Time Asymptotic Behavior Research Articles

Exit times for multivariate autoregressive processes

We study exit times from a set for a family of multivariate autoregressive processes with normally distributed noise. By using the large deviation principle, and other methods, we show that the asymptotic behavior of the exit time depends only on the set itself and on the covariance matrix of the stationary distribution of the process. The results are extended to exit times from intervals for the univariate autoregressive process of order n, where the exit time is of the same order of magnitude as the exponential of the inverse of the variance of the stationary distribution.

Asymptotic behavior for critical Patlak–Keller–Segel model and a repulsive–attractive aggregation equation

In this paper we study the long time asymptotic behavior for a class of diffusion–aggregation equations. Most results except the ones in Section 3.3 concern radial solutions. The main tools used in the paper are maximum principle type arguments on mass concentration of solutions, as well as energy method. For the Patlak–Keller–Segel problem with critical power m=2−2/d, we prove that all radial solutions with critical mass would converge to a family of stationary solutions, while all radial solutions with subcritical mass converge to a self-similar dissipating solution algebraically fast. For non-radial solutions, we obtain convergence towards the self-similar dissipating solution when the mass is sufficiently small. We also apply the mass comparison method to another aggregation model with repulsive–attractive interaction, and prove that radial solutions converge to the stationary solution exponentially fast.

Transient dynamics by continuous-spectrum perturbations in stratified shear flows

AbstractThe transient dynamics of the linearized Euler–Boussinesq equations governing parallel stratified shear flows is presented and analysed. Solutions are expressed as integral superpositions of generalized eigenfunctions associated with the continuous-spectrum component of the Taylor–Goldstein linear stability operator, and reveal intrinsic dynamics not captured by its discrete-spectrum counterpart. It is shown how continuous-spectrum perturbations are generally characterized by non-normal energy growth and decay with algebraic asymptotic behaviour in either time or space. This behaviour is captured by explicit long-time/far-field expressions from rigorous asymptotic analysis, and it is illustrated with direct numerical simulations of the whole (non-Boussinesq) stratified Euler system. These results can be helpful in understanding recent numerical observations for parallel and non-parallel perturbed stratified shear flows.

One-dimensional nonlinear chromatography system and delta-shock waves

The Riemann problem for the nonlinear chromatography system is considered. Existence and admissibility of δ-shock type solution in both variables are established for this system. By the interactions of δ-shock wave with elementary waves, the generalized Riemann problem for this system is presented, the global solutions are constructed, and the large time-asymptotic behavior of the solutions are analyzed. Moreover, by studying the limits of the solutions as perturbed parameter \({\varepsilon}\) tends to zero, one can observe that the Riemann solutions are stable for such perturbations of the initial data.

On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics

In this paper, we study Landau-Lifshitz equations of ferromagnetism with a total energy that does not include a so-called exchange energy. Many problems, including existence, stability, regularity and asymptotic behaviors, have been extensively studied for such equations of models with the exchange energy. The problems turn out quite different and challenging for Landau-Lifshitz equations of no-exchange energy models because the usual methods based on certain compactness do not apply. We present a new method for the existence of global weak solution to the Landau-Lifshitz equation of no-exchange energy models based on the existence of regular solutions for smooth data and certain stability of the solutions. We also study higher time regularity, energy identity and asymptotic behaviors in some special cases for weak solutions.

Qualitative analysis of some PDE models of traffic flow

We review our previous results on partial differential equation(PDE) models of traffic flow. These models include the first order PDE models, a nonlocal PDE traffic flow model with Arrhenius look-ahead dynamics, and the second order PDE models, a discrete model which captures the essential features of traffic jams and chaotic behavior. We study the well-posedness of such PDE problems, finite time blow-up, front propagation, pattern formation and asymptotic behavior of solutions including the stability of the traveling fronts. Traveling wave solutions are wave front solutions propagating with a constant speed and propagating against traffic.

