• Honam Mathematical Journal
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• v.42 no.4
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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.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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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.

• Journal of the Korean Mathematical Society
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• v.33 no.3
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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.

• Bulletin of the Korean Mathematical Society
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• v.42 no.4
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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.

• Honam Mathematical Journal
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• v.14 no.1
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• pp.67-84
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• 1992

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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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.

• Communications of the Korean Mathematical Society
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• v.14 no.2
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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.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.39 no.4
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• pp.447-451
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• 2015

The objective of this study is to predict the structural characteristics of a heat exchanger mounted on an aircraft engine using finite element analysis. The plastic fracture and life of the heat exchanger were estimated by a thermo-mechanical analysis. Tensile tests were conducted under high temperature conditions (700, 800, 900, 1000 K) using five specimens to obtain the mechanical properties of the Inconel 625 tubes. To assess the structural characteristics of the heat exchanger, the full and partial models were applied under the operating conditions given by the thermo-mechanical and inertial load. As a result, the case, tubesheet, flange, and mounting components have a reasonable safety margin to the allowable stress assuming a fatigue strength of Inconel 625 of 10000 cycles under 1000 K.