Pointwise wave behavior of the Navier-Stokes equations in half space

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.Comment: Comments and references are added and some typos are corrected. Accepted by DCDS-

Pointwise wave behavior of the Navier-Stokes equations in half space

In this paper, we investigate the pointwise behavior of the solution for the compressible Navier-Stokes equations with mixed boundary condition in half space. Our results show that the leading order of Green's function for the linear system in half space are heat kernels propagating with sound speed in two opposite directions and reflected heat kernel (due to the boundary effect) propagating with positive sound speed. With the strong wave interactions, the nonlinear analysis exhibits the rich wave structure: the diffusion waves interact with each other and consequently, the solution decays with algebraic rate.Comment: Comments and references are added and some typos are corrected. Accepted by DCDS-

Boundary Wave and Interior Wave Propagations

Ph.DDOCTOR OF PHILOSOPH

Bilateral boundary control of an input delayed 2-D reaction-diffusion equation

In this paper, a delay compensation design method based on PDE backstepping is developed for a two-dimensional reaction-diffusion partial differential equation (PDE) with bilateral input delays. The PDE is defined in a rectangular domain, and the bilateral control is imposed on a pair of opposite sides of the rectangle. To represent the delayed bilateral inputs, we introduce two 2-D transport PDEs that form a cascade system with the original PDE. A novel set of backstepping transformations is proposed for delay compensator design, including one Volterra integral transformation and two affine Volterra integral transformations. Unlike the kernel equation for 1-D PDE systems with delayed boundary input, the resulting kernel equations for the 2-D system have singular initial conditions governed by the Dirac Delta function. Consequently, the kernel solutions are written as a double trigonometric series with singularities. To address the challenge of stability analysis posed by the singularities, we prove a set of inequalities by using the Cauchy-Schwarz inequality, the 2-D Fourier series, and the Parseval's theorem. A numerical simulation illustrates the effectiveness of the proposed delay-compensation method.Comment: 11 pages, 3 figures(including 8 sub-figures

Hopf bifurcation analysis in the 1-D Lengyel–Epstein reaction–diffusion model

AbstractThe Lengyel–Epstein model with diffusion and homogeneous Neumann boundary condition is considered in this paper. We give the existence of multiple spatially non-homogeneous periodic solutions though all the parameters of the system are spatially homogeneous

Emergence of a periodically rotating one-point cluster in a thermodynamic Cucker-Smale ensemble

We study emergent behaviors of thermomechanical Cucker-Smale (TCS) ensemble confined in a harmonic potential field. In the absence of external force field, emergent dynamics of TCS particles has been extensively studied recently under various frameworks formulated in terms of initial configuration, system parameters and network topologies. Moreover, the TCS model does not exhibit rotating motions in the absence of an external force field. In this paper, we show the emergence of periodically rotating onepoint cluster for the TCS model in a harmonic potential field using elementary energy estimates and continuity argument. We also provide several numerical simulations and compare them with analytical results.N

Emergence of a periodically rotating one-point cluster in a thermodynamic Cucker-Smale ensemble

We study emergent behaviors of thermomechanical Cucker-Smale (TCS) ensemble confined in a harmonic potential field. In the absence of external force field, emergent dynamics of TCS particles has been extensively studied recently under various frameworks formulated in terms of initial configuration, system parameters and network topologies. Moreover, the TCS model does not exhibit rotating motions in the absence of an external force field. In this paper, we show the emergence of periodically rotating onepoint cluster for the TCS model in a harmonic potential field using elementary energy estimates and continuity argument. We also provide several numerical simulations and compare them with analytical results.N

Protective Effects of Bifidobacterium on Intestinal Barrier Function in LPS-Induced Enterocyte Barrier Injury of Caco-2 Monolayers and in a Rat NEC Model

<div><p>Zonulin protein is a newly discovered modulator which modulates the permeability of the intestinal epithelial barrier by disassembling intercellular tight junctions (TJ). Disruption of TJ is associated with neonatal necrotizing enterocolitis (NEC). It has been shown bifidobacterium could protect the intestinal barrier function and prophylactical administration of bifidobacterium has beneficial effects in NEC patients and animals. However, it is still unknown whether the zonulin is involved in the gut barrier dysfunction of NEC, and the protective mechanisms of bifidobacterium on intestinal barrier function are also not well understood. The present study aims to investigate the effects of bifidobacterium on intestinal barrier function, zonulin regulation, and TJ integrity both in LPS-induced enterocyte barrier injury of Caco-2 monolayers and in a rat NEC model. Our results showed bifidobacterium markedly attenuated the decrease in transepithelial electrical resistance and the increase in paracellular permeability in the Caco-2 monolayers treated with LPS (<i>P</i> < 0.01). Compared with the LPS group, bifidobacterium significantly decreased the production of IL-6 and TNF-α (<i>P</i> < 0.01) and suppressed zonulin release (<i>P</i> < 0.05). In addition, bifidobacterium pretreatment up-regulated occludin, claudin-3 and ZO-1 expression (<i>P</i> < 0.01) and also preserved these proteins localization at TJ compared with the LPS group. In the in vivo study, bifidobacterium decreased the incidence of NEC from 88 to 47% (<i>P</i> < 0.05) and reduced the severity in the NEC model. Increased levels of IL-6 and TNF-α in the ileum of NEC rats were normalized in bifidobacterium treated rats (<i>P</i> < 0.05). Moreover, administration of bifidobacterium attenuated the increase in intestinal permeability (<i>P</i> < 0.01), decreased the levels of serum zonulin (<i>P</i> < 0.05), normalized the expression and localization of TJ proteins in the ileum compared with animals with NEC. We concluded that bifidobacterium may protect against intestinal barrier dysfunction both in vitro and in NEC. This protective effect is associated with inhibition of proinflammatory cytokine secretion, suppression of zonulin protein release and improvement of intestinal TJ integrity.</p></div

Bifidobacterium prevented the disruption of TJ in a rat NEC model.

<p><b>(A)</b> TJ proteins localization was evaluated by Immunohistochemical staining in the terminal ileum of neonatal rats. Representative slides for control, NEC, and BIF were shown. Magnification: ×40, n = 3 to 6 per group. <b>(B)</b> Western blot for TJ proteins. Terminal ilea were subjected to immunoblotting for ZO-1, occludin claudin-3 and β-actin. Representative results of one experiment are shown. Similar results were obtained in three independent experiments: control group, NEC group, BIF group. <b>(C)</b> The intensity of the bands was quantified by scanning densitometry, standardized with respect to β-actin protein and expressed as mean ± SD fold change compared with control animals.*<i>P</i> < 0.01, ^##<i>P</i> < 0.05 vs the control group, ^#<i>P</i> < 0.01, **<i>P</i> < 0.05 vs the NEC group.</p

Bifidobacterium reduced the severity and incidence of NEC in a rat NEC model.

<p><b>(A)</b> Body weight changes. *<i>P</i> < 0.01 vs the control group, ^#<i>P</i> < 0.01 vs the NEC group, n = 9–11 animals per group. Three independent experiments were performed in duplicate. <b>(B)</b> Macroscopic appearance of the gastrointestinal tract. In rat pups with NEC, dilation, significant hemorrhage, and discoloration were seen in the terminal ileum. <b>(C)</b> Images of H&E staining using light microscopy. The histological changes in the terminal ilea (representative images) in the control, NEC and BIF groups. Magnification: ×20. <b>(D)</b> Intestinal histological score. *<i>P</i> < 0.01 vs the control group, ^#<i>P</i> < 0.01 vs the NEC group, n = 6 animals per group.</p