• 호남수학학술지
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• 제42권4호
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• pp.821-828
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• 2020

In this work, we prove an existence of an inertial manifold for a system of differential equations with a non-self adjoint leading part. The result follows from the existence and uniqueness of negatively bounded solutions. In fact, we show that a sharp spectral condition is sufficient for the proof.

• 대한수학회지
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• 제58권5호
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• pp.1261-1277
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• 2021

An inertial manifold is often constructed as a graph of a function from low Fourier modes to high ones and one used to consider backward bounded (in time) solutions for that purpose. We here show that the proof of the uniqueness of such solutions is crucial in the existence theory of inertial manifolds. Avoiding contraction principle, we mainly apply the Arzela-Ascoli theorem and Laplace transform to prove their existence and uniqueness respectively. A non-self adjoint example is included, which is related to a differential system arising after Kwak transform for Navier-Stokes equations.

• 대한수학회지
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• 제33권3호
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• pp.693-707
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• 1996

Consider a specific class of scalar-valued reaction diffusion equations of the form $$ (1.1) u_t = \nu\Delta u + f(u), u \in R $$ where $\nu$ < 0 is viscosity parameter and $f : R \to R$ is sufficiently smooth.

• 대한수학회보
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• 제42권4호
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• pp.829-836
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• 2005

In this note, we introduce a new proof of the unique-ness and existence of a negatively bounded solution for a parabolic partial differential equation. The uniqueness in particular implies the finiteness of the Fourier spanning dimension of the global attractor and the existence allows a construction of an inertial manifold.

• 호남수학학술지
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• 제14권1호
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• pp.67-84
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• 1992

• 대한수학회보
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• 제58권3호
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• pp.669-688
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• 2021

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

• 대한수학회논문집
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• 제14권2호
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• pp.337-352
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• 1999

In this paper we find a geometric condition for the weaker principle of spatial averaging (PSA) for a class of polyhedral domains. Let \ulcorner be a polyhedron in R\ulcorner, n$\leq$3. If all dihedral angles of \ulcorner are submultiples of $\pi$, then there exists a parallelopiped \ulcorner generated by n linearily independent vectors {\ulcorner}\ulcorner in R\ulcorner containing \ulcorner so that solutions of $\Delta$u+λu=0 in \ulcorner with either the boundary condition u=0 or ∂u/∂n=0 are expressed by linear combinations of those of $\Delta$u+λn=0 in \ulcorner with periodic boundary condition. Moreover, if {\ulcorner}\ulcorner satisfies rational condition, we guarantee the weaker PSA for the domain \ulcorner.

• 대한기계학회논문집A
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• 제39권4호
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• pp.447-451
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• 2015

본 연구의 목적은 유한요소 해석을 이용하여 항공기 엔진에 장착된 미세 튜브형 열교환기의 작동하중에 대한 특성을 파악하고 구조안전성을 평가하는 것이다. 작동 하중은 열-기계하중이 고려되었다. 항공기 엔진 부품의 특수성으로 인하여 구조적 강성확보뿐만 아니라 고효율, 저중량, 최소체적을 만족하는 형상 설계가 요구 된다. 브레이징한 미세튜브의 기계적물성치 확보를 위해 고온 인장 실험을 수행하였다. Ansys 12.1의 자체 모델러를 이용하여 형상을 모델링하고 유한요소해석을 수행하였다. 열-구조 연성해석을 통하여 튜브를 제외한 모듈형(modular type) 열교환기에 대한 설계방법을 제시하고 구조적 건전성을 평가하였다.