Jaiok Roh

THE GLOBAL ATTRACTOR OF THE 2D G-NAVIER-STOKES EQUATIONS ON SOME UNBOUNDED DOMAINS

In this paper, we study the two dimensional g-Navier- Stokes equations on some unbounded domain › ‰ R 2 : We prove the existence of the global attractor for the two dimensional g-Navier- Stokes equations under suitable conditions. Also, we estimate the dimension of the global attractor. For this purpose, we exploit the concept of asymptotic compactness used by Rosa for the usual Navier-Stokes equations.

Hyers-Ulam stability of the time independent Schrödinger equations

Abstract In Jung and Roh (2017), we investigated some properties of approximate solutions of the second-order inhomogeneous linear differential equations. In this paper as an application of above paper we will study the stability of the time independent Schrodinger equation with a potential box of finite walls.

The linear differential equations with complex constant coefficients and Schrödinger equations

Abstract We investigate some properties of approximate solutions for the second-order inhomogeneous linear differential equations, y ′ ′ ( x ) + α y ′ ( x ) + β y ( x ) = r ( x ) , with complex constant coefficients. And, as an application of our results, we will see the time independent Schrodinger equations. This paper was motivated by the paper, Li and Shen (2010).

Functional Inequalities Associated with Additive Mappings

The functional inequality is investigated, where is a group divisible by and are mappings, and is a Banach space. The main result of the paper states that the assumptions above together with (1) and (2) , or , imply that is additive. In addition, some stability theorems are proved.

- Stability of the Incompressible Flows with Nonzero Far-Field Velocity

We consider the stability of stationary solutions 𝐰 for the exterior Navier-Stokes flows with a nonzero constant velocity 𝐮∞ at infinity. For 𝐮∞=0 with nonzero stationary solution 𝐰, Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability in 𝐿𝑝 spaces for 1 1 and obtain 𝐿𝑟-𝐿𝑝 stability as Kozono and Ogawa and Borchers and Miyakawa obtained for 𝐮∞=0.

Asymptotic aspect of derivations in Banach algebras

We prove that every approximate linear left derivation on a semisimple Banach algebra is continuous. Also, we consider linear derivations on Banach algebras and we first study the conditions for a linear derivation on a Banach algebra. Then we examine the functional inequalities related to a linear derivation and their stability. We finally take central linear derivations with radical ranges on semiprime Banach algebras and a continuous linear generalized left derivation on a semisimple Banach algebra.

Approximation by First-Order Linear Differential Equations with an Initial Condition

We will consider a continuously differentiable function satisfying the inequality for all and for some and some . Then we will approximate by a solution of the linear equation with .

Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity

Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in space and by Shibata (1999) and Enomoto-Shibata (2005) in spaces for . However, their results did not include enough information to find the spatial decay. So, Bae-Roh (2010) improved Enomoto-Shibatas results in some sense and estimated the spatial decay even though their results are limited. In this paper, we will prove temporal decay with a weighted function by using decay estimates obtained by Roh (2011). Bae-Roh (2010) proved the temporal rate becomes slower by if a weighted function is for . In this paper, we prove that the temporal decay becomes slower by where if a weighted function is . For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted -norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).

Approximation Property of the Stationary Stokes Equations with the Periodic Boundary Condition

In this paper, we will consider the stationary Stokes equations with the periodic boundary condition and we will study approximation property of the solutions by using the properties of the Fourier series. Finally, we will discuss that our estimation for approximate solutions is optimal.

The properties of the solutions of the incompressible flows on an exterior domain

Abstract In this paper, we want to see the properties of the smooth solutions u of the incompressible flows on an exterior domain Ω of R 2 . Specially, when the vorticity ω = ∇ × u has a bounded support, with suitable conditions we will show that there exists a constant C ( p , q ) such that ∥ u ∥ L p ( Ω ) ≤ C ∥ u ∥ L q ( Ω ) for 1 p ≤ q ≤ ∞ .