Convergence rates to travelling waves for a nonconvex relaxation model

1998 ◽  
Vol 128 (5) ◽  
pp. 1053-1068 ◽  
Author(s):  
Ming Mei ◽  
Tong Yang
Keyword(s):  
Convergence Rates ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Decay Rates ◽  
Relaxation Model ◽  
Weighted Energy Method ◽  
Time Decay Rates ◽  
Algebraic Forms ◽  
The Stability ◽  
Nonconvex Relaxation

In this paper we study the asymptotic behaviour of the solution for a nonconvex relaxation model. The time decay rates in both the exponential and algebraic forms of the travelling wave solutions are shown by the weighted energy method. Our results develop and improve the stability theory in [8,9].

CONVERGENCE RATES TO SUPERPOSITION OF TWO TRAVELLING WAVES OF THE SOLUTIONS TO A RELAXATION HYPERBOLIC SYSTEM WITH BOUNDARY EFFECTS

2001 ◽  
Vol 11 (07) ◽  
pp. 1143-1168 ◽  
Author(s):  
LING HSIAO ◽  
HAILIANG LI ◽  
MING MEI
Keyword(s):  
Hyperbolic System ◽  
Convergence Rates ◽  
Travelling Waves ◽  
Initial Boundary ◽  
Travelling Wave ◽  
Asymptotic Behavior Of Solutions ◽  
Shift Function ◽  
Special Choice ◽  
Weighted Energy Method ◽  
Algebraic Forms

This paper is to study the asymptotic behavior of solutions for an initial–boundary value problem to Jin–Xin's 2×2 relaxation hyperbolic system. When the initial data are small perturbation of the superposition of two travelling waves at t=0, subsequent to the previous work,6 we further show the convergence rates of the IBVP solutions to the superposition of two waves. Precisely, when the initial perturbations decay in the exponential or algebraic forms, we prove that the corresponding solutions tend to the superposition of two waves time-asymptotically in the exponential or algebraic forms, respectively. The method adopted is the weighted energy method. The use of a shift function for the forward travelling wave and the special choice of shift functions for backward travelling plays a key role to overcome the difficulties caused by the boundary and degeneration.

STABILITY OF SHOCK PROFILES FOR NONCONVEX SCALAR VISCOUS CONSERVATION LAWS

1995 ◽  
Vol 05 (03) ◽  
pp. 279-296 ◽  
Author(s):  
MING MEI
Keyword(s):  
Conservation Laws ◽  
Decay Rates ◽  
Asymptotically Stable ◽  
Time Decay ◽  
Shock Condition ◽  
Viscous Conservation Laws ◽  
Time Decay Rates ◽  
Shock Profiles ◽  
Degenerate Shock ◽  
The Stability

This paper is to study the stability of shock profiles for nonconvex scalar viscous conservation laws with the nondegenerate and the degenerate shock conditions by means of an elementary energy method. In both cases, the shock profiles are proved to be asymptotically stable for suitably small initial disturbances. Moreover, in the case of nondegenerate shock condition, time decay rates of asymptotics are also obtained.

The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien–Tanner system with magnetic field

2021 ◽  
pp. 1-34
Author(s):  
Yuhui Chen ◽  
Minling Li ◽  
Qinghe Yao ◽  
Zheng-an Yao
Keyword(s):  
Magnetic Field ◽  
Initial Data ◽  
Convergence Rates ◽  
Splitting Method ◽  
Mhd Flow ◽  
Decay Rates ◽  
Upper And Lower Bounds ◽  
Time Decay ◽  
Large Solutions ◽  
Time Decay Rates

In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture L 2 -convergence result via auxiliary logarithmic time decay estimates with the initial data in L p ( R 2 ) ∩ L 2 ( R 2 ). Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized.

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

2011 ◽  
Vol 141 (6) ◽  
pp. 1141-1173 ◽  
Author(s):  
Jared C. Bronski ◽  
Mathew A. Johnson ◽  
Todd Kapitula
Keyword(s):  
Blow Up ◽  
Kdv Equation ◽  
Travelling Waves ◽  
Classical Mechanics ◽  
Index Theorem ◽  
Travelling Wave ◽  
Kdv Equations ◽  
The Kdv Equation ◽  
De Vries ◽  
The Stability

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

scholarly journals Thermal solitons: travelling waves in combustion

2013 ◽  
Vol 469 (2150) ◽  
pp. 20120587 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Travelling Waves ◽  
Reaction System ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Governing Equations ◽  
Competitive Reaction ◽  
Chemical Reagent ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
The Stability

