• 대한수학회논문집
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• 제37권3호
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• pp.735-748
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• 2022

The global well-posedness for the fourth-order modified Zakharov equations in 1-D, which is a system of PDE in two variables describing interactions between quantum Langmuir and quantum ionacoustic waves is studied. In this paper, it is proven that the system is globally well-posed in (u, n) ∈ L² × L² by making use of Bourgain restriction norm method and L² conservation law in u, and controlling the growth of n via appropriate estimates in the local theory. In particular, this improves on the well-posedness results for this system in [9] to lower regularity.

• 대한수학회보
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• 제56권6호
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• pp.1601-1615
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• 2019

We show the existence of ground state and orbital stability of standing waves of nonlinear $Schr{\ddot{o}}dinger$ equations with singular linear potential and essentially mass-subcritical power type nonlinearity. For this purpose we establish the existence of ground state in $H^1$. We do not assume symmetry or monotonicity. We also consider local and global well-posedness of Strichartz solutions of energy-subcritical equations. We improve the range of inhomogeneous coefficient in [5, 12] slightly in 3 dimensions.

• 대한수학회보
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• 제58권6호
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• pp.1521-1537
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• 2021

In this paper we consider the global well-posedness of compressible magnetohydrodynamic system in ℝ^d with d ≥ 2, in the framework of the critical Besov spaces. We can show that if the initial data, the shear viscosity and the magnetic diffusion coefficient are small comparing with the volume viscosity, then the compressible magnetohydrodynamic system has a unique global solution. Our result improves the previous one by Danchin and Mucha [10] who considered the compressible Navier-Stokes equations.

• 대한수학회보
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• 제57권4호
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• pp.891-907
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• 2020

In this paper, we consider the Cauchy problem to the tropical climate model. We establish the global regularity for the 2D tropical climate model with generalized nonlocal dissipation of the barotropic mode and obtain a multi-logarithmical vorticity blow-up criterion for the 2D tropical climate model without any dissipation of the barotropic mode.

• 대한기계학회논문집A
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• 제21권5호
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• pp.850-858
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• 1997

A new a posteriori error estimator is introduced and applied to variational inequalities occurring in problems of flow through porous media. In order to construct element-wise a posteriori error estimates the global error is localized by a special mixed formulation in which continuity conditions at interfaces are treated as constraints. This approach leads to error indicators which provide rigorous upper bounds of the element errors. A discussion of a compatibility condition for the well-posedness of the local error analysis problem is given. Two numerical examples are solved to check the compatibility of the local problems and convergence of the effectivity index both in a local and a global sense with respect to local refinements.

• Nuclear Engineering and Technology
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• 제27권1호
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• pp.45-52
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• 1995

수평 유로를 통과하는 이상류 내에서 기체상의 난류 응력이 이상류 시스템의 전반적 거동 특성에 미치는 영향을 연구하였다. 이를 위하여 이미 오랜 연구 노력으로 입증된 평균 이상류 거동 방정식을 채택하여 수평관에서 흔히 발생하는 성층류형 이상류에 대하여 적절히 모델하였다. 모델의 고유치를 계산하고 그의 특성 방정식으로부터 이상류 시스템의 well-posedness 및 다른 이상 유동 영역으로의 천이 경계를 새로이 결정하였다. 그 결과로 지금까지 무시해 왔던 기체상의 난류웅력이 이상류 시스템의 안정성에 커다란 영향을 주는 것이 발견되었다. 본 연구에서는 그 영향이 정량화 되었으며 그외의 효과도 정량화 하는 방법을 정립하였다. 본 연구는 원자로 시스템 해석 코드에 쓰이고 있는 열수력 모델 중 불확실성이 많은 모델인 수평관에서의 이상유동 영역 천이 조건식을 개선함으써 시스템 해석 결과의 신뢰도를 높이는 데 이용할 수 있다.

• 충청수학회지
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• 제37권1호
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• pp.27-40
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• 2024

In this paper, we consider the higher-order HartreeFock equations. The higher-order linear Schrödinger equation was introduced in [5] as the formal finite Taylor expansion of the pseudorelativistic linear Schrödinger equation. In [13], the authors established global-in-time Strichartz estimates for the linear higher-order equations which hold uniformly in the speed of light c ≥ 1 and as their applications they proved the convergence of higher-order Hartree-Fock equations to the corresponding pseudo-relativistic equation on arbitrary time interval as c goes to infinity when the Taylor expansion order is odd. To achieve this, they not only showed the existence of solutions in L² space but also proved that the solutions stay bounded uniformly in c. We address the remaining question on the convergence of higherorder Hartree-Fock equations when the Taylor expansion order is even. The distinguished feature from the odd case is that the group velocity of phase function would be vanishing when the size of frequency is comparable to c. Owing to this property, the kinetic energy of solutions is not coercive and only weaker Strichartz estimates compared to the odd case were obtained in [13]. Thus, we only manage to establish the existence of local solutions in H^s space for s > $\frac{1}{3}$ on a finite time interval [-T, T], however, the time interval does not depend on c and the solutions are bounded uniformly in c. In addition, we provide the convergence result of higher-order Hartree-Fock equations to the pseudo-relativistic equation with the same convergence rate as the odd case, which holds on [-T, T].