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Hyung Ju Hwang

Publications and source records attributed to Hyung Ju Hwang.

43 records · Page 3Linked to original sources
arXiv2008

On the Existence of Exponentially Decreasing Solutions of the Nonlinear Landau Damping Problem

In this paper we prove the existence of a large class of periodic solutions of the Vlasov-Poisson in one space dimension that decay exponentially as t goes to infinity. The exponential decay is well known for the linearized version of the Landau damping problem and it has been proved in [4] for a class of solutions of the Vlasov-Poisson system that behaves asymptotically as free streaming solutions and are sufficiently flat in the space of velocities. The results in this paper enlarge the class of possible asymptotic limits, replacing the fatness condition in [4] by a stability condition for the linearized problem.

math.AP↗
arXiv2006

Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data

The Vlasov-Poisson system describes interacting systems of collisionless particles. For solutions with small initial data in three dimensions it is known that the spatial density of particles decays like $t^{-3}$ at late times. In this paper this statement is refined to show that each derivative of the density which is taken leads to an extra power of decay so that in $N$ dimensions for $N\ge 3$ the derivative of the density of order $k$ decays like $t^{-N-k}$. An asymptotic formula for the solution at late times is also obtained.

math.AP↗
arXiv2006

Global existence results for complex hyperbolic models of bacterial chemotaxis

Bacteria are able to respond to environmental signals by changing their rules of movement. When we take into account chemical signals in the environment, this behaviour is often called chemotaxis. At the individual-level, chemotaxis consists of several steps. First, the cell detects the extracellular signal using receptors on its membrane. Then, the cell processes the signal information through the intracellular signal transduction network, and finally it responds by altering its motile behaviour accordingly. At the population level, chemotaxis can lead to aggregation of bacteria, travelling waves or pattern formation, and the important task is to explain the population-level behaviour in terms of individual-based models. It has been previously shown that the transport equation framework is suitable for connecting different levels of modelling of bacterial chemotaxis. In this paper, we couple the transport equation for bacteria with the (parabolic/elliptic) equation for the extracellular signals. We prove global existence of solutions for the general hyperbolic chemotaxis models of cells which process the information about the extracellular signal through the intracellular biochemical network and interact by altering the extracellular signal as well. The conditions for global existence in terms of the properties of the signal transduction model are given.

math.AP↗
arXiv2005

Pattern formation (II): The Turing Instability

We consider the classical Turing instability in a reaction-diffusion system as the secend part of our study on pattern formation. We prove that nonlinear dynamics of a general perturbation of the Turing instability is determined by the finite number of linear growing modes over a time scale of $ln(1/δ)$, where &δ$ is the strength of the initial perturbation.

math.AP↗
arXiv2005

Variational approach to nonlinear gravity-driven instabilities in a MHD setting

We establish a variational framework for nonlinear instabilities in a setting of the ideal magnetohydrodynamic (MHD) equations. We apply a variational method to various kind of smooth steady states which are shown to be nonlinearly unstable for both incompressible and compressible ideal MHD equations. Destabilizing effect of compressibility is justified as well as stabilizing effect of magnetic field lines arising in MHD dynamics, which distinguishes from the Rayleigh-Taylor instability in the absence of magnetic field lines.

math.AP↗
arXiv2005

Pattern formation (I): The Keller-Segel Model

We investigate nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Given any general perturbation of magnitude $δ$, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of $ln(1/δ)$. Our result can be interpreted as a rigourous mathematical characterization for early pattern formation in the Keller-Segel model.

math.AP↗