A competitive reaction system is considered, under which some chemical reagent decays by means of two simultaneous chemical reactions to form two separate inert products. One reaction is exothermic, and the other is endothermic. The governing equations for the model are presented, and a weakly nonlinear theory is then generated using the method of strained coordinates. Travelling-wave solutions are possible in the model, and the temperature is found to have a classical sech-squared profile. The stability of these moderate-amplitude temperature solitons is confirmed both analytically and numerically.

scholarly journals An energy-based stability criterion for solitary travelling waves in Hamiltonian lattices

2018 ◽  
Vol 376 (2117) ◽  
pp. 20170192 ◽  
Author(s):  
Haitao Xu ◽  
Jesús Cuevas-Maraver ◽  
Panayotis G. Kevrekidis ◽  
Anna Vainchtein
Keyword(s):  
Nonlinear Waves ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Point Of View ◽  
Floquet Multiplier ◽  
Theme Issue ◽  
Future Directions ◽  
Infinite Dimensional ◽  
The Stability ◽  
Delay Differential

In this work, we revisit a criterion, originally proposed in Friesecke & Pego (Friesecke & Pego 2004 Nonlinearity 17 , 207–227. ( doi:10.1088/0951715/17/1/013 )), for the stability of solitary travelling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-travelling frame, as well as at that of the Floquet problem arising when considering the travelling wave as a periodic orbit modulo shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the travelling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

Stability Analysis of the Nanjung Water Diversion Twin Tunnels based on Convergence Measurement

2019 ◽  
Vol 1 (1) ◽  
pp. 49-60
Author(s):  
Simon Heru Prassetyo ◽  
Ganda Marihot Simangunsong ◽  
Ridho Kresna Wattimena ◽  
Made Astawa Rai ◽  
Irwandy Arif ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Convergence Rate ◽  
Critical Strain ◽  
Convergence Rates ◽  
Strain Analysis ◽  
Water Diversion ◽  
Tunnel Face ◽  
Tunnel Stability ◽  
The Stability ◽  
Twin Tunnels

This paper focuses on the stability analysis of the Nanjung Water Diversion Twin Tunnels using convergence measurement. The Nanjung Tunnel is horseshoe-shaped in cross-section, 10.2 m x 9.2 m in dimension, and 230 m in length. The location of the tunnel is in Curug Jompong, Margaasih Subdistrict, Bandung. Convergence monitoring was done for 144 days between February 18 and July 11, 2019. The results of the convergence measurement were recorded and plotted into the curves of convergence vs. day and convergence vs. distance from tunnel face. From these plots, the continuity of the convergence and the convergence rate in the tunnel roof and wall were then analyzed. The convergence rates from each tunnel were also compared to empirical values to determine the level of tunnel stability. In general, the trend of convergence rate shows that the Nanjung Tunnel is stable without any indication of instability. Although there was a spike in the convergence rate at several STA in the measured span, that spike was not replicated by the convergence rate in the other measured spans and it was not continuous. The stability of the Nanjung Tunnel is also confirmed from the critical strain analysis, in which most of the STA measured have strain magnitudes located below the critical strain line and are less than 1%.

scholarly journals Experimental and theoretical progress in pipe flow transition

2008 ◽  
Vol 366 (1876) ◽  
pp. 2671-2684 ◽  
Author(s):  
A.P Willis ◽  
J Peixinho ◽  
R.R Kerswell ◽  
T Mullin
Keyword(s):  
Pipe Flow ◽  
Quantitative Agreement ◽  
Travelling Wave ◽  
Turbulent Pipe Flow ◽  
Transition To Turbulence ◽  
Reynolds Number Flow ◽  
Threshold Amplitude ◽  
Number Flow ◽  
The Stability ◽  
Laminar State

There have been many investigations of the stability of Hagen–Poiseuille flow in the 125 years since Osborne Reynolds' famous experiments on the transition to turbulence in a pipe, and yet the pipe problem remains the focus of attention of much research. Here, we discuss recent results from experimental and numerical investigations obtained in this new century. Progress has been made on three fundamental issues: the threshold amplitude of disturbances required to trigger a transition to turbulence from the laminar state; the threshold Reynolds number flow below which a disturbance decays from turbulence to the laminar state, with quantitative agreement between experimental and numerical results; and understanding the relevance of recently discovered families of unstable travelling wave solutions to transitional and turbulent pipe flow.

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