• 대한수학회지
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• 제54권3호
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• pp.909-943
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• 2017

This paper is devoted to studying abstract stochastic semilinear evolution equations with additive noise in Hilbert spaces. First, we prove the existence of unique local mild solutions and show their regularity. Second, we show the regular dependence of the solutions on initial data. Finally, some applications to stochastic partial differential equations are presented.

• 대한수학회지
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• 제49권6호
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• pp.1273-1299
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• 2012

We prove the global regularity of weak solutions to a conormal derivative boundary value problem for quasilinear elliptic equations in divergence form on Lipschitz domains under the controlled growth conditions on the low order terms. The leading coefficients are in the class of BMO functions with small mean oscillations.

• 대한수학회지
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• 제40권6호
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• pp.1031-1050
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• 2003

In this paper, we assume a density with integrability on the space $L^{\infty}$(0, T; $L^{q_{0}}$) for some $q_{0}$ and T > 0. Under the assumption on the density, we obtain a regularity result for the weak solutions to the compressible Navier-Stokes equations. That is, the supremum of the density is finite and the infimum of the density is positive in the domain $T^3$ ${\times}$ (0, T). Moreover, Moser type iteration scheme is developed for $L^{\infty}$ norm estimate for the velocity.

• Kyungpook Mathematical Journal
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• 제49권2호
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• pp.195-202
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• 2009

This paper deals with very weak solutions of the A-harmonic equation $divA(x,{\nabla}u)$ = 0 (*) with the operator $A:{\Omega}{\times}R^n{\rightarrow}R^n$ satisfies some coercivity and controllable growth conditions with Muckenhoupt weight. By using the Hodge decomposition with weight, a regularity property is proved: There exists an integrable exponent $r_1=r_1({\lambda},n,p)$ < p, such that every very weak solution $u{\in}W_{loc}^{1,r}({\Omega},{\omega})$ with $r_1$ < r < p belongs to $W_{loc}^{1,p}({\Omega},{\omega})$. That is, u is a weak solution to (*) in the usual sense.

• 대한수학회지
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• 제45권2호
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• pp.537-558
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• 2008

Consider a weak solution u of the Navier-Stokes equations in the class $L^2((0,T);X_1(\mathbb{R}^d)^d)$. We establish a new approach to treat the regularity problem for the Navier-Stokes equation in term of the multiplier space $X_1(\mathbb{R}^d)$.

• 충청수학회지
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• 제20권3호
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• pp.327-332
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• 2007

A variational inequality defined by the second-order cone is considered, and it is shown that a necessary and sufficient condition for a solution of the variational inequality holds under some regularity condition.

• East Asian mathematical journal
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• 제37권5호
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• pp.591-598
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• 2021

In this paper we study the Cauchy problem for the Chern-Simons gauged O(3) sigma model. We prove the local existence of solutions with low regularity initial data, observing null forms of the system and applying bilinear estimates for wave-Sobolev space H^{s, b}.

• 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.

• 한국지진공학회:학술대회논문집
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• 한국지진공학회 2003년도 춘계 학술발표회논문집
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• pp.85-92
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• 2003

Hysteretic behaviors of a seismic isolator are identified by using the regularized output error estimator (OEE) based on the secant stiffness model. A proper regularity condition of tangent stiffness for the current OEE is proposed considering the regularity condition of Duhem hysteretic operator. The proposed regularity condition is defined by 12-norm of the tangent stiffness with respect to time. The secant stiffness model for the OEE is obtained by approximating the tangent stiffness under the proposed regularity condition by the secant stiffness at each time step. A least square method is employed to minimize the difference between the calculated response and measured response for the OEE. The regularity condition of the secant stiffness is utilized to alleviate ill-posedness of the OEE and to yield numerically stable solutions through the regularization technique. An optimal regularization factor determined by geometric mean scheme (GMS) is used to yield appropriate regularization effects on the OEE.

• 대한수학회지
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• 제55권5호
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• pp.1269-1283
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• 2018

We are concerned with elliptic equations in ${\mathbb{R}}^N$, driven by a non-local integro-differential operator, which involves the fractional Laplacian. The main aim of this paper is to prove the existence of small solutions for our problem with negative energy in the sense that the sequence of solutions converges to 0 in the $L^{\infty}$-norm by employing the regularity type result on the $L^{\infty}$-boundedness of solutions and the modified functional method.