Time Asymptotic Convergence Research Articles

Longtime behavior of a second order finite element scheme simulating the kinematic effects in liquid crystal dynamics

<p style='text-indent:20px;'>We consider an unconditional fully discrete finite element scheme for a nematic liquid crystal flow with different kinematic transport properties. We prove that the scheme converges towards a unique critical point of the elastic energy subject to the finite element subspace, when the number of time steps go to infinity while the time step and mesh size are fixed. A Lojasiewicz type inequality, which is the key for getting the time asymptotic convergence of the whole sequence furnished by the numerical scheme, is also derived.</p>

Time-asymptotic convergence rates towards discrete steady states of a nonlocal selection-mutation model

This paper is concerned with large time behavior of solutions to a semi-discrete model involving nonlinear competition that describes the evolution of a trait-structured population. Under some threshold assumptions, the steady solution is shown unique and strictly positive, and also globally stable. The exponential convergence rate to the steady state is also established. These results are consistent with the results in [P.-E. Jabin, H. L. Liu. Nonlinearity 30 (2017) 4220–4238] for the continuous model.

Convergence to equilibrium for linear Fokker-Planck equations

In this note, we investigate the spectral analysis and long time asymptotic convergence of semigroups associated to discrete, fractional and classical Fokker-Planck equations in some regime where the corresponding operators are close. We successively deal with the discrete and the classical Fokker-Planck model and the fractional and the classical Fokker-Planck model. In each case, we present results of uniform convergence to equilibrium based on perturbation and/or enlargement arguments and obtained in collaboration with S. Mischler in [7].

Sliding Mode Controlfor Robust Regulationof Chemical Processes

Objective: This paper implies a comparison between sliding mode control and linear state feedback controller. When the linear systems are given bounded disturbance their robustness properties are studied. Method: Pole placement, classical sliding mode control using constant rate reaching law and constant plus proportional rate reaching law, super-twisting controller. Findings: Finite time asymptotic convergence is not achieved in linear feedback controller, Sliding Mode Control when used produced finite time asymptotic convergence. The measure of the control variable is obtained by taking norm of the control input. Applications: Process Control, Aerodynamics.

Robust output‐feedback attitude control of a three‐degree‐of‐freedom helicopter via sliding‐mode observation technique

A robust output-feedback attitude controller for three-degree-of-freedom laboratory helicopter is proposed in this study. The laboratory helicopter has three rotational degree-of-freedom and preserves essential control challenges in the attitude control of actual helicopter. An output injection sliding-mode observer is developed to estimate the angular velocities and the external disturbances in finite time. The estimations are used to construct a robust LQR controller that requires angular position measurements only and compensates the external disturbances. The finite time asymptotic convergence of the observed states and disturbances to the actual values is proved and the results are verified numerically through simulation.

Time-asymptotic convergence rates towards the discrete evolutionary stable distribution

This paper is concerned with the discrete dynamics of an integro-differential model that describes the evolution of a population structured with respect to a continuous trait. Various time-asymptotic convergence rates towards the discrete evolutionary stable distribution (ESD) are established. For some special ESD satisfying a strict sign condition, the exponential convergence rates are obtained for both semi-discrete and fully discrete schemes. Towards the general ESD, the algebraic convergence rate that we find is consistent with the known result for the continuous model.

An explicit solution of Burgers equation with stationary point source

Existence, uniqueness and regularity of the global weak solution to the Burgers equation with a reaction term is shown when the reaction term is given as a time independent point source and produces heat constantly. An explicit solution is obtained and used to show the long time asymptotic convergence of the solution to a steady state. For the heat equation case without any convection the solution diverges everywhere as time increases and hence it is the first order convection term that gives the compactness of the solution trajectory of the Burgers equation with reaction.

On the linear Boltzmann equation in evolutionary domains with an absorbing boundary

We consider the linear Boltzmann equation in , , under the effect of an absorbing moving barrier. We prove the existence and uniqueness of the solution and consider the problem of the time-asymptotic convergence to equilibrium. We then propose a numerical strategy to study the problem and provide quantitative results concerning some relevant test cases.

Stability and convergence at infinite time of several fully discrete schemes for a Ginzburg-Landau model for nematic liquid crystal flows

In this paper, we study a conditional long-time stable fully discrete finite element scheme for a Ginzburg-Landau model for nematic liquid crystal flow. We also obtain its time asymptotic convergence (when number of time steps go to infinity, fixed time step and mesh size) towards a unique critical point of the elastic energy subject to the finite element subspace. Finally, we estimate some convergence rates towards this limit critical point. To prove convergence of the whole sequence, a Lojasiewicz type inequality is used. <br>&nbsp; &nbsp Moreover, we extend these results to other schemes given in [3] and [10].

