Large Time Asymptotic Behaviors Research Articles

Large-time and small-time behaviors of the spectral heat content for time-changed stable processes

We study the large-time and small-time asymptotic behaviors of the spectral heat content for time-changed stable processes, where the time change belongs to a large class of inverse subordinators. For the large-time behavior, the spectral heat content decays polynomially with the decay rate determined by the Laplace exponent of the underlying subordinator, which is in sharp contrast to the exponential decay observed in the case when the time change is a subordinator. On the other hand, the small-time behavior exhibits three different decay regimes, where the decay rate is determined by both the Laplace exponent and the index of the stable process.

Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions

<p style="text-indent:20px;">We are concerned with the large-time asymptotic behaviors towards the planar rarefaction wave to the three-dimensional (3D) compressible and isentropic Navier-Stokes equations in half space with Navier boundary conditions. It is proved that the planar rarefaction wave is time-asymptotically stable for the 3D initial-boundary value problem of the compressible Navier-Stokes equations in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^+\times \mathbb{T}^2 $\end{document}</tex-math></inline-formula> with arbitrarily large wave strength. Compared with the previous work [<xref ref-type="bibr" rid="b17">17</xref>, <xref ref-type="bibr" rid="b16">16</xref>] for the whole space problem, Navier boundary conditions, which state that the impermeable wall condition holds for the normal velocity and the fluid tangential velocity is proportional to the tangential component of the viscous stress tensor on the boundary, are crucially used for the stability analysis of the 3D initial-boundary value problem.</p>

The Perturbed Riemann Problem for the Aw-Rascle Model with Modified Chaplygin Gas Pressure

This paper is concerned with the perturbed Riemann problem for the Aw-Rascle model with the modified Chaplygin gas pressure. We obtain constructively the solutions when the initial values are three piecewise constant states. The global structure and the large-time asymptotic behaviors of the solutions are discussed case by case. Further, we obtain the stability of the corresponding Riemann solutions as the initial perturbed parameter tends to zero.

INTERACTIONS OF DELTA SHOCK WAVES FOR THE AW-RASCLE TRAFFIC MODEL WITH SPLIT DELTA FUNCTIONS

This paper is concerned with the interactions of δ-shock waves for the Aw-Rascle traffic model with split delta functions. The solutions are obtained constructively when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter ε → 0.

Interactions of Elementary Waves for the Nonlinear Chromatography Equations

In this article, we study the global solution of the elementary waves interaction problem for the nonlinear chromatography equations. We constructively obtain the solutions when the initial data are three piecewise constant states. The global structures and large time-asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, it is easy to see that the solutions of the perturbed Riemann problem converge to nothing but the corresponding Riemann solutions as \(\epsilon \rightarrow 0\), from which the stability of the Riemann solutions with respect to this local small perturbation of the Riemann initial data are obtained.

Interactions of delta shock waves for the relativistic Chaplygin Euler equations with split delta functions

In this article, we are concerned with the interactions of delta shock waves with contact discontinuities for the relativistic Euler equations for Chaplygin gas by using split delta functions method. The solutions are obtained constructively and globally when the initial data consists of three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. During the process of the interaction, the strengths of delta shock waves are computed completely. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with special initial data by letting perturbed parameter e tends to zero. Copyright © 2014 John Wiley & Sons, Ltd.

Interactions of delta shock waves for the Chaplygin gas equations with split delta functions

In this article, we study the interactions of delta shock waves for the one-dimensional Euler equations for Chaplygin gas with split delta functions. We constructively obtain the solutions when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, we obtain the stability of solutions by letting perturbed parameter ϵ tend to zero.

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.

Structural stability of solutions to the Riemann problem for a scalar conservation law

The aim of this paper is to study the structural stability of solutions to the Riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients. It is proved that the Riemann solution is possibly instable when one of the Riemann initial data is at the vacuum. Furthermore, we point out that the Riemann solution is also possibly instable even when the Riemann initial data stay far away from vacuum. In order to deal with it, we perturb the Riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively. It is also proved that the Riemann solutions are unstable in some certain situations under the local small perturbations of the Riemann initial data by letting the perturbed parameter ε tend to zero. In addition, the interaction of the delta standing wave and the contact vacuum state is considered which appear in the Riemann solutions.

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143–1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.

Exact solution of a two-type branching process: clone size distribution in cell divisionkinetics

We study a two-type branching process which provides an excellent description ofexperimental data on cell dynamics in skin tissue. The model involves only a single type ofprogenitor cell, that is it does not require support from a self-renewed population of stemcells. The progenitor cells divide and may differentiate into post-mitotic cells. We derivean exact solution for the generating function of the number of cells of differenttypes. The generating function is then used to numerically determine the timedependent probability distribution of the numbers of the two types of cells. We alsodeduce large time asymptotic behaviors drawing on exact results and on a diffusionapproximation.

Large Time Asymptotic Behaviors for Periodic Solutions of Generalized Burgers Equations With Spherical Symmetry or Linear Damping

Using the method of balancing arguments, large time asymptotic behaviors for the periodic solutions of generalized Burgers equations ut + u3ux + ju/2t =δ/2uxx and ut + u3ux +λu =δ/2uxx subject to the periodic initial condition and the vanishing boundary conditions u(0, t) = u(l, t) = 0, t ≥ 0 or t0, where A, A1, δ, λ, l, t0, ∈R+ and j = 1, 2, are obtained.

Wave interactions and stability of the Riemann solutions for the chromatography equations

We prove that the Riemann solutions are stable for the chromatography system under the local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis. It is noteworthy that both the propagation directions of the shock wave S and rarefaction wave R are unchanged when they interact with the contact discontinuity J. Moreover, the global structures and large time asymptotic behaviors of the perturbed Riemann solutions are constructed and analyzed case by case.

Interactions of delta shock waves for the transport equations with split delta functions

This paper is concerned with the interactions of δ-shock waves and the vacuum states between the two contact discontinuities for the transport equations. The solutions are obtained constructively when the initial data are three piecewise constant states. The global structure and large time-asymptotic behaviors of the solutions are analyzed case by case. Moreover, it can be found that the Riemann solutions are stable for such small perturbations with initial data by studying the limits of the solutions when the perturbed parameter ε tends to zero.

Random walks on abelian-by-cyclic groups

We describe the large time asymptotic behaviors of the probabilities p 2t (e, e) of return to the origin associated to finite symmetric generating sets of abelian-by-cyclic groups. We characterize the different asymptotic behaviors by simple algebraic properties of the groups.