Minkyu Kwak

Finite-dimensional description of convective Reaction-Diffusion equations

We are concerned with the asymptotic dynamics of a certain type of semilinear parabolic equation, namely,ut=uxx+(f(u))x+g(u)+h(x) on the interval [0,L]. Under the general condition we prove that this equation admits a