Nonlinear instability for nonhomogeneous incompressible viscous fluids

We investigate the nonlinear instability of a smooth steady density profile solution to the three-dimensional nonhomogeneous incompressible Navier-Stokes equations in the presence of a uniform gravitational field, including a Rayleigh-Taylor steady-state solution with heavier density with increasing height (referred to the Rayleigh-Taylor instability). We first analyze the equations obtained from linearization around the steady density profile solution. Then we construct solutions to the linearized problem that grow in time in the Sobolev space H ^k , thus leading to a global instability result for the linearized problem. With the help of the constructed unstable solutions and an existence theorem of classical solutions to the original nonlinear equations, we can then demonstrate the instability of the nonlinear problem in some sense. Our analysis shows that the third component of the velocity already induces the instability, which is different from the previous known results.     

Sharp criteria of global existence and blow-up for a type of nonlinear parabolic equations

We study a type of nonlinear parabolic equations. In terms of the variational characterization of the corresponding nonlinear elliptic equations and the invariant flow arguments, we establish the sharp criteria for global existence and blow-up. Furthermore, we also get the instability of the steady states and the global existence with small initial data.     

Nonlinear dynamics of open marine population with space-limited recruitment: The case of mortality control

In this paper we develop a nonlinear extension for the open marine population model which has been proposed by Roughgarden et al. [Ecology 66 (1985) 54-67]. To avoid the negative population density, which is a drawback of the original model, we introduce a nonlinear mechanism that the mortality rate depends on the size of area occupied by the adult population. Then we give a rigorous mathematical framework to analyse the model equation, and we show sufficient conditions for stability and instability of the steady state. Our instability result suggests, as was proposed by Roughgarden et al., that there exists a sustained oscillation of the population density.     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION

According to the variational analysis and the potential well argument,we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations.Furthermore,we give its application in the instability of the steady states.     

Nonlinear stability and instability of relativistic Vlasov‐Maxwell systems

We prove the nonlinear stability or instability of certain periodic equilibria of the 1½D relativistic Vlasov‐Maxwell system. In particular, for a purely magnetic equilibrium with vanishing electric field, we prove its nonlinear stability under a sharp criterion by extending the usual Casimir‐energy method in several new ways. For a general electromagnetic equilibrium we prove that nonlinear instability follows from linear instability. The nonlinear instability is macroscopic, involving only the L¹‐norms of the electromagnetic fields. © 2006 Wiley Periodicals, Inc.     

Transverse nonlinear instability for two-dimensional dispersive models

We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrödinger equation.     

Stability for steady states of Navier-Stokes-Poisson equations

In this paper, we study the stationary solution and nonlinear stability of Navier-Stokes-Poisson equations. Using variational method, we construct steady states of the N-S-P system as minimizers of a suitably defined energy functional, then show their dynamical stability against general, i.e. not necessarily spherically symmetric perturbation.     

Multiscale analysis for diffusion-driven neutrally stable states

The Turing instabilities for reaction–diffusion systems are studied from the Fourier normal modes which appear by searching the solution obtained from linearization of the reaction–diffusion system at the spatially homogeneous steady state. The linear stability analysis is only appropriate when the temporal eigenvalues associated to every given spatial eigenvalue have non-zero real part. If the real part of the temporal eigenvalue in a normal mode is equal to zero there is no enough information coming from the linearized system. Given an arbitrary spatial eigenvalue, by equating to zero the real part of the corresponding temporal eigenvalue will lead to a neutral stability manifold in the parameter space. If for a given spatial eigenvalue the other parameters in the reaction–diffusion process drive the system to the neutral manifold, then neither stability nor instability can be warranted by the usual linear analysis. In order to give a sketch of the nonlinear analysis we use a multiple scales method. As an application, we analyze the behavior of solutions to the Schnakenberg trimolecular reaction kinetics in the presence of diffusion.     

具有复杂边界条件的杆的振动分析

本文研究一端带有集中质量并支以弹簧另一端作支承运动的杆的纵向振动.由于这个问题的边界条件比较复杂,且要考虑阻尼,因此本文只求稳态周期解.首先分析线性系统;然后考虑材料非线性,用摄动法求具有非线性边界条件的非线性方程的近似解析解.     

The asymptotic analysis of wave interactions and numerical calculations of transonic nozzle flow

Transonic flow through a duct of variable cross section for which nonlinear resonance effects are important is considered. The complicated local interactions of nonlinear waves are resolved through asymptotic analysis and this is then used to construct a random choice method to calculate general unsteady flowfields. The method produces sharp shocks without oscillations, is accurate in smooth regions, and converges to a stable steady flow.     

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.     