Vacuum Behaviors around Rarefaction Waves to 1D Compressible Navier--Stokes Equations with Density-Dependent Viscosity

In this paper, we study the large time asymptotic behavior toward rarefaction waves for solutions to the one-dimensional compressible Navier--Stokes equations with density-dependent viscosities for general initial data whose far fields are connected by a rarefaction wave to the corresponding Euler equations with one end state being vacuum. First, a global-in-time weak solution around the rarefaction wave is constructed by approximating the system and regularizing the initial data with general perturbations, and some a priori uniform-in-time estimates for the energy and entropy are obtained. Then it is shown that the density of any weak solution satisfying the natural energy and entropy estimates will converge to the rarefaction wave connected to vacuum with arbitrary strength in sup-norm time-asymptotically. Our results imply, in particular, that the initial vacuum at far field will remain for all the time, which is in sharp contrast to the case of nonvacuum rarefaction waves studied in [Q. S. Jiu, Y. Wang, and Z. P. Xin, Comm. Partial Differential Equations, 36 (2011), pp. 602--634], where all the possible vacuum states will vanish in finite time. Finally, it is proved that the weak solution becomes regular away from the vacuum region of the rarefaction wave.

Thermally assisted spin-transfer torque magnetization reversal in uniaxial nanomagnets

We simulate the stochastic Landau-Lifshitz-Gilbert dynamics of a uniaxial nanomagnet out to sub-millisecond timescales using a graphical processing unit based micromagnetic code and determine the effect of geometrical tilts between the spin-current and uniaxial anisotropy axes on the thermally assisted reversal dynamics. The asymptotic behavior of the switching time (I→0, 〈τ〉∝exp(−ξ(1−I)2)) is approached gradually, indicating a broad crossover regime between the ballistic and thermally assisted spin transfer reversal. Interestingly, the functional form of the mean switching time is shown to be independent of the angle between the spin current and magnet's uniaxial axes. These results have important implications for modeling the energetics of thermally assisted magnetization reversal of spin transfer magnetic random access memory bit cells.

Asymptotic behaviour of solutions to the Keller–Segel model for chemotaxis with prevention of overcrowding

This paper deals with the Keller–Segel chemotaxis model of parabolic–elliptic type with the volume-filling effect studied by Burger et al (2006 The Keller–Segel model for chemotaxis with prevention of overcrowding: linear versus nonlinear diffusion SIAM J. Math. Anal. 38 1288–315). In their discussion on the large time asymptotic behaviour of solutions, the diffusion rate of ρ (the density of cells) had to be assumed to be large with . While for the nonlinear diffusion model, it was proved that the asymptotic behaviour of solutions is fully determined by the diffusion constant being larger or smaller than the threshold value ε = 1. The same ‘large ε-restriction’ was also made for studying the parallel parabolic–parabolic model in Di Francesco and Rosado (2008 Fully parabolic Keller–Segel model for chemotaxis with prevention of overcrowding Nonlinearity 21 2715–30), where it was pointed out that ‘Whether this condition is necessary to have large time decay (and consequently a self-similar behaviour) for ρ is still an open problem even in the (simpler) parabolic–elliptic case’. The aim of the paper is to answer this problem for the parabolic–elliptic model. We prove the mentioned time decay estimate without the restriction . The main technique used in this paper is the Lp-Lq estimate method.

Wave interactions and stability of the Riemann solutions for a scalar conservation law with a discontinuous flux function

This paper is devoted to studying the interactions of elementary waves for a model of a scalar conservation law with a flux function involving discontinuous coefficients. In order to cover all the situations completely, we take the initial data as three piecewise constant states and the middle region is regarded as the perturbed region with small distance. It is proved that the Riemann solutions are stable under the local small perturbations of the Riemann initial data by letting the perturbed parameter tend to zero. The proof is based on the detailed analysis of the interactions of stationary wave discontinuities with shock waves and rarefaction waves. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.

Transversal and longitudinal mixing in compound channels

An experimental campaign, based on particle image velocimetry (PIV) measurements of free‐surface velocities, forms the basis for an analysis of the mixing processes which occur in a compound‐channel flow. The flow mixing is characterized in terms of Lagrangian statistics (absolute dispersion and diffusivity) and of the related mean flow characteristics. Mixing properties strongly depend on the ratiorh between the main channel flow depth (h*mc) and the floodplain depth (h*fp), and three flow classes can be identified, namely shallow, intermediate, and deep flows. In the present study the large time asymptotic behavior of the mixing characteristics is analyzed in terms of the absolute diffusivity in order to characterize typical values of longitudinal and transversal diffusivity coefficients. Various sets of experiments, which cover a wide range of the governing physical parameters, have been performed and the asymptotic values of the absolute diffusivity have been evaluated. The results are then compared with several studies of flow dispersion for both the longitudinal diffusivity coefficient and the transversal turbulent mixing coefficient. The present results highlight a stronger dependence of such coefficients with the flow‐depth ratio than with the flow regime (Froude number).

