Nontrivial Stationary Solutions Research Articles

The stability of a class of 2D non-newtonian fluid equations with unbounded delays

ABSTRACT We address the stability of stationary solutions to a class of 2D non-newtonian fluid equations, when the external force contains hereditary characteristics involving unbounded delays. Firstly, when the unbounded variable delay is driven by a continuously differential function, we establish the stability of nontrivial weak stationary solutions and the asymptotic stability of trivial stationary solution. Then when the general unbounded delay is continuous with respect to time, the stability of nontrivial strong stationary solutions is also obtained. Eventually, when the proportional delay is considered, the polynomial stability of trivial stationary solution is verified.

A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field.

Time reversal of spinal processes for linear and non-linear branching processes near stationarity

We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time T, which is the path constituted of the traits of the ancestors of this individual in past times t≤T. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of T). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.

Investigation of Nonlinear Thermo-Elastic Behavior of Fluid Conveying Piezoelectric Microtube Reinforced by Functionally Distributed Carbon Nanotubes on Viscoelastic-Hetenyi Foundation

In this paper, nonlinear and nonlocal thermo-elastic behavior of a microtube reinforced by Functionally Distributed Carbon Nanotubes, with internal and external piezoelectric layers, in the presence of nonlinear viscoelastic-Hetenyi foundation, and axial fluid flow inside the microtube is studied. Nonlinear partial differential equations governing the system are derived using Reddy’s third-order shear deformations theory along with the Von-Karman theory including the effect of fluid viscosity. Then, the equations are converted to time-dependent ordinary nonlinear equations using the Galerkin method. Afterward, the governing equations of the microtube’s lateral displacements are solved using the multiple scales method. The analysis of the piezoelectric’s parametric resonance is performed by obtaining trivial and nontrivial stationary solutions and plotting characteristic curves of the frequency response and voltage response. At the end, the effect of different parameters including the flow velocity, excitation voltage, parameters of the foundation, viscosity parameter, thermal loading and nanotubes’ volume fraction index on the nonlinear behavior of the system, under parametric resonance condition, is investigated.

Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace

<p style='text-indent:20px;'>A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. This article presents an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.</p>

An Arbitrary High Order Well-Balanced ADER-DG Numerical Scheme for the Multilayer Shallow-Water Model with Variable Density

In this work, we present a novel numerical discretization of a variable pressure multilayer shallow water model. The model can be written as a hyperbolic PDE system and allows the simulation of density driven gravity currents in a shallow water framework. The proposed discretization consists in an unlimited arbitrary high order accurate (ADER) Discontinuous Galerkin (DG) method, which is then limited with the MOOD paradigm using an a posteriori subcell finite volume limiter. The resulting numerical scheme is arbitrary high order accurate in space and time for smooth solutions and does not destroy the natural subcell resolution inherent in the DG methods in the presence of strong gradients or discontinuities. A numerical strategy to preserve non-trivial stationary solutions is also discussed. The final method is very accurate in smooth regions even using coarse or very coarse meshes, as shown in the numerical simulations presented here. Finally, a comparison with a laboratory test, where empirical data are available, is also performed.

Non-uniqueness of Steady-State Weak Solutions to the Surface Quasi-Geostrophic Equations

We show the existence of nontrivial stationary weak solutions to the surface quasi-geostrophic equations on the two dimensional periodic torus.

Stability analysis of nontrivial stationary solution and constant equilibrium point of reaction–diffusion neural networks with time delays under Dirichlet zero boundary value

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of globally exponentially stable (positive) stationary solution of delayed reaction–diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria in the classical meaning. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods. It is worth mentioning that some interesting mathematical problems are originally put forward in the last chapter.

A Consistent Estimator of Nontrivial Stationary Solutions of Dynamic Neural Fields

Dynamics of neural fields are tools used in neurosciences to understand the activities generated by large ensembles of neurons. They are also used in networks analysis and neuroinformatics in particular to model a continuum of neural networks. They are mathematical models that describe the average behavior of these congregations of neurons, which are often in large numbers, even in small cortexes of the brain. Therefore, change of average activity (potential, connectivity, firing rate, etc.) are described using systems of partial different equations. In their continuous or discrete forms, these systems have a rich array of properties, among which is the existence of nontrivial stationary solutions. In this paper, we propose an estimator for nontrivial solutions of dynamical neural fields with a single layer. The estimator is shown to be consistent and a computational algorithm is proposed to help carry out implementation. An illustrations of this consistency is given based on different inputs functions, different kernels, and different pulse emission rate functions.

