Bongsuk Kwon

A hydrodynamic model for the interaction of Cucker–Smale particles and incompressible fluid

We present a new hydrodynamic model for the interactions between collision-free Cucker–Smale flocking particles and a viscous incompressible fluid. Our proposed model consists of two hydrodynamic models. For the Cucker–Smale flocking particles, we employ the pressureless Euler system with a non-local flocking dissipation, whereas for the fluid, we use the incompressible Navier–Stokes equations. These two hydrodynamic models are coupled through a drag force, which is the main flocking mechanism between the particles and the fluid. The flocking mechanism between particles is regulated by the Cucker–Smale model, which accelerates global flocking between the particles and the fluid. We show that this model admits the global-in-time classical solutions, and exhibits time-asymptotic flocking, provided that the initial data is appropriately small. In the course of our analysis for the proposed system, we first consider the hydrodynamic Cucker–Smale equations (the pressureless Euler system with a non-local flocking dissipation), for which the global existence and the time-asymptotic behavior of the classical solutions are also investigated.

Emergent Dynamics for the Hydrodynamic Cucker--Smale System in a Moving Domain

We study the emergent dynamics for the hydrodynamic Cucker--Smale system arising in the modeling of flocking dynamics in interacting many-body systems. Specifically, the initial value problem with a moving domain is considered to investigate the global existence and time-asymptotic behavior of classical solutions, provided that the initial mass density has bounded support and the initial data are in an appropriate Sobolev space. In order to show the emergent behavior of flocking, we make use of an appropriate Lyapunov functional that measures the total fluctuation in the velocity relative to the mean velocity. In our analysis, we present the local well-posedness of the smooth solutions via Lagrangian coordinates, and we extend to the global-in-time solutions by establishing the uniform flocking estimates.

ASYMPTOTIC BEHAVIOR OF MULTIDIMENSIONAL SCALAR RELAXATION SHOCKS

We establish pointwise bounds for the Green function and consequent linearized stability for multidimensional planar relaxation shocks of general relaxation systems whose equilibrium model is scalar, under the necessary assumption of spectral stability. Moreover, we obtain nonlinear L2 asymptotic behavior/sharp decay rate of perturbed weak shocks of general simultaneously-symmetrizable relaxation systems, under small L1 ∩ H[d/2]+3 perturbations with first moment in the normal direction to the front.

Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations

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.

Structural conditions for full MHD equations

In this paper, we investigate the characteristic structure of the full equations of magnetohydrodynamics (MHD) and show that it satisfies the hypotheses of a general variable-multiplicity stability framework introduced by Metivier and Zumbrun, thereby extending to the general case various results obtained by Metivier and Zumbrun for the isentropic equations of MHD.

Quasi-neutral limit for the Euler–Poisson system in the presence of plasma sheaths with spherical symmetry

The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of a ball-shaped material immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study existence and the quasi-neutral limit behavior of the stationary spherical symmetric solutions for the Euler–Poisson equations in a three-dimensional annular domain. We first propose a suitable condition on the velocity at the sheath edge, referred as to Bohm criterion for the annulus, and under this condition together with the constant Dirichlet boundary conditions for the potential, we show that there exists a unique stationary spherical symmetric solution. Moreover, we study the quasi-neutral limit behavior by establishing L2 estimate of the difference of the solutions to the Euler–Poisson equations and its quasi-neutral limiting equations, incorporated with the correctors for the boundary layers. The quasi-neutral limit analysis employing the correctors and their pointwise estimates enables us to obtain detailed asymptotic behaviors including the convergence rates in L2 and H1 norms as well as the thickness of the boundary layers as a consequence of the pointwise estimates.