非线性算子方程的泰勒展式算法

本文的目的是给出一种解Ｈｉｌｂｅｒｔ空间中非线性方程的ｋ阶泰勒展式算法（ｋ１）．标准Ｇａｌｅｒｋｉｎ方法可以看作１阶泰勒展式算法，而最优非线性Ｇａｌｅｒｋｉｎ方法可视为２阶泰勒展式算法．我们应用这种算法于定常的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程的数值逼近．在一定情景下，最优非线性Ｇａｌｅｒｋｉｎ方法提供比标准Ｇａｌｅｒｋｉｎ方法和非线性Ｇａｌｅｒｋｉｎ方法更高阶的收敛速度．     

The Onset of Linear Instabilities in a Solid Combustion Model

This article concerns the onset of linear instability in a simple model of solid combustion in a semi-infinite two-dimensional strip of width l . The free boundary problem that describes the model involves initial and boundary conditions, including a nonlinear kinetic condition at the interface. The linear problem governing perturbations to a basic solution is solved by the method of images with the reaction front perturbation satisfying an integro-differential equation. This equation is then solved using Laplace transforms. Finally, we perform a stability analysis for the model by studying the solution of the reaction front perturbation. The inclusion of initial conditions enables us to show the development of linear instability from arbitrary initial small disturbances.     

Complete set of periodic solutions of a discontinuous recurrence equation

A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang‐bang control is studied. We show that each solution is periodic and its prime period can be determined by three of its consecutive terms. By means of this periodicity property, we may then solve the steady state problem which to our knowledge is not solved by other means.     

Nonlinear Physiologically Structured Population Models with Two Internal Variables

First-order hyperbolic partial differential equations with two internal variables have been used to model biological and epidemiological problems with two physiological structures, such as chronological age and infection age in epidemic models, age and another physiological character (maturation, size, stage) in population models, and cell-age and molecular content (cyclin content, maturity level, plasmid copies, telomere length) in cell population models. In this paper, we study nonlinear double physiologically structured population models with two internal variables by applying integrated semigroup theory and non-densely defined operators. We consider first a semilinear model and then a nonlinear model, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii’s fixed point theorem to obtain the existence of a steady state, and study the stability of the steady state by estimating the essential growth bound of the semigroup. Finally, we generalize the techniques to investigate a nonlinear age-size structured model with size-dependent growth rate.

     

An explicit series solution of the squeezing flow between two infinite plates by means of the homotopy analysis method

In this paper, we investigated an axisymmetric Newtonian fluid squeezed between two parallel plates. The steady nonlinear governing equations are reduced to a single differential equation using integrability condition. Homotopy analysis method (HAM) is used to solve the nonlinear differential equation analytically. Numerical solutions indicate this method is satisfactory.     

On Stabilizing the Strongly Nonlinear Internal Wave Model

A strongly nonlinear asymptotic model describing the evolution of large amplitude internal waves in a two-layer system is studied numerically. While the steady model has been demonstrated to capture correctly the characteristics of large amplitude internal solitary waves, a local stability analysis shows that the time-dependent inviscid model suffers from the Kelvin–Helmholtz instability due to a tangential velocity discontinuity across the interface accompanied by the interfacial deformation. An attempt to represent the viscous effect that is missing in the model is made with eddy viscosity, but this simple ad hoc model is shown to fail to suppress unstable short waves. Alternatively, when a smooth low-pass Fourier filter is applied, it is found that a large amplitude internal solitary wave propagates stably without change of form, and mass and energy are conserved well. The head-on collision of two counter-propagating solitary waves is studied using the filtered strongly nonlinear model and its numerical solution is compared with the weakly nonlinear asymptotic solution.     

Funzioni regolari di più variabili quaternioniche

Summary In this paper the stability analysis of an incompressible toroidal Hall Current plasma with resistivity and viscosity on the basis of the linearization of the governing equations and boundary condition is rigorously justified. A nonlinear local existence theorem for an initial-boundary value problem is first proved, and local stable and unstable invariant manifolds of a nonlinear resolving operator are then constructed. It is shown that linear stability implies nonlinear stability and global existence, and linear instability implies nonlinear instability in some sense.     

Nonlinear dynamics and stability analysis of piezo-visco medium nanoshell resonator with electrostatic and harmonic actuation

In this paper, nonlinear dynamics, vibration and stability analysis of piezo-visco medium nanoshell resonator (PVM-NSR) based on functionally graded (FG) cylindrical nanoshell integrated with two piezoelectric layers subjected to visco-pasternak medium, electrostatic and harmonic excitations is investigated. Nonclassical method of the electro-elastic Gurtin–Murdoch surface/interface theory with von-Karman–Donnell's shell model as well as Hamilton's principle, the assumed mode method combined with Lagrange–Euler's are considered. Complex averaging method combined with arc-length continuation is used to achieve a numerical solution for the steady state vibrations of the system. The stability analysis of the steady state response is performed. The parametric studies such as the effects of different boundary conditions, different geometric ratios, structural parameters, electrostatic and harmonic excitation on the nonlinear frequency response and stability analysis are studied. The results indicate that near the natural frequency of the nanoshell, it will lead to resonance and will have large motion amplitude and near the resonant frequency, the nanoshell shows a softening type of nonlinear behavior, and the nanoshell bandwidth increases due to nonlinear factors. In this range, nanoshell has three different ranges of motion, of which two are stable and the other unstable, and so the jump phenomenon and saddle-node bifurcation are visible in the behavior of the system. Also piezoelectric voltage influences on static deformation and resonant frequency but has no significant effect on nonlinear behavior and bandwidth and also system very sensitive to the damping coefficient and due to decrease of nano shell stiffness, natural frequency decreases. And also, increasing or decreasing of some parameters lead to increasing or decreasing the resonance amplitude, resonant frequency, the system's instability, nonlinear behavior and bandwidth.