A new approach to the stabilization of a fluid–structure interaction model

This article is a continuation of our work on a linear fluid–structure interaction model [Grobbelaar-Van Dalsen, On a fluid–structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech. 10(3) (2008), pp. 388–401; Grobbelaar-Van Dalsen, Strong stability for a fluid––structure model, Math. Methods Appl. Sci., 32(2009) pp. 1452–1466]. The model describes the interaction between a 3-D incompressible fluid and a 2-D plate, the interface, which coincides with a flat flexible part of the surface of the vessel containing the fluid. The mathematical model comprises the Stokes equations and the equations for the longitudinal deflections of the plate with the inclusion of the shear stress that the fluid exerts on the plate. A dissipative damping mechanism of Kelvin–Voigt type is applied to the interior of the plate. While our earlier work shows that weak solutions in a space of finite energy are strongly asymptotically stable under no-slip transmission conditions at the interface with uniform exponential stability only attainable under an additional domination condition, the present research is directed at achieving uniform exponential stability of weak solutions without imposing the domination condition. Using energy methods we establish uniform exponential decay under a modified transmission condition at the interface. This condition entails that the fluid velocity at the interface is coupled to a linear combination of the plate velocity and displacement.     

On the high temperature region of the Sherrington–Kirkpatrick model

We prove the validity of the replica-symmetric formulas for the Sherrington–Kirkpatrick (SK) model in a region of parameters that (probably) coincides with the region predicted by the physicists.     

On the ground states of the one-dimensional Falicov–Kimball model

The ground states of the one-dimensional Falicov–Kimball model are studied in the grand canonical ensemble for large values of the interaction strength U. The quantum particle chemical potential μ_e is chosen in the interval −U+4<μ_e<0, such that, then, these states are neutral states and depend only on the sum of the two chemical potentials, μ=μ_i+μ_e. Consequences of this study are, among others, the following results. If ρ=p/q (p and q relatively prime) is a rational number we prove that, for U⩾U₀(q) (where U₀(q) is a specific function), there is an interval on the μ-axis, of length larger than qU^−2q+3, such that for any μ in this interval, the ground state has density ρ. In this interval the ground state is unique, up to translations, and the corresponding classical particle configuration is described by the characteristic sequence associated with the rational number ρ.     

A new method of finding the fractional Euler–Lagrange and Hamilton equations within Caputo fractional derivatives

In this paper, we have investigated the fractional Caputo derivative of a composition function. The obtained results were applied to investigate the fractional Euler–Lagrange and Hamilton equations for constrained systems. The approach was applied within an illustrative.     

On one model problem for the reaction–diffusion–advection equation

The asymptotic behavior of the solution with boundary layers in the time-independent mathematical model of reaction–diffusion–advection arising when describing the distribution of greenhouse gases in the surface atmospheric layer is studied. On the basis of the asymptotic method of differential inequalities, the existence of a boundary-layer solution and its asymptotic Lyapunov stability as a steady-state solution of the corresponding parabolic problem is proven. One of the results of this work is the determination of the local domain of the attraction of a boundary-layer solution.     

On the dynamics of the Lengyel–Epstein model with forcing intensity

This paper is concerned with the Lengyel–Epstein model for interacting chemicals under Dirichlet–Neumann boundary data. This model describe the reaction between iodide, malonic and clorite acid (CIMA reaction). In particular the Lengyel–Epstein model that takes into account the effect of illumination of the reaction cell is investigated. It is shown that the solutions are bounded. The linear stability of the steady states is discussed. Conditions guaranteeing the nonlinear stability are also obtained.

     

A predator–prey model with disease in the prey and two impulses for integrated pest management

In this paper, a predator–prey model with disease in the prey is constructed and investigated for the purpose of integrated pest management. In the first part of the main results, the sufficient condition for the global stability of the susceptible pest-eradication periodic solution is obtained, which means if the release amount of infective prey and predator satisfy the condition, then the pest will be controlled. The sufficient condition for the permanence of the system is also obtained subsequently, which means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. At last, we interpret our mathematical results.     