Viscous Conservation Laws with Boundary

In this paper, we continue the scale separation approach, introduced in [S.-H. Yu, J. Amer. Math. Soc., 23 (2010), pp. 1041--1118] for the Boltzmann equation, to construct the linear wave propagation over a viscous shock profile. We have obtained exponentially sharp pointwise structure of the linear wave propagation and apply it to yield pointwise time-asymptotic convergence rate to a viscous shock profile for initial-boundary value problem.

Stability, Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres

We consider the spatially homogeneous Boltzmann equation for {\em inelastic hard spheres}, in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$. In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive $\alpha_*>0$) we prove uniqueness of the self-similar profile for given values of the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the initial datum lies in $L^1_3$, and under some smallness condition on $(1-\alpha_*)$ depending on the mass, energy and $L^1_3$ norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called {\em homogeneous cooling state}). These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the "quasi-elastic self-similar temperature" and the rate of convergence towards self-similarity at first order in terms of $(1-\alpha)$, are obtained from our study. These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity.

Invariance property of a conservation law without convexity

The main goal of this paper is to investigate the mechanism of a conservation law that gives the N-wave like asymptotics. It turns out that the positivity of the flux function provides a certain invariance of solution which singles out the right asymptotics among two parameter family of N-waves. Two kinds of long time asymptotic convergence orders in L 1 -norm to this N-wave are proved using a potential comparison technique. The first one is of the magnitude of the N-wave itself and the second one is of order 1/t. We observe that these asymptotic convergence orders are related to space and time translations of potentials.

Global existence and stability of mild solutions to the Boltzmann system for gas mixtures

We present the global existence and stability of mild solutions to the Boltzmann system with inverse power molecular interactions for a binary gas mixture, when initial data are sufficiently small and decay exponentially in phase space. For the existence and stability of mild solutions, we employ a modified Kaniel-Shinbrot’s scheme and a weighted nonlinear functional approach. Time-asymptotic convergence toward the free molecular motion is established using a weighted collision potential, and we show that the weighted L 1 L^1 -distance between two mild solutions is uniformly controlled by that of initial data.

Convergence rates toward the travelling waves for a model system of the radiating gas

AbstractThe present paper is concerned with an asymptotics of a solution to the model system of radiating gas. The previous researches have shown that the solution converges to a travelling wave with a rate (1 + t)−1/4 as time t tends to infinity provided that an initial data is given by a small perturbation from the travelling wave in the suitable Sobolev space and the perturbation is integrable. In this paper, we make more elaborate analysis under suitable assumptions on initial data in order to obtain shaper convergence rates than previous researches. The first result is that if the initial data decays at the spatial asymptotic point with a certain algebraic rate, then this rate reflects the time asymptotic convergence rate. Precisely, this convergence rate is completely same as the spatial convergence rate of the initial perturbation. The second result is that if the initial data is given by the Riemann data, an admissible weak solution, which has a discontinuity, converges to the travelling wave exponentially fast. Both of two results are proved by obtaining decay estimates in time through energy methods with suitably chosen weight functions. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Lyapunov functionals for the Enskog-Boltzmann equation

We present two Lyapunov functionals measuring future interactions between particles with different velocities, and L 1 -distance between two classical solutions for the Enskog-Boltzmann equation, when initial datum is a small perturbation of a vacuum and tends to zero fast enough at infinity in the phase space. These Lyapunov functionals yield time-asymptotic convergence of classical solutions to the collision free motion and the L 1 stability of classical solutions.

A nonlinear oblique derivative boundary value problem for the heat equation: Analogy with the porous medium equation

The problem under investigation is the heat equation in the upper half-plane, to which the diffusion in the longitudinal direction has been suppressed, and augmented with a nonlinear oblique derivative condition. This paper proves global existence and qualitative properties to the Cauchy problem for this model, furthering the study [18] of the self-similar solutions. The qualitative behaviour of the solutions exhibits a strong analogy with the porous medium equation: propagation with compact support and finite speed, free boundary relation and time-asymptotic convergence to self-similar solutions.

Transient Bounds and Time-Asymptotic Behavior of Solutions to Nonlinear Equations of Fisher Type

A study is made of the transient behavior of solutions to certain classes of semilinear parabolic initial value problems. First, families of upper and lower bounds for these solutions are explicitly constructed and compared. Secondly, the question of time-asymptotic convergence of these solutions towards certain traveling wave solutions (of the underlying parabolic equation) is discussed. It is shown, in particular, that this long time behavior depends crucially and in an interesting manner on the behavior of initial data at infinity.