial time derivative and variable time delays is investigated. The existence of a global attracting set is established. It is shown to be a singleton set under a certain condition on the Lipschitz constant

<em>Helicobacter pylori</em> Infection and Endothelial Dysfunction

Endothelial cells play a critical role in maintaining the integrity of vascular structure and function. Endothelial dysfunction is closely associated with the development and progression of cardiovascular diseases (CVDs) like hypertension (HTN) and atherosclerosis. Gut microorganisms significantly contribute to atherosclerosis and related CVDs. Helicobacter pylori (H. pylori) colonizes in human gastric epithelium in a significant portion of general population in the world. Patients with H. pylori infection have significantly increased risk for CVDs including atherosclerosis, HTN, coronary heart disease, and cerebrovascular disease especially in younger patients (< 65 years old). H. pylori infection significantly impairs vascular endothelial function through multiple mechanisms including increased reactive oxygen species production and oxidative stress, inflammation, decreased nitric oxide formation, modification of the expression of cytokines and microRNAs, abnormalities of lipid and glucose metabolisms, and exosomes-mediated pathways. Endothelial dysfunction associated with H. pylori infection is reversible in both animal model and human subjects. Accumulating data suggests that H. pylori infection is an important risk factor for endothelial dysfunction and CVDs especially in young patients. Screening young male population for H. pylori infection and treating accordingly could be an effective approach for early prevention of CVDs especially premature atherosclerosis associated with H. pylori infection

Application Value of Limb Ischemic Preconditioning in Preventing Intradialytic Hypotension during Maintenance Hemodialysis

Introduction: The aim of this study was to investigate the efficacy and safety of limb ischemia preconditioning (LIPC) in the treatment of intradialytic hypotension (IDH) in patients with maintenance hemodialysis (MHD). Methods: This was a single-center, prospective, and randomized controlled case study. A total of 38 patients with MHD who met the inclusion criteria from September 2021 to August 2022 were selected from the Blood Purification Center of our hospital. They were randomly divided into the LIPC group (n = 19) and the control group (n = 19). For patients in the LIPC group, the femoral artery blood flow was blocked with an LIPC instrument for 5 min (pressurized to 200 mm Hg) before each dialysis, and they were reperfused for 5 min. The cycle was repeated five times, with a total of 50 min for 12 weeks. The control group was pressurized to 20 mm Hg with an LIPC instrument, and the rest was the same as the LIPC group. The blood pressure of 0 h, 1 h, 2 h, 3 h, 4 h, and body weight before and after hemodialysis were measured in the two groups during hemodialysis, the incidence of IDH and the changes of serum troponin I (TNI) and creatine kinase isoenzyme MB (CK-MB) levels before and after the intervention were observed, and the ultrafiltration volume and ultrafiltration rate were recorded. Results: At the 8th and 12th week after intervention, the MAP in the LIPC group was higher than that in the control group (103.28 ± 12.19 mm Hg vs. 93.18 ± 11.11 mm Hg, p = 0.04; 101.81 ± 11.36 mm Hg vs. 91.81 ± 11.92 mm Hg, p = 0.047). The incidence of IDH in the LIPC group was lower than that in the control group (36.5% vs. 43.1%, p = 0.01). The incidence of clinical treatment in IDH patients in the LIPC group was lower than that in the control group (6.3% vs. 12.4%, p = 0.00). The incidence of early termination of hemodialysis in the LIPC group was lower than that in the control group (1.6% vs. 3.8%, p = 0.01). The levels of TNI and CK-MB in the LIPC group after the intervention were lower than those in the control group (322.30 ± 13.72 ng/dL vs. 438.50 ± 24.72 ng/dL, p = 0.00; 159.78 ± 8.48 U/dL vs. 207.00 ± 8.70 U/dL, p = 0.00). The changes of MAP before and after the intervention were negatively correlated with the changes of TNI and CK-MB before and after the intervention (r = −0.473, p = 0.04; r = −0.469, p = 0.04). There were no differences in dry body mass and ultrafiltration rate between the two groups before and after the LIPC intervention (p &gt; 0.05). Multiple linear regression analysis shows that TNI is the main influencing factor of ΔMAP. No LIPC-related adverse events were found during the study period. Conclusion: LIPC can effectively reduce the incidence of IDH in patients with MHD and may be associated with the alleviation of myocardial damage