We study the global solvability and the large-time behavior of solutions to the inhomogeneous Vla... more We study the global solvability and the large-time behavior of solutions to the inhomogeneous Vlasov–Navier–Stokes equations. When the initial data is sufficiently small and regular, we first show the unique existence of the global strong solution to the kinetic-fluid equations, and establish the a priori estimates for the large-time behavior using an appropriate Lyapunov functional. More specifically, we show that the velocities of particles and fluid tend to be aligned together exponentially fast, provided that the local density of the particles satisfies a certain integrability condition.
Communications in Nonlinear Science and Numerical Simulation, May 1, 2021
In this article, we construct the approximate solutions to the Euler & ndash;Poisson syst... more In this article, we construct the approximate solutions to the Euler & ndash;Poisson system in an annular domain, that arises in the study of dynamics of plasmas. Due to a small parameter (proportional to the square of the Debye length) multiplied to the Laplacian operator, together with unmatched boundary conditions, we find that the solutions exhibit sharp transition layers near the boundaries, which makes the associated limit problem singular. To investigate this singular behavior, we explicitly construct the approximate solutions composed of the outer and inner solutions by the method of asymptotic expansions in appropriate order of the small parameter, turned out to be the Debye length. The equations to single out the boundary layers are determined by the inner expansions, for which we effectively treat nonlinear terms using the Taylor polynomial expansions with multinomials. We can obtain estimates showing that the approximate solutions are close enough to the original ones. We also provide numerical evidences demonstrating that the approximate solutions converge to those of the Euler & ndash;Poisson system as the parameter goes to zero. (c) 2021 Elsevier B.V. All rights reserved
Abstract We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poiss... more Abstract We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poisson system with radial symmetry in an annular domain. Under physically relevant conditions at the boundary, referred to as the Bohm criterion, we first construct the approximate solutions by the method of asymptotic expansion in the limit parameter, the square of the rescaled Debye length, whose detailed derivation and analysis are carried out in our companion paper [8] . By establishing H m -norm, ( m ≥ 2 ) , estimate of the difference between the original and approximation solutions, provided that the well-prepared initial data is given, we show that the local-in-time solution exists in the time interval, uniform in the quasi-neutral limit, and we prove the difference converges to zero with a certain convergence rate validating the formal expansion order. Our results mathematically justify the quasi-neutrality of a plasma in the regime of plasma sheath, indicating that a plasma is electrically neutral in bulk, whereas the neutrality may break down in a scale of the Debye length.
We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed... more We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time-delay and phase lag effects. We also present several numerical experiments supporting our main results.
We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed... more We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time-delay and phase lag effects. We also present several numerical experiments supporting our main results.
We consider the asymptotic behavior of perturbations of transition front solutions arising in Cah... more We consider the asymptotic behavior of perturbations of transition front solutions arising in Cahn-Hilliard systems on R. Such equations arise naturally in the study of phase separation, and systems describe cases in which three or more phases are possible. When a Cahn-Hilliard system is linearized about a transition front solution, the linearized operator has an eigenvalue at 0 (due to shift invariance), which is not separated from essential spectrum. In cases such as this, nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate. Our main result asserts that spectral stability - a necessary condition for stability, defined in terms of an appropriate Evans function - implies nonlinear stability
We discuss existence, time-asymptotic behavior, and quasi-neutral limit for the Euler-Poisson equ... more We discuss existence, time-asymptotic behavior, and quasi-neutral limit for the Euler-Poisson equations. Specifically we construct the global-in-time solution near the plasma sheath, and investigate the properties of the solution. If time permits, some key features of the proof and related problems will be discussed. This is joint work with C.-Y. Jung (UNIST) and M. Suzuki (Nagoya Inst. Tech.)
We study the global solvability and the large-time behavior of solutions to the inhomogeneous Vla... more We study the global solvability and the large-time behavior of solutions to the inhomogeneous Vlasov–Navier–Stokes equations. When the initial data is sufficiently small and regular, we first show the unique existence of the global strong solution to the kinetic-fluid equations, and establish the a priori estimates for the large-time behavior using an appropriate Lyapunov functional. More specifically, we show that the velocities of particles and fluid tend to be aligned together exponentially fast, provided that the local density of the particles satisfies a certain integrability condition.
