• 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.

• 대한수학회보
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• 제45권1호
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• pp.169-190
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• 2008

Second-order parabolic equations with variable coefficients are considered on $\mathbb{R}^d$ and $C^1$ domains. Existence and uniqueness results are given in $L_q(L_p)$-spaces, where it is allowed for the powers of summability with respect to space and time variables to be different.

• 대한수학회지
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• 제43권4호
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• pp.733-763
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• 2006

We study quasi-stationary Stokes flow in a dihedral domain arising from a study of a free boundary problem of viscous fluid in a container. We construct an exact solution of quasi-stationary Stokes equations and derive its estimates with norm in a weighted Sobolev spaces.

• 대한수학회지
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• 제37권6호
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• pp.1071-1084
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• 2000

We show that a weak solution of the Cauchy problem for he nonlinear Schrodinger equation, {i∂(sub)t u + ∂$^2$(sub)x u = f(u,u), t∈(-T,T), x∈R, u(0,x) = ø(x).} in the negative solbolev space H(sup)s has a smoothing effect up to real analyticity if the initial data only have a single point singularity such as the Dirac delta measure. It is shown that for H(sup)s (R)(s>-3/4) data satisfying the condition (※Equations, See Full-text) the solution is analytic in both space and time variable. The argument is based on the recent progress on the well-posedness result by Bourgain [2] and Kenig-Ponce-Vega [18] and previous work by Kato-Ogawa [12]. We give an improved new argument in the regularity argument.

• 대한수학회지
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• 제38권6호
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• pp.1205-1234
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• 2001

We consider the time local well-posedness of the Benjamin equation. Like the result due to Keing-Ponce-Vega [10], [12], we show that the initial value problem is time locally well posed in the Sobolev space H$^{s}$ (R) for s>-3/4.

• 대한수학회지
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• 제33권4호
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• pp.1039-1046
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• 1996

In the theory of partial differential equations with given initial values and boundary values one usually investigates to examine the well-posedness, that is, the unique existence of the solution as well as its continuous dependence on the data. This theory is strong enough for us to determine the situation anywhere and anytime provided that the initial data are actually given. However, in many cases the data are not completely known for us. Then in those situations arise the new problem to determine the unknown initial data by taking other conditions for the solutions.

• 대한수학회지
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• 제55권1호
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• pp.211-224
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• 2018

We consider a certain three dimensional Euler flow with infinite energy, which is sometimes called the columnar or two and half dimensional flow. We prove the global smoothness of such flow in ${\mathbb{R}}^3$ when the initial data is in some Sobolev or Besov spaces and ${\partial}_3u_3$ is nonnegative.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 1989년도 한국자동제어학술회의논문집; Seoul, Korea; 27-28 Oct. 1989
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• pp.947-951
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• 1989

A Class of nonlinear distributed parameter control problems is first stated in a partial differential equation form in multi-index notion and then converted into an integral equation form. Necessary conditions for optimality in the form of maximum principle are then derived in Sobolev space W$^{l}$, p/(1 leq. p .leq. .inf.)..

• East Asian mathematical journal
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• 제31권5호
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• pp.685-693
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• 2015

In this paper, we introduce fully discrete mixed discontinuous Galerkin approximations for parabolic problems. And we analyze the error estimates in $l^{\infty}(L^2)$ norm for the primary variable and the error estimates in the energy norm for the primary variable and the flux variable.

• 충청수학회지
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• 제21권4호
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• pp.471-481
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• 2008

We develop a new function space $L_{\alpha}(X)$ that generalizes the classical Lebesgue space $L^p(X)$. The generalization is focused on a better explanation of the flux terms arising from many dynamics.