Mathematics of Fluid Motion II: Theory and Computation
December 2628th, 2018 KIAS 1423

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SungJin Oh
Title: On the Cauchy problem for the Hall MHD equations without resistivity
Abstract: In this talk, I will describe recent work with I.J. Jeong on the Cauchy problem for the Hall MHD equation without resistivity. This PDE, first investigated by Lighthill, is a onefluid description of magnetized plasma with a quadratic secondorder correction term, called the Hall current term, that takes into account the motion of electrons relative to positive ions. We demonstrate both ill and wellposedness of the Cauchy problem depending on the initial data.
Takashi Sakajo
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InJee Jeong
Title: Dynamics of Singular Vortex Patches
Abstract: Vortex patches are solutions to the 2D Euler equations that are given by the characteristic function of a bounded domain that moves with time. It is wellknown that if initially the boundary of the domain is smooth, the boundary remains smooth for all time. On the other hand, we consider patches with corner singularities. It turns out that, depending on whether the initial patch satisfies an appropriate rotational symmetry condition or not, the corner structure may propagate for all time or lost immediately. In the rotationally symmetric case, we are able to construct patches with interesting dynamical behavior as time goes to infinity. When the symmetry is absent, we present a simple yet formal evolution equation which describes the dynamics of the boundary. It suggests that the angle cusps instantaneously for t>0.
This is joint work with Tarek Elgindi.
Tomoyuki Miyaji
Title: Computerassisted proof of the existence of unimodal solutions of the ProudmanJohnson equation
Abstract: We consider unimodal solutions of the ProudmanJohnson equation which comes from a representative of the twodimensional NavierStokes equation related to fluid flow. The unimodal solution is a model of a large coherent vortex appearing in 2D NavierStokes flows at large Reynolds numbers. Although such a largescale structure is often observed numerically and experimentally, its existence has not been proved from a viewpoint of mathematics. We formulate the multipleshooting method for the stationary ProudmanJohnson equation and solve it via the interval Newton method. As a result, we obtain a computerassisted proof that a unimodal solution exists at a moderately large Reynolds number. In this talk, we present a brief review of interval analysis, a method of computerassisted proof based on interval analysis, and its application to our problem. This is a joint work with Hisashi Okamoto of Gakushuin University
SunChul Kim
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Byungjoon Lee
Title and Abstract (file download)
Tsuyoshi Yoneda
Title: Instantaneous vortex stretching and energy cascade on the incompressible 3D Euler equations
Abstract: By DNS of NavierStokes turbulence, GotoSaitoKawahara (2017) showed that the turbulence consists of a selfsimilar hierarchy of antiparallel pairs of vortex tubes, in particular, stretching in largerscale strain fields create smallerscale vortices. Inspired by their numerical result, in this talk, we examine the GotoSaitoKawahara type of vortextubes behavior by incompressible 3D Euler equations, and show that such vortextubes behavior induces instantaneous energycascade (on a single vortextube) without scaleinteraction.
This is a joint work with InJee Jeong.
SungIk Sohn
Title: Vortex models for hovering wings and application to insect flights
Abstract: The interaction of vortex and body is one of fundamental problems in fluid dynamics. In this talk, we discuss recent development of vortex models for aerodynamic wings and address some challenging problems. Applications of the models to insect flights will be also presented.
Donghyun Lee
Title: Fluids with freesurface and its vanishing viscosity limits
Abstract: We discuss vanishing viscosity limit of freeboundary problem of NavierStokes to obtain freeboundary Euler. To control singular behavior, we introduce Sobolev conormal space. In particular, we will compare differences between surface tension and nonsurface tension cases. In the end of this talk, a partial result about freeboundary MHD will be mentioned.
Joonhyun La
Title: On a kinetic model of polymeric fluids
Abstract: In this talk, we prove global wellposedness of a system describing behavior of dilute flexible polymeric fluids. This model is based on kinetic theory, and a main difficulty for this system is its multiscale nature. A new function space, based on moments, is introduced to address this issue, and this function space allows us to deal with larger initial data.
Kyungkeun Kang
Title: On Caccioppoli's inequalities of Stokes and NavierStokes equations up to boundary
Abstract: We are concerned with Caccioppolis inequalities of the nonstationary Stokes system and Navier Stokes equations. It is known that the Caccioppolis inequalities of the Stokes system and the NavierStokes equations are true known in the interior case. We prove that the Caccioppolis inequalities of the Stokes system and the NavierStokes equations may, however, fail near boundary, when only local analysis is considered at the at flat boundary. This is a joint work with Dr. TongKeun Chang.
Bongsuk Kwon
Title: Small Debye length limit for the EulerPoisson system
Abstract: We discuss existence, timeasymptotic behavior, and quasineutral limit for the EulerPoisson equations. Specifically, under the Bohm's criterion, we construct the globalintime solution in the regime of the plasma sheath and investigate the properties of the solution including the timeasymptotic behavior and small Debye length limit. 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 Tech.).