Breather solutions of the cubic Klein–Gordon equation

We obtain real-valued, time-periodic and radially symmetric solutions of the cubic Klein–Gordon equation which are weakly localized in space. Various families of such ‘breather’ solutions are shown to bifurcate from any given nontrivial stationary solution. The construction of weakly localized breathers in three space dimensions is, to the author’s knowledge, a new concept and based on the reformulation of the cubic Klein–Gordon equation as a system of coupled nonlinear Helmholtz equations involving suitable conditions on the far field behavior.

Pth Moment Stability of a Stationary Solution for a Reaction Diffusion System with Distributed Delays

In this paper, the Sobolev embedding theorem, Holder inequality, the Lebesgue contrl convergence theorem, the operator norm estimation technique, and critical point theory are employed to prove the existence of nontrivial stationary solution for p-Laplacian diffusion system with distributed delays. Furthermore, by giving the definition of pth moment stability, the authors use the Lyapunovfunctional method and Kamke function to derive the stability of nontrivialstationary solution. Moreover, a numerical example illuminates the effectiveness of the proposed methods. Finally, an interesting further thought is put forward, which is conducive to the in-depth study of the problem.

Random attractors for stochastic semilinear degenerate parabolic equations with delay

In this paper, we consider a stochastic semilinear degenerate parabolic equation with delay in a bounded domain in RN and the nonlinearity satisfying an arbitrary polynomial growth condition. The random dynamical system generated by the equation is shown to have a random attractor in C([−τ,0],Lp(O)∩D01(O,σ)), which is a compact and invariant tempered set and attracts every tempered random subset of C([−τ,0],L2(O)) in the topology of C([−τ,0],Lp(O)). In a particular case, the random attractor consists of singleton sets (i.e., a random fixed point), which generates an exponentially stable non-trivial stationary solution. This theoretical result improves some recent ones for stochastic semilinear degenerate parabolic equations.

High-order fully well-balanced Lagrange-projection scheme for shallow water

In this work we propose a novel strategy to define high-order fully well-balanced Lagrange-Projection finite volume solvers for balance laws. In particular, we focus on the 1D shallow water system as it is a reference system of balance laws with non-trivial stationary solutions. Nevertheless, the strategy proposed here could be extended to other interesting balance laws. By fully well-balanced, it is meant that the scheme is able to preserve stationary smooth solutions. Following [6], we exploit the idea of using a high-order well-balanced reconstruction operator for the Lagrangian step. Nevertheless, this is not enough to achieve well-balanced high-order during the projection step. We propose here a new projection step that overcomes this difficulty and that reduces to the standard one in case of conservation laws. Finally, some numerical experiments illustrate the good behaviour of the scheme.

Bifurcation analysis of the Degond–Lucquin-Desreux–Morrow model for gas discharge

The main purpose of this paper is to investigate mathematically gas discharge. Townsend discovered α- and γ-mechanisms which are essential for ionization of gas, and then derived a threshold of voltage at which gas discharge can happen. In this derivation, he used some simplification such as discretization of time. Therefore, it is an interesting problem to analyze the threshold by using the Degond–Lucquin-Desreux–Morrow model and also to compare the results of analysis with Townsend's theory. Note that gas discharge never happens in Townsend's theory if γ-mechanism is not taken into account. In this paper, we study an initial–boundary value problem to the model with α-mechanism but no γ-mechanism. This problem has a trivial stationary solution of which the electron and ion densities are zero. It is shown that there exists a threshold of voltage at which the trivial solution becomes unstable from stable. Then we conclude that gas discharge can happen for a voltage greater than this threshold even if γ-mechanism is not taken into account. It is also of interest to know the asymptotic behavior of solutions to this initial–boundary value problem for the case that the trivial solution is unstable. To this end, we establish bifurcation of non-trivial stationary solutions by applying Crandall and Rabinowitz's Theorem, and show the linear stability and instability of those non-trivial solutions.

Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise

Let $\xi (t,x)$ denote space–time white noise and consider a reaction–diffusion equation of the form \[\dot{u}(t,x)=\frac{1}{2}u"(t,x)+b\big(u(t,x)\big)+\sigma \big(u(t,x)\big)\xi (t,x),\] on $\mathbb{R}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon >0$ such that $\vert b(z)\vert \ge \vert z\vert (\log \vert z\vert )^{1+\varepsilon }$ for all sufficiently-large values of $\vert z\vert $. When $\sigma \equiv 0$, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209–215] have recently shown that there is finite-time blowup when $\sigma $ is a nonzero constant. In this paper, we prove that the Bonder–Groisman condition is unimprovable by showing that the reaction–diffusion equation with noise is “typically” well posed when $\vert b(z)\vert =O(\vert z\vert \log_{+}\vert z\vert )$ as $\vert z\vert \to \infty $. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.

