Browsing by Author "Lin, Chi-Kun"

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An Asymptotic Limit of a Navier-Stokes System with Capillary Effects
(2014-07-01)
Juengel, Ansgar
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Lin, Chi-Kun
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Wu, Kung-Chien
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應用數學系
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數學建模與科學計算所(含中心)
A combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes equations for smooth solutions is proved. The equations are considered on the two-dimensional torus with well prepared initial data. The momentum equation contains a rotational term originating from a Coriolis force, a general Korteweg-type tensor modeling capillary effects, and a density-dependent viscosity. The limiting model is the viscous quasi-geostrophic equation for the "rotated" velocity potential. The proof of the singular limit is based on the modulated energy method with a careful choice of the correction terms.
ANELASTIC APPROXIMATION OF THE GROSS-PITAEVSKII EQUATION FOR GENERAL INITIAL DATA
(2014)
Lin, Chi-Kun
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Wu, Kung-Chien
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應用數學系
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數學建模與科學計算所(含中心)
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Department of Applied Mathematics
We perform a rigorous analysis of the anelastic approximation for the Gross-Pitaevskii equation with x-dependent chemical potential. For general initial data and periodic boundary condition, we show that as epsilon -> 0, equivalently the Planck constant tends to zero, the density vertical bar psi(epsilon)vertical bar(2) converges toward the chemical potential rho(0)(x) and the velocity field converges to the anelastic system.
EXPONENTIAL STABILITY OF NONMONOTONE TRAVELING WAVES FOR NICHOLSON'S BLOWFLIES EQUATION
(2014-01-01)
Lin, Chi-Kun
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Lin, Chi-Tien
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Lin, Yanping
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Mei, Ming
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應用數學系
This paper is concerned with Nicholson's blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 < p/d <= e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p/d > e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e < p/d <= e(2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, where c(*) > 0 is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed c > c(*) and any size of the time-delay r > 0; however, when p/d > e(2) with a small time-delay r < [pi-arctan root ln p/d(ln p/d -2)]/d root ln p/d(ln p/d -2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, too. As a corollary, we also prove the uniqueness of traveling waves in the case of p/d > e(2), which to the best of our knowledge was open. Finally, some numerical simulations are carried out. When e < p/d <= e(2), we demonstrate numerically that after a long time the solution behaves like a monotone traveling wave for a small time-delay, and behaves like an oscillatory traveling wave for a big time-delay. When p/d > e(2), if the time-delay is small, then the solution numerically behaves like a monotone/nonmonotone traveling wave, but if the time-delay is big, then the solution is numerically demonstrated to be chaotically oscillatory but not an oscillatory traveling wave. These either confirm and support our theoretical results or open up some new phenomena for future research.
EXPONENTIAL STABILITY OF NONMONOTONE TRAVELING WAVES FOR NICHOLSON\'S BLOWFLIES EQUATION
(2014-01-01)
Lin, Chi-Kun
;
Lin, Chi-Tien
;
Lin, Yanping
;
Mei, Ming
;
應用數學系
This paper is concerned with Nicholson\'s blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 < p/d <= e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p/d > e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e < p/d <= e(2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, where c(*) > 0 is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed c > c(*) and any size of the time-delay r > 0; however, when p/d > e(2) with a small time-delay r < [pi-arctan root ln p/d(ln p/d -2)]/d root ln p/d(ln p/d -2), all noncritical traveling waves phi(x + ct) with c > c(*) > 0 are exponentially stable, too. As a corollary, we also prove the uniqueness of traveling waves in the case of p/d > e(2), which to the best of our knowledge was open. Finally, some numerical simulations are carried out. When e < p/d <= e(2), we demonstrate numerically that after a long time the solution behaves like a monotone traveling wave for a small time-delay, and behaves like an oscillatory traveling wave for a big time-delay. When p/d > e(2), if the time-delay is small, then the solution numerically behaves like a monotone/nonmonotone traveling wave, but if the time-delay is big, then the solution is numerically demonstrated to be chaotically oscillatory but not an oscillatory traveling wave. These either confirm and support our theoretical results or open up some new phenomena for future research.
GUIDING CENTER DRIFT INDUCED BY HOMOGENIZATION
(2012-05-01)
Jiang, Jiann-Sheng
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Lin, Chi-Kun
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應用數學系
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數學建模與科學計算所(含中心)
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Department of Applied Mathematics
The guiding center drift induced by the homogenization of the Lorentz forces is studied. It generates memory effects. The memory (or nonlocal) kernel is described by the Volterra integral equation. The memory kernel can be characterized explicitly in terms of a Radon measure. It describes the extra velocity drift. By way of velocity drift, we view the Gauss's law with polarization charges.
