Gichan Bae (Dalian University of Technology)
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Gichan Bae (Dalian University of Technology)
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Junsik Bae (Kyungpook National Univ.)
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Younghun Hong (Chung-Ang University)
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Junyeong Jang (Chung-Ang University)
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In-Jee Jeong (KIAS)
Title: Stability of vortex dipoles
Abstract: We consider the existence, uniqueness, and geometric properties of vortex dipoles, which are pairs of steadily-translating and counter-rotating vortices. First, we make precise the notion that the Lamb dipole, an explicit traveling wave solution described by Lamb back in 1895, is the dipole with "maximal mass" under enstrophy and impulse constraints. Then we discuss rigidity of dipoles near the Lamb dipole. Lastly, we discuss stability of multiple Lamb dipoles, allowing for dipoles with mixed signs. This is based on joint works with Ken Abe, Kyudong Choi, Guolin Qin, Yao Yao, and Weicheng Zhan.
Jinwook Jung (Hanyang Univeristy)
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Dong-Ha Kim (Seoul National University)
Title: A Lagrangian approach to uniqueness for active scalars equations in generalized mean oscillation spaces
Abstract: We prove uniqueness for a class of two-dimensional inviscid active scalar equations in generalized mean-oscillation spaces via a Lagrangian approach. Our unified framework covers the Euler equation, $\alpha$-SQG equation, and SQG equation in spaces which are slightly rougher than $\mathrm{BMO}$, $C^\alpha$, and $C^{0,1}$, respectively. In each case, the corresponding modulus is enlarged by a factor of $\log\log(1/|x-y|)$. It also yields analogous results for the IPM equation and the logarithmically perturbed Euler equation. This work is joint with Junha Kim, Hyunjin In, and Seungjae Lee.
Junha Kim (Ajou University)
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Junsung Kim (POSTECH)
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Joonhyun La (KIAS)
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Jinyeop Lee (Kyung Hee University)
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Sungbin Park (POSTECH)
Title: Exponential-Type Ill-Posedness for the BGK Model and Well-Posedness for the Boltzmann Equation in a Half-Space with Inflow Boundary Conditions
Abstract: The Boltzmann equation and the BGK model are widely used to describe the nonequilibrium statistical behavior of dilute gases. Although the two models share fundamental features, such as the conservation of macroscopic quantities, they exhibit different behavior when their solutions are considered in exponentially weighted spaces. Extending previous results from the whole-space setting, we study both models in a half-space with an inflow boundary condition and vacuum initial data. We establish ill-posedness for the BGK model and well-posedness for the Boltzmann equation in suitable exponentially weighted spaces. Unlike in the whole-space setting, the analysis of the half-space problems requires different exponential weights for the two models, reflecting their distinct responses to the boundary inflow. This is a joint work with Donghyun Lee, Sung-Jun Son, and Seok-Bae Yun.
Jinsol Seo (KIAS)
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Jinmyoung Seok (Seoul National University)
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