• 대한수학회지
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• 제37권5호
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• pp.871-884
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• 2000

Polyharmonic operator with Dirichlet boundary condition is considered in a n-dimensional cone. The regularity properties of weak solutions are studied. In particular, it is proved the Holder contionuity of solutions near the vertex of the cone for dimensions n=2m+3,2m+4, where 2m is the order of the polyharmonic operator.

• 충청수학회지
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• 제27권1호
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• pp.17-27
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• 2014

In order to study the propagation of singularities for solutions to second order quasilinear strictly hyperbolic equations with boundary, we have to consider the regularity of solutions of first order nonlinear equations satisfied by a characteristic hyper-surface. In this paper, we study the regularity compositions of the form v(${\varphi}$(x), x) with v and ${\varphi}$ assumed to have limited Sobolev regularities and we use it to prove the regularity of solutions of the first order nonlinear equations.

• East Asian mathematical journal
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• 제38권1호
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• pp.129-141
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• 2022

We obtain interior regularity estimates in the weighted Orlicz spaces for viscosity solutions of fully nonlinear uniformly parabolic equations u_t - F(D² u, x, t) = f(x, t) in Q₁ under relaxed structure conditions on the nonlinear operator F.

• 충청수학회지
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• 제22권3호
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• pp.557-569
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• 2009

We present a new regularity criterion for suitable weak solutions of the steady-state Navier-Stokes equations near boundary in dimension five. We show that suitable weak solutions are regular up to the boundary if the scaled $L^{\frac{5}{2}}$-norm of the solution is small near the boundary. Our result is also valid in the interior.

• East Asian mathematical journal
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• 제30권5호
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• pp.631-637
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• 2014

In this paper, we construct some results on the existence and regularity for solutions of neutral functional differential equations with unbounded principal operators in Hilbert spaces. In order to establish the existence and regularity for solutions of the neutral system by using fractional power of operators and the local Lipschtiz continuity of nonlinear term without using many of the strong restrictions considering in the previous literature.

• 대한수학회지
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• 제53권5호
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• pp.1019-1036
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• 2016

In this paper, we study the existence of solutions and $L^2$-regularity for fractional order retarded neutral functional differential equations in Hilbert spaces. We no longer require the compactness of structural operators to prove the existence of continuous solutions of the non-linear differential system, but instead we investigate the relation between the regularity of solutions of fractional order retarded neutral functional differential systems with unbounded principal operators and that of its corresponding linear system excluded by the nonlinear term. Finally, we give a simple example to which our main result can be applied.

• 대한수학회지
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• 제37권6호
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• pp.957-975
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• 2000

We find a regularity criterion for the Ostwald-de Waele models like Serrin's condition to the Navier-Stokes equations. Moreover, we show short time existence and estimate the Hausdorff dimension of the set of singular times for the weak solutions.

• Journal of applied mathematics & informatics
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• 제32권3_4호
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• pp.547-554
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• 2014

In this paper, we consider the existence and uniqueness of global weak solution for a sixth-order classical surface-diffusion equation in one spatial dimension. Moreover, the regularity and blow-up of solutions are also studied.

• 대한수학회보
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• 제33권1호
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• pp.17-28
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• 1996

• 충청수학회지
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• 제28권1호
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• pp.151-159
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• 2015

This paper investigates temporal regularity of solutions for the incompressible Euler equations in a critical Besov space $B^{\frac{d}{p}+1}_{p,1}(\mathbb{R}^d)$ for $1{\leq}p{\leq}d$.