On the Su–Schrieffer–Heeger model of electron transport: Low-temperature optical conductivity by the Mellin transform

We describe the low-temperature optical conductivity as a function of frequency for a quantum-mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su–Schrieffer–Heeger tight-binding Hamiltonian for noninteracting spinless electrons on a one-dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo-type formulation for the optical conductivity with a nonzero damping due to microscopic collisions, (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone, and (iii) evaluation of this integral in terms of elementary functions via the three-dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations.     

On some free boundary problems of the prey–predator model

In this paper we investigate some free boundary problems for the Lotka–Volterra type prey–predator model in one space dimension. The main objective is to understand the asymptotic behavior of the two species (prey and predator) spreading via a free boundary. We prove a spreading–vanishing dichotomy, namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment, or they fail to establish and die out in the long run. The long time behavior of solution and criteria for spreading and vanishing are also obtained. Finally, when spreading successfully, we provide an estimate to show that the spreading speed (if exists) cannot be faster than the minimal speed of traveling wavefront solutions for the prey–predator model on the whole real line without a free boundary.     

On the classical Maki–Thompson rumour model in continuous time

In this paper, the Maki–Thompson model is slightly refined in continuous time, and a new general solution is obtained for each dynamics of spreading of a rumour. It is derived an equation for the size of a stochastic rumour process in terms of transitions. We give new lower and upper bounds for the proportion of total ignorants who never learned a rumour and the proportion of total stiflers who either forget the rumour or cease to spread the rumour when the rumour process stops, under general initial conditions. Simulation results are presented for the analytical solutions. The model and these numerical results are capable to explain the behaviour of the dynamics of any other dynamical system having interactions similar to the ones in the stochastic rumour process and requiring numerical interpretations to understand the real phenomena better. The numerical process in the differential equations of the model is investigated by using error-estimates. The estimated error is calculated by the Runge–Kutta method and found either negligible or zero for a relatively small size of the population. This pioneering paper introduces a new mathematical method into Operations research, motivated by various areas of scientific, social and daily life, it presents numerical computations, discusses structural frontiers and invites the interested readers to future research.     

On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.     

On a model for the Navier–Stokes equations using magnetization variables

It is known that in a classical setting, the Navier–Stokes equations can be reformulated in terms of so-called magnetization variables w that satisfy

(1)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+{(\mathrm{?}\mathbb{P}w)}^{?}w?\mathrm{\Delta }w=0,$

and relate to the velocity u via a Leray projection $u=\mathbb{P}w$. We will prove the equivalence of these formulations in the setting of weak solutions that are also in $L^{\infty }(0,T;H^{1/2})\cap L^2(0,T;H^{3/2})$ on the 3-dimensional torus.Our main focus is the proof of global well-posedness in $H^{1/2}$ for a new variant of (1), where $\mathbb{P}w$ is replaced by w in the second nonlinear term:

(2)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+\frac{1}{2}\mathrm{?}|w{|}^2?\mathrm{\Delta }w=0.$

This is based on a maximum principle, analogous to a similar property of the Burgers equations.     

A new variant of the Schwarz–Pick–Ahlfors Lemma

We prove a “general shrinking lemma” that resembles the Schwarz–Pick–Ahlfors Lemma and its many generalizations, but differs in applying to maps of a finite disk into a disk, rather than requiring the domain of the map to be complete. The conclusion is that distances to the origin are all shrunk, and by a limiting procedure we can recover the original Ahlfors Lemma, that all distances are shrunk. The method of proof is also different in that it relates the shrinking of the Schwarz–Pick–Ahlfors-type lemmas to the comparison theorems of Riemannian geometry. Received: 26 May 1998 / Revised version: 4 May 1999     

On slow flows of the full nonlinear Doi–Edwards polymer model

The full Doi–Edwards model constitutive equation derived by Palierne (Phys Rev Lett 93:136001-1–136001-4, (2004) is discussed in detail. The corresponding configurational probability equation is next solved for slow flows, and the solution is used to calculate the material constants: zero-shear viscosity and the normal stress differences.