Homogenization of the Maxwell's system for conducting media
(2008-07-01)
Jiang, Jiann-Sheng
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Lin, Chi-Kun
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Liu, Chi-Hua
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應用數學系
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Department of Applied Mathematics
This paper is devoted to the study of the memory effect induced by homogenization of the Maxwell system for conducting media. The memory kernel is described by the Volterra integral equation. Furthermore, it can be characterized explicitly in terms of Young's measure, and the kinetic formulation of the homogenized equation is also obtained. The kinetic formulation allows us to obtain the homogenization of the energy density and the associated conservation law with the Poynting vector. The interesting interaction phenomenon of the microscopic and macroscopic scales is also discussed and the memory effect explains qualitatively something about irreversibility.
Hydrodynamic limits of the nonlinear Klein-Gordon equation
(2012-09-01)
Lin, Chi-Kun
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Wu, Kung-Chien
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應用數學系
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數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
We perform the mathematical derivation of the compressible and incompressible Euler equations from the modulated nonlinear Klein-Gordon equation. Before the formation of singularities in the limit system, the nonrelativistic-semiclassical limit is shown to be the compressible Euler equations. If we further rescale the time variable, then in the semiclassical limit (the light speed kept fixed), the incompressible Euler equations are recovered. The proof involves the modulated energy introduced by Brenier (2000) [1]. (C) 2012 Elsevier Masson SAS. All rights reserved.
Numerical computations for long-wave short-wave interaction equations in semi-classical limit
(2008-10-01)
Chang, Qianshun
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Wong, Yau-Shu
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Lin, Chi-Kun
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應用數學系
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Department of Applied Mathematics
This paper presents and compares various numerical techniques for the long-wave short-wave interaction equations. In addition to the standard explicit, implicit schemes and the spectral methods, a novel scheme SRK which is based on a time-splitting approach combined with the Runge-Kutta method is presented. We demonstrate that not only the SRK scheme is efficient compared to the split step spectral methods, but it can apply directly to problems with general boundary conditions. The conservation properties of the numerical schemes are discussed. Numerical simulations are reported for case studies with different types of initial data. The present study enhances our understanding of the behavior of nonlinear dispersive waves in the semi-classical limit. (C) 2008 Elsevier Inc. All rights reserved.
ON THE FLUID DYNAMICAL APPROXIMATION TO THE NONLINEAR KLEIN-GORDON EQUATION
(2012-06-01)
Lin, Chi-Kun
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Wu, Kung-Chien
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應用數學系
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數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
We study the nonrelativistic, semiclassical and nonrelativistic-semiclassical limits of the (modulated) nonlinear Klein-Gordon equations from its hydrodynamical structure via WKB analysis. The nonrelativistic-semiclassical limit is proved rigorously by modulated energy method.
On travelling wavefronts of Nicholson's blowflies equation with diffusion
(2010)
Lin, Chi-Kun
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Mei, Ming
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應用數學系
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Department of Applied Mathematics
This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x -> -infinity, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.
Quasineutral Limit of the Schrodinger-Poisson System in Coulomb Gauge
(2011)
Lin, Chi-Kun
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Wong, Yau-Shu
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Wu, Kung-Chien
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應用數學系
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數學建模與科學計算所(含中心)
The zero Debye length asymptotic of the Schrodinger-Poisson system in Coulomb gauge for ill-prepared initial data is studied. We prove that when the scaled Debye length lambda -> 0, the current density defined by the solution of the Schrodinger-Poisson system in the Coulomb gauge converges to the solution of the rotating incompressible Euler equation plus a fast singular oscillating gradient vector field.
Singular Limits of the Klein-Gordon Equation
(2010-08-01)
Lin, Chi-Kun
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Wu, Kung-Chien
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應用數學系
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數學建模與科學計算所(含中心)
;
Department of Applied Mathematics
We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein-Gordon equation. For the semiclassical limit, h -> 0, we show that the limit wave function of the modulated defocusing cubic nonlinear Klein-Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c -> infinity, of the modulated defocusing nonlinear Klein-Gordon equation is the defocusing nonlinear Schrodinger equation. The nonrelativistic-semiclassical limit, h -> 0, c = h (-alpha) -> infinity for some alpha > 0, of the modulated defocusing cubic nonlinear Klein-Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.
The initial value problem for some hyperbolic-dispersive system
(2012-01-30)
Kawashima, Shuichi
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Lin, Chi-Kun
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Segata, Jun-ichi
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應用數學系
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數學建模與科學計算所(含中心)
We consider the initial value problem for some nonlinear hyperbolic-dispersive systems in one space dimension. Combining the classical energy method and the smoothing estimates for the Airy equation, we guarantee the time local well-posedness for this system. We also discuss the extension of our results to more general hyperbolic-dispersive system. Copyright (C) 2011 John Wiley & Sons, Ltd.