On the Time Asymptotic Behavior of a Transport Operator with Diffuse Reflection Boundary Condition

This article is concerned with the spectral properties of a transport operator with diffuse reflection boundary condition arising in L 1-spaces. Furthermore, a practical way to study asymptotic behavior of the solution of the transport operator without restriction on the initial data is given.

The Cauchy problem of a focusing energy-critical nonlinear Schrödinger equation

In this paper, we combine variational methods and harmonic analysis to discuss the Cauchy problem of a focusing nonlinear Schrödinger equation. We study the global well-posedness, finite time blowup and asymptotic behavior of this problem. By Hamiltonian property, we establish two types of invariant evolution flows. Then from one flow and the stability of classical energy-critical nonlinear Schrödinger equation, we find that the solution exists globally and scattering occurs. Finally, we get a precise blowup criterion of this problem for positive energy initial data via the other flow.

Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized (L 1) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

A hybrid model for opinion formation

This paper presents a hybrid model for opinion formation in a large group of agents exposed to the persuasive action of a small number of strong opinion leaders. The model is defined by coupling a finite difference equation for the dynamics of leaders opinion with a continuous integro-differential equation for the dynamics of the others. Such a definition stems from the idea that the leaders are few and tend to retain original opinions, so that their dynamics occur on a longer time scale with respect to the one of the other agents. A general well-posedness result is established for the initial value problem linked to the model. The asymptotic behavior in time of the related solution is characterized for some general parameter settings, which mimic distinct social scenarios, where different emerging behaviors can be observed. Analytical results are illustrated and extended through numerical simulations.

Asymptotic analysis of continuous opinion dynamics models under bounded confidence

This paper deals with the asymptotic behavior of mathematical models for opinion dynamics under bounded confidence of Deffuant-Weisbuch type.Focusing on the Cauchy Problem related to compromise models with homogeneous bound of confidence, a general well-posedness result is provided and a systematic study of the asymptotic behavior in time of the solution is developed. More in detail, we prove a theorem that establishes the weak convergence of the solution to a sum of Dirac masses and characterizes the concentration points for different values of the model parameters. Analytical results are illustrated by means of numerical simulations.

Decay of solutions to fractal parabolic conservation laws with large initial data

In this paper, we study the time-asymptotic behavior of solutions to the Cauchy problem for multi-dimensional parabolic conservation laws with fractional dissipation. For arbitrarily large initial data, we obtain the optimal decay rates in $L^2$ and homogeneous Sobolev spaces for solutions to the equation with the power of Laplacian $\frac{1}{2} < \alpha \le 1$ by using the time-frequency decomposition method and the energy method. The argument is based on a maximum principle.

The asymptotic behavior of the solution of a doubly degenerate parabolic equation with the convection term

The objective of this article is to study the large time asymptotic behavior of the nonnegative weak solution of the following nonlinear parabolic equation with initial condition u(x, 0) = u0(x). By using Moser iteration technique, assuming that the uniqueness of the Barenblatt-type solution E c of the equation u t = div(|Du m |p-2Du m ) is true, then the solution u may satisfy which is uniformly true on the sets . Here B(u m ) = (b1(u m ), b2(u m ), ..., b N (u m )) satisfies some growth order conditions, the exponents m and p satisfy m(p - 1) > 1. Mathematics Subject Classification 2000: 35K55; 35K65; 35B40.

Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type

This paper is concerned with the time-asymptotic behavior toward strong rarefaction waves of solutions to one-dimension compressible fluid models of Korteweg type, which governs the motions of the compressible fluids with internal capillarity. Assume that the corresponding Riemann problem to the compressible Euler system can be solved by rarefaction waves (VR,UR)(t,x). If the initial data is a small perturbation of an approximate rarefaction wave for (VR,UR)(t,x), we show that the corresponding Cauchy problem admits a unique global smooth solution which tends to (VR,UR)(t,x) time asymptotically. Since we do not require the strength of the rarefaction waves to be small, this result gives the nonlinear stability of strong rarefaction waves for the one-dimensional compressible fluid models of Korteweg type. The analysis is based on the elementary L2 energy method together with continuation argument.

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.