• Title/Summary/Keyword: Global well-posedness

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AN IMPROVED GLOBAL WELL-POSEDNESS RESULT FOR THE MODIFIED ZAKHAROV EQUATIONS IN 1-D

  • Soenjaya, Agus L.
    • Communications of the Korean Mathematical Society
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    • v.37 no.3
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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.

ON THE ORBITAL STABILITY OF INHOMOGENEOUS NONLINEAR SCHRÖDINGER EQUATIONS WITH SINGULAR POTENTIAL

  • Cho, Yonggeun;Lee, Misung
    • Bulletin of the Korean Mathematical Society
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    • v.56 no.6
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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.

GLOBAL LARGE SOLUTIONS FOR THE COMPRESSIBLE MAGNETOHYDRODYNAMIC SYSTEM

  • Li, Jinlu;Yu, Yanghai;Zhu, Weipeng
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.6
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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 ℝd 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.

ON WELL-POSEDNESS AND BLOW-UP CRITERION FOR THE 2D TROPICAL CLIMATE MODEL

  • Zhou, Mulan
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.4
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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.

Finite Element Analysis and Local a Posteriori Error Estimates for Problems of Flow through Porous Media (다공매체를 통과하는 유동문제의 유한요소해석과 부분해석후 오차계산)

  • Lee, Choon-Yeol
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.21 no.5
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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.

Influence of Vapor Phase Turbulent Stress to the Onset of Slugging in a Horizontal Pipe (기체상의 난류 응력이 수평 유동관 내에서의 Slugging에 미치는 영향에 관한 연구)

  • Park, Jee-Won
    • Nuclear Engineering and Technology
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    • v.27 no.1
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    • pp.45-52
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    • 1995
  • In influence of the vapor phase turbulent stress (i.e., the too-phase Reynolds stress) to the characteristics of two-phase system in a horizontal pipe has been theoretically investigated. The average two-fluid model has been constituted with closure relations for stratified flow in a horizontal pipe. A vapor phase turbulent stress model for the regular interface geometry has been included. It is found that the second order waves propagate in opposite direction with almost the same speed in the moving frame of reference of the liquid phase velocity. Using the well-posedness limit of the two-phase system, the dispersed-stratified How regime boundary has been modeled. Two-phase Froude number has been found to be a convenient parameter in quantifying the onset of slugging as a function of the global void fraction. The influence of the taper phase turbulent stress was found to stabilize the flow stratification.

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LOW REGULARITY SOLUTIONS TO HIGHER-ORDER HARTREE-FOCK EQUATIONS WITH UNIFORM BOUNDS

  • Changhun Yang
    • Journal of the Chungcheong Mathematical Society
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    • v.37 no.1
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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 Hs 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].