We propose a phenomenological yet general model in a form of extended complex Ginzburg-Landau equ... more We propose a phenomenological yet general model in a form of extended complex Ginzburg-Landau equation to understand edge-localized modes (ELMs), a class of quasi-periodic fluid instabilities in the boundary of toroidal magnetized high-temperature plasmas. The model reproduces key dynamical features of the ELMs (except the final explosive relaxation stage) observed in the high-confinement state plasmas on the Korea Superconducting Tokamak Advanced Research: quasi-steady states characterized by field-aligned filamentary eigenmodes, transitions between different quasi-steady eigenmodes, and rapid transition to non-modal filamentary structure prior to the relaxation. It is found that the inclusion of time-varying perpendicular sheared flow is crucial for reproducing all of the observed dynamical features.
In this paper, we consider an electrified thin film equation with periodic boundary conditions. W... more In this paper, we consider an electrified thin film equation with periodic boundary conditions. When an applied voltage is sufficiently small after a finite time, we prove the global existence of unique solutions around positive constant steady states and study the asymptotic behavior of the solutions. On the other hand, when the applied voltage is constant and sufficiently large, we prove that the solutions around the constant steady states are unstable. Moreover, we prove the existence of infinitely many curves of nontrivial steady states of the electrified thin film equation around positive constant solutions at certain positive values of the voltage. Finally, as the applied voltage passes through the first bifurcation value, we obtain a unique global-in-time solution with an initially perturbed domain around nontrivial steady states which come from the first bifurcation curve, and we show that the solutions exponentially converge to the nontrivial steady-state solutions as time goes to infinity.
In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathemati... more In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathematical model, a reaction-diffusion-advection system, describing a cross-talk between antigens and immune cells via chemokines. We analyze the stability and instability arising in our chemotaxis model and find their conditions for different chemotactic strengths by using energy estimates, spectral analysis, and bootstrap argument. Numerical simulations are also performed to the model, by using the finite volume method in order to deal with the chemotaxis term, and the fractional step methods are used to solve the whole system. From the analytical and numerical results for our model, we explain not only the effective attraction of immune cells toward the site of infection but also hypersensitivity when chemotactic strength is greater than some threshold.
We propose a phenomenological yet general model in a form of extended complex Ginzburg-Landau equ... more We propose a phenomenological yet general model in a form of extended complex Ginzburg-Landau equation to understand edge-localized modes (ELMs), a class of quasi-periodic fluid instabilities in the boundary of toroidal magnetized high-temperature plasmas. The model reproduces key dynamical features of the ELMs (except the final explosive relaxation stage) observed in the high-confinement state plasmas on the Korea Superconducting Tokamak Advanced Research: quasi-steady states characterized by field-aligned filamentary eigenmodes, transitions between different quasi-steady eigenmodes, and rapid transition to non-modal filamentary structure prior to the relaxation. It is found that the inclusion of time-varying perpendicular sheared flow is crucial for reproducing all of the observed dynamical features.
In this paper, we consider an electrified thin film equation with periodic boundary conditions. W... more In this paper, we consider an electrified thin film equation with periodic boundary conditions. When an applied voltage is sufficiently small after a finite time, we prove the global existence of unique solutions around positive constant steady states and study the asymptotic behavior of the solutions. On the other hand, when the applied voltage is constant and sufficiently large, we prove that the solutions around the constant steady states are unstable. Moreover, we prove the existence of infinitely many curves of nontrivial steady states of the electrified thin film equation around positive constant solutions at certain positive values of the voltage. Finally, as the applied voltage passes through the first bifurcation value, we obtain a unique global-in-time solution with an initially perturbed domain around nontrivial steady states which come from the first bifurcation curve, and we show that the solutions exponentially converge to the nontrivial steady-state solutions as time goes to infinity.
In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathemati... more In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathematical model, a reaction-diffusion-advection system, describing a cross-talk between antigens and immune cells via chemokines. We analyze the stability and instability arising in our chemotaxis model and find their conditions for different chemotactic strengths by using energy estimates, spectral analysis, and bootstrap argument. Numerical simulations are also performed to the model, by using the finite volume method in order to deal with the chemotaxis term, and the fractional step methods are used to solve the whole system. From the analytical and numerical results for our model, we explain not only the effective attraction of immune cells toward the site of infection but also hypersensitivity when chemotactic strength is greater than some threshold.
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