The influence of the perturbation of the wheel rotation speed on the stability of a railway bogie on steady curve sections of a track

ABSTRACTBased on the theory of non-linear dynamic systems, the influence of the perturbation of the wheel rotation speed on the quasi-steady curved motions of a two-axle railway bogie system with a realistic wheel/rail contact relation is investigated in this paper. Since the wheel/rail contact relation is non-linear, it is tabulated in a wheel/rail contact table. The bifurcation diagram of the bogie system is constructed with gradually increasing and decreasing speed in the speed range . A supercritical Hopf bifurcation, where the stable non-trivial stationary solution loses its stability, is found in both models with and without perturbation of the wheel rotation speed. In the model without perturbation of the wheel rotation speed, the first chaotic motions develop at the speed where the wheel flange contact starts. A period-doubling cascade of the bogie system through pitchfork bifurcations, which explains the transition from periodic solutions to chaotic motions of the bogie system, is found. Several jumps happen at higher speeds because of the coexistence of multiple attracting solutions, which should be avoided. A comprehensive investigation of the hysteresis phenomena is made. However, in the model with a perturbation of the wheel rotation speed, no chaotic motions are found.

Nonexistence of Nontrivial Stationary Solutions with Decay Order for Some Nonlinear Evolution Equations

We show that there are no nontrivial stationary solutions of certain decay order for some applied nonlinear evolution equations which include the thin epitaxial film model with slope selection and the square phase field crystal (SPFC) equation. The method is to use the Morawetz multiplier and the Gagliardo-Nirenberg inequality.

Stationary Solutions and Connecting Orbits for p-Laplace Equation

We deal with one dimensional p-Laplace equation of the form ut=(|ux|p-2ux)x+f(x,u),x∈(0,l),t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} u_t = (|u_x|^{p-2} u_x )_x + f(x,u), \\ x\\in (0,l), \\ t>0, \\end{aligned}$$\\end{document}under Dirichlet boundary condition, where p>2 and f:[0,l]times {mathbb {R}}rightarrow {mathbb {R}} is a continuous function with f(x,0)=0. We will prove that if there is at least one eigenvalue of the p-Laplace operator between lim _{urightarrow 0} f(x,u)/|u|^{p-2}u and lim _{|u|rightarrow +infty } f(x,u)/|u|^{p-2}u, then there exists a nontrivial stationary solution. Moreover we show the existence of a connecting orbit between stationary solutions. The results are based on Conley index and detect stationary states even when those based on fixed point theory do not apply. In order to compute the Conley index for nonlinear semiflows deformation along p is used.

Global existence and asymptotic behavior of solutions to a nonlocal Fisher–KPP type problem

In this work, we consider a nonlocal Fisher–KPP reaction–diffusion problem with Neumann boundary condition and nonnegative initial data in a bounded domain in Rn(n≥1), with reaction term uα(1−m(t)), where m(t) is the total mass at time t. With the help of Pohožaev’s identity, the non-existence of nontrivial stationary solutions with Dirichlet boundary conditions is being shown. When α≥1 and the initial mass is greater than or equal to one, the problem has nonnegative classical solutions. While if the initial mass is less than one, then the problem admits global solutions for n=1,2 with any 1≤α<2 or n≥3 with any 1≤α<1+2/n. Moreover, the asymptotic convergence to the solution of the heat equation is proved.

Stability of a nonlinear second order equation under parametric bounded noise excitation

The motivation for the following work is a structural column under dynamic axial loads with both deterministic (harmonic transmitted forces from the surrounding structure) and random (wind and/or earthquake) loading components. The bounded noise used herein is a sinusoid with an argument composed of a random (Wiener) process deviation about a mean frequency. By this approach, a noise parameter may be used to investigate the behavior through the spectrum from simple harmonic forcing, to a bounded random process with very little harmonic content.The stability of both the trivial and non-trivial stationary solutions of an axially-loaded column (which is modeled as a second order nonlinear equation) under parametric bounded noise excitation is investigated by use of Lyapunov exponents. Specifically the effect of noise magnitude, amplitude of the forcing, and damping on stability of a column is investigated. First order averaging is employed to obtain analytical approximations of the Lyapunov exponents of the trivial solution. For the non-trivial stationary solution however, the Lyapunov exponents are obtained via Monte Carlo simulation as the stability equations become analytically intractable.