Traveling wavefronts for time-delayed reaction-diffusion equation: (I) Local nonlinearity
(2009-07-15)
Mei, Ming
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Lin, Chi-Kun
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Lin, Chi-Tien
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So, Joseph W. -H.
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應用數學系
In this paper, we study a class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate. For all wave-fronts with the speed c > c(*), where c(*) > 0 is the critical wave speed, we prove that these wavefronts are asymptotically stable, when the initial perturbation around the traveling waves decays exponentially as x -> -infinity, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results obtained by Mei. So. Li and Shen [M. Mei, J.W.-H. So, M.Y. Li, S.S.R Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594] for the speed c > 2 root D(m) (epsilon p - d(m)) with small initial perturbation and by Lin and Mei [C.-K. Lin, M. Mei, On travelling wavefronts of the Nicholson's blowflies equations with diffusion, submitted for publication] for c > c(*) with sufficiently small delay time r approximate to 0. The approach adopted in this paper is the technical weighted energy method used in [M. Mei, J.W.-H. So, M.Y. Li, S.S.R Shen, Asymptotic stability of traveling waves for the Nicholson's blowflies equation with diffusion, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 579-594], but inspired by Gourley [S.A. Gourley, Linear stability of travelling fronts in an age-structured reaction-diffusion Population model, Quart. J. Mech. Appl. Math. 58 (2005) 257-268] and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any c > c(*) and for an arbitrary time-delay r > 0. (C) 2009 Elsevier Inc. All rights reserved.
Traveling wavefronts for time-delayed reaction-diffusion equation: (II) Nonlocal nonlinearity
(2009-07-15)
Mei, Ming
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Lin, Chi-Kun
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Lin, Chi-Tien
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So, Joseph W. -H.
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應用數學系
This is the second part of a series of study oil the stability of traveling wavefronts of reaction-diffusion equations with time delays. In this paper we will consider a nonlocal time-delayed reaction-diffusion equation. When the initial perturbation around the traveling wave decays exponentially as x -> -infinity (but the initial perturbation call be arbitrarily large in other locations), we prove the asymptotic stability of all traveling waves for the reaction-diffusion equation, including even the slower waves whose speed are close to the critical speed. This essentially improves the previous stability results by Mei and So [M. Mei, J.W.-H. So, Stability of strong traveling waves for a nonlocal time-delayed reaction-diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008) 551-568] for the speed c > 2 root D(m)(3 epsilon p-2d(m)) with a small initial perturbation. The approach we use here is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank-Nicholson scheme, we present some numerical results which confirm Our theoretical Study. (C) 2009 Elsevier Inc. All rights reserved.
Weak turbulence plasma induced by two-scale homogenization
(2014-02-15)
Jiang, Jiann-Sheng
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Lin, Chi-Kun
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應用數學系
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Department of Applied Mathematics
This paper is devoted to the homogenization of the quasilinear theory of the plasma turbulence described by the Vlasov-Poisson system. It is shown that the homogenization limit, in the sense of two-scale limit, of the distribution function satisfies the linear Vlasov-Poisson equations. Moreover, the limit distribution function can be decomposed into the mean and the fluctuation parts and the mean part (the equilibrium distribution function) is shown to be the solution of the nonlocal quasilinear velocity-space diffusion equation. We also investigate the Landau damping from the point of view of homogenization through the two-scale limit. Crown Copyright (C) 2013 Published by Elsevier Inc. All rights reserved.
WKB analysis of the Schrodinger-KdV system
(2014-06-01)
Lin, Chi-Kun
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Segata, Jun-ichi
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應用數學系
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數學建模與科學計算所(含中心)
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Department of Applied Mathematics
We consider the behavior of solutions to the water wave interaction equations in the limit epsilon -> 0(+). To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system. (c) 2014 Elsevier Inc. All rights reserved.
Zero-dispersion limit of the short-wave-long-wave interaction equations
(2006-09-01)
Lin, Chi-Kun
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Wong, Yau-Shu
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應用數學系
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Department of Applied Mathematics
The purpose of this paper is to study the zero-dispersion limit of the water wave interaction equations which arise in modelling surface waves in the present of both gravity and capillary modes. This topic is also of interest in plasma physics. For the smooth solution, the limiting equation is given by the compressible Euler equation with a nonlocal pressure caused by the long wave. For weak solution, when the coupling coefficient lambda is small order of epsilon, lambda = o(epsilon), the wave map equation is derived and the scattering sound wave is shown to satisfy a linear wave equation. (c) 2006 Elsevier Inc. All rights reserved.