• 한국수학교육학회지시리즈B:순수및응용수학
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• 제8권1호
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• pp.15-23
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• 2001

This paper gives the sufficient conditions for the existence of positive solution of a quasilinear elliptic with homogeneous Dirichlet boundary conditon.

• 대한수학회지
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• 제57권6호
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• pp.1451-1470
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• 2020

We are concerned with the following elliptic equations: $$\{(-{\Delta})^s_pu={\lambda}f(x,u)\;{\text{in {\Omega}}},\\u=0\;{\text{on {\mathbb{R}}^N{\backslash}{\Omega}},$$ where λ are real parameters, (-∆)^s_p is the fractional p-Laplacian operator, 0 < s < 1 < p < + ∞, sp < N, and f : Ω × ℝ → ℝ satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L^∞(Ω) of any possible weak solution by applying the bootstrap argument.

• 대한수학회지
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• 제56권6호
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• pp.1529-1560
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• 2019

We are concerned with the following elliptic equations: $$(-{\Delta})^s_pu+V (x){\mid}u{\mid}^{p-2}u={\lambda}g(x,u){\text{ in }}{\mathbb{R}}^N$$, where $(-{\Delta})_p^s$ is the fractional p-Laplacian operator with 0 < s < 1 < p < $+{\infty}$, sp < N, the potential function $V:{\mathbb{R}}^N{\rightarrow}(0,{\infty})$ is a continuous potential function, and $g:{\mathbb{R}}^N{\times}{\mathbb{R}}{\rightarrow}{\mathbb{R}}$ satisfies a $Carath{\acute{e}}odory$ condition. We show the existence of at least one weak solution for the problem above without the Ambrosetti and Rabinowitz condition. Moreover, we give a positive interval of the parameter ${\lambda}$ for which the problem admits at least one nontrivial weak solution when the nonlinearity g has the subcritical growth condition.

• 대한수학회보
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• 제58권5호
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• pp.1109-1127
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• 2021

We study the long-term dynamics for a family of incompressible three-dimensional Leray-α-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-α model when varying two nonnegative parameters 𝜃₁ and 𝜃₂. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-α-like models into a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제26권2호
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• pp.76-107
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• 2022

In this paper, we study the temporal decay of global weak solutions to the inhomogeneous Hall-magnetohydrodynamic (Hall-MHD) equations. First, an approximation problem and its weak solutions are obtained via the Caffarelli-Kohn-Nirenberg retarded mollification technique. Then, we prove that the approximate solutions satisfy uniform decay estimates. Finally, using the weak convergence method, we construct weak solutions with optimal decay rates to the inhomogeneous Hall-MHD equations.

• 대한수학회보
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• 제61권3호
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• pp.637-669
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• 2024

The goal of this paper is to study decay properties of weak solutions to Cauchy problem of the non-stationary fractional Navier-Stokes equations. By using the Fourier splitting method, we give the time L²-decay rate of weak solutions, which reveals that L²-decay is generally determined by its linear generalized Stokes flow. In second part, we establish various decay results and the uniqueness of the two dimensional fractional Navier-Stokes flows. In the end of this article, as an appendix, the existence of global weak solutions is given by making use of Galerkin' method, weak and strong compact convergence theorems.

• East Asian mathematical journal
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• 제39권1호
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• pp.51-63
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• 2023

We establish the interior L^q regularity estimates for spatial gradient of weak solutions to nonlinear parabolic equations with the inhomogeneity which is given by the divergence and nondivergence data.

• 대한수학회지
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• 제54권3호
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• pp.945-966
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• 2017

The paper develops a rigorous mathematical framework for the behavior of arch and membrane like structures. Our main goal is to incorporate moving point loads. Both the weak and the strong damping cases are considered. First, we prove the existence and the uniqueness of the solutions. Then it is shown that the solution in the weak damping case is the limit of the strong damping solutions, as the strong damping vanishes. The theory is applied to a car moving on a bridge.

• 대한수학회보
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• 제50권2호
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• pp.375-388
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• 2013

In this paper, we introduce an iterative sequence for finding solution of a system of mixed equilibrium problems and the set of fixed points of a nonexpansive mapping in Hilbert spaces. Then, the weak and strong convergence theorems are proved under some parameters controlling conditions. Moreover, we apply our result to fixed point problems, system of equilibrium problems, general system of variational inequalities, mixed equilibrium problem, equilibrium problem and variational inequality.

• 대한수학회지
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• 제53권3호
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• pp.597-621
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• 2016

We present regularity conditions for suitable weak solutions of the Navier-Stokes equations with slip boundary data near the curved boundary. To be more precise, we prove that suitable weak solutions become regular in a neighborhood boundary points, provided the scaled mixed norm $L^{p,q}_{x,t}$ with 3/p + 2/q = 2, $1{\leq}q$ < ${\infty}$ is sufficiently small in the neighborhood.