Communications in Nonlinear Science and Numerical Simulation, May 1, 2021
In this article, we construct the approximate solutions to the Euler & ndash;Poisson syst... more In this article, we construct the approximate solutions to the Euler & ndash;Poisson system in an annular domain, that arises in the study of dynamics of plasmas. Due to a small parameter (proportional to the square of the Debye length) multiplied to the Laplacian operator, together with unmatched boundary conditions, we find that the solutions exhibit sharp transition layers near the boundaries, which makes the associated limit problem singular. To investigate this singular behavior, we explicitly construct the approximate solutions composed of the outer and inner solutions by the method of asymptotic expansions in appropriate order of the small parameter, turned out to be the Debye length. The equations to single out the boundary layers are determined by the inner expansions, for which we effectively treat nonlinear terms using the Taylor polynomial expansions with multinomials. We can obtain estimates showing that the approximate solutions are close enough to the original ones. We also provide numerical evidences demonstrating that the approximate solutions converge to those of the Euler & ndash;Poisson system as the parameter goes to zero. (c) 2021 Elsevier B.V. All rights reserved
Abstract We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poiss... more Abstract We investigate the quasi-neutral limit (the zero Debye length limit) for the Euler-Poisson system with radial symmetry in an annular domain. Under physically relevant conditions at the boundary, referred to as the Bohm criterion, we first construct the approximate solutions by the method of asymptotic expansion in the limit parameter, the square of the rescaled Debye length, whose detailed derivation and analysis are carried out in our companion paper [8] . By establishing H m -norm, ( m ≥ 2 ) , estimate of the difference between the original and approximation solutions, provided that the well-prepared initial data is given, we show that the local-in-time solution exists in the time interval, uniform in the quasi-neutral limit, and we prove the difference converges to zero with a certain convergence rate validating the formal expansion order. Our results mathematically justify the quasi-neutrality of a plasma in the regime of plasma sheath, indicating that a plasma is electrically neutral in bulk, whereas the neutrality may break down in a scale of the Debye length.
We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed... more We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time-delay and phase lag effects. We also present several numerical experiments supporting our main results.
We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed... more We investigate the synchronized collective behavior of the Kuramoto oscillators with time-delayed interactions and phase lag effect. Both the phase and frequency synchronization are in view. We first prove the frequency synchronization for both semi-delay and full-delay models with heterogeneous time-delays and phase lags. We also prove the complete and partial phase synchronization for both models with the uniform time-delay and phase lag. Our results show that the Kuramoto models incorporated with small variation of time-delays and/or phase lag effect still exhibit the synchronization. These support that the original Kuramoto model (i.e., no time-delays/phase lags) is qualitatively robust in the perturbation of time-delay and phase lag effects. We also present several numerical experiments supporting our main results.
We consider the asymptotic behavior of perturbations of transition front solutions arising in Cah... more We consider the asymptotic behavior of perturbations of transition front solutions arising in Cahn-Hilliard systems on R. Such equations arise naturally in the study of phase separation, and systems describe cases in which three or more phases are possible. When a Cahn-Hilliard system is linearized about a transition front solution, the linearized operator has an eigenvalue at 0 (due to shift invariance), which is not separated from essential spectrum. In cases such as this, nonlinear stability cannot be concluded from classical semigroup considerations and a more refined development is appropriate. Our main result asserts that spectral stability - a necessary condition for stability, defined in terms of an appropriate Evans function - implies nonlinear stability
We discuss existence, time-asymptotic behavior, and quasi-neutral limit for the Euler-Poisson equ... more We discuss existence, time-asymptotic behavior, and quasi-neutral limit for the Euler-Poisson equations. Specifically we construct the global-in-time solution near the plasma sheath, and investigate the properties of the solution. If time permits, some key features of the proof and related problems will be discussed. This is joint work with C.-Y. Jung (UNIST) and M. Suzuki (Nagoya Inst. Tech.)
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Papers by Bongsuk Kwon