Mathematics of Fluid Motion II: Theory and Computation




December 26-28th, 2018               KIAS 1423


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Sung-Jin 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 one-fluid description of magnetized plasma with a quadratic second-order 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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In-Jee 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 well-known 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: Computer-assisted proof of the existence of unimodal solutions of the Proudman-Johnson equation 
Abstract: We consider unimodal solutions of the Proudman-Johnson equation which comes from a representative of the two-dimensional Navier-Stokes equation related to fluid flow. The unimodal solution is a model of a large coherent vortex appearing in 2D Navier-Stokes flows at large Reynolds numbers. Although such a large-scale structure is often observed numerically and experimentally, its existence has not been proved from a viewpoint of mathematics. We formulate the multiple-shooting method for the stationary Proudman-Johnson equation and solve it via the interval Newton method. As a result, we obtain a computer-assisted 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 computer-assisted proof based on interval analysis, and its application to our problem. This is a joint work with Hisashi Okamoto of Gakushuin University




Sun-Chul Kim

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Byungjoon Lee

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Tsuyoshi Yoneda
Title: Instantaneous vortex stretching and energy cascade on the incompressible 3D Euler equations
Abstract: By DNS of Navier-Stokes turbulence, Goto-Saito-Kawahara (2017) showed that the turbulence consists of a self-similar hierarchy of anti-parallel pairs of vortex tubes, in particular, stretching in larger-scale strain fields create smaller-scale vortices. Inspired by their numerical result, in this talk, we examine the Goto-Saito-Kawahara type of vortex-tubes behavior by incompressible 3D Euler equations, and show that such vortex-tubes behavior induces instantaneous energy-cascade (on a single vortex-tube) without scale-interaction.
This is a joint work with In-Jee Jeong.




Sung-Ik 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 free-surface and its vanishing viscosity limits
Abstract: We discuss vanishing viscosity limit of free-boundary problem of Navier-Stokes to obtain free-boundary Euler. To control singular behavior, we introduce Sobolev co-normal space. In particular, we will compare differences between surface tension and non-surface tension cases. In the end of this talk, a partial result about free-boundary MHD will be mentioned.




Joonhyun La
Title: On a kinetic model of polymeric fluids
Abstract: In this talk, we prove global well-posedness 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 multi-scale 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 Navier-Stokes equations up to boundary
Abstract: We are concerned with Caccioppolis inequalities of the non-stationary Stokes system and Navier- Stokes equations. It is known that the Caccioppolis inequalities of the Stokes system and the Navier-Stokes equations are true known in the interior case. We prove that the Caccioppolis inequalities of the Stokes system and the Navier-Stokes equations may, however, fail near boundary, when only local analysis is considered at the at flat boundary. This is a joint work with Dr. Tong-Keun Chang.




Bongsuk Kwon
Title: Small Debye length limit for the Euler-Poisson system 
Abstract: We discuss existence, time-asymptotic behavior, and quasi-neutral limit for the Euler-Poisson equations. Specifically, under the Bohm's criterion, we construct the global-in-time solution in the regime of the plasma sheath and investigate the properties of the solution including the time-asymptotic 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.).