DSpace Community: KAIST Dept. of Mathematical SciencesKAIST Dept. of Mathematical Scienceshttp://hdl.handle.net/10203/5272023-12-05T03:22:27Z2023-12-05T03:22:27ZTHE MODIFIED SCATTERING FOR DIRAC EQUATIONS OF SCATTERING-CRITICAL NONLINEARITYCho, YonggeunKwon, SoonsikLee, KiyeonYang, Changhunhttp://hdl.handle.net/10203/3145332023-11-14T01:00:12Z2024-03-01T00:00:00ZTitle: THE MODIFIED SCATTERING FOR DIRAC EQUATIONS OF SCATTERING-CRITICAL NONLINEARITY
Authors: Cho, Yonggeun; Kwon, Soonsik; Lee, Kiyeon; Yang, Changhun
Abstract: In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type non -linearity with a long-range potential as |x|-1. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the space-time resonance argument of Germain-Masmoudi-Shatah ([14, 15, 16]), as well as the spinorial null-structure. On the way, we recognize a long-range interaction which is responsible for a logarithmic phase correction in the modified scattering statement. This result was obtained by Cloos in his dissertation [9], via a different technique (see Remark 1.2).2024-03-01T00:00:00ZSpectral analysis of the Neumann-Poincare operator on the crescent-shaped domain and touching disks and analysis of plasmon resonanceJung, YounghoonLim, Mikyounghttp://hdl.handle.net/10203/3114582023-08-14T03:00:16Z2023-12-01T00:00:00ZTitle: Spectral analysis of the Neumann-Poincare operator on the crescent-shaped domain and touching disks and analysis of plasmon resonance
Authors: Jung, Younghoon; Lim, Mikyoung
Abstract: We consider the Neumann-Poincare operator on a planar domain enclosed by two touching circular boundaries. This domain, which is a crescent-shaped domain or touching disks, has a cusp at the touching point of two circles. We analyze the operator via the Fourier transform on the boundary circles of the domain. In particular, we define a Hilbert space on which the operator is bounded, self-adjoint. We then obtain the complete spectral resolution of the Neumann- Poincare operator. On both the crescent-shaped domain and touching disks, the Neumann-Poincare operator has only absolutely continuous spectrum on the closed interval [-1/2, 1/2]. As an application, we analyze the plasmon resonance on the crescent-shaped domain.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).2023-12-01T00:00:00ZDetached Shock Past a Blunt BodyBae, MyoungjeanXiang, Weihttp://hdl.handle.net/10203/3152352023-11-27T07:00:37Z2023-12-01T00:00:00ZTitle: Detached Shock Past a Blunt Body
Authors: Bae, Myoungjean; Xiang, Wei
Abstract: In R-2, a symmetric blunt body W-b is fixed by smoothing out the tip of a symmetric wedge W(0 )with the half-wedge angle theta(w )is an element of (0, pi/2). We first show that if a horizontal supersonic flow of uniform state moves toward W(0 )with a Mach number M-infinity >1 being sufficiently large depending on theta(w), then the half-wedge angle theta(w) is less than the detachment angle so that there exist two shock solutions, a weak shock solution and a strong shock solution, with the shocks being straight and attached to the vertex of the wedge W-0. Such shock solutions are given by a shock polar analysis, and they satisfy entropy conditions. The main goal of this work is to construct a detached shock solution of the steady Euler system for inviscid compressible irrotational flow in R-2 \ W-b. Especially, we seek a shock solution with the far-field state given as the strong shock solution obtained from the shock polar analysis. Furthermore, we prove that the detached shock forms a convex curve around the blunt body W-b if the Mach number of the incoming supersonic flow is sufficiently large, and if the boundary of W(b )is convex.2023-12-01T00:00:00ZEssential dimension of semisimple groups of type BBaek, SanghoonKim, Yeongjonghttp://hdl.handle.net/10203/3114592023-08-14T03:00:24Z2023-11-01T00:00:00ZTitle: Essential dimension of semisimple groups of type B
Authors: Baek, Sanghoon; Kim, Yeongjong
Abstract: We determine the essential dimension of an arbitrary semisim-ple group of type B of the form G = (Spin(2n1 + 1) x & BULL; & BULL; & BULL; x Spin(2nm + 1))/& mu; over a field of characteristic 0, for all n1, ... , nm & GE; 7, and a central subgroup & mu; of Spin(2n1 + 1) x & BULL; & BULL; & BULL; x Spin(2nm + 1) not containing the center of Spin(2ni + 1) as a direct factor for every i = 1, ... , m.& COPY; 2023 Elsevier Inc. All rights reserved.2023-11-01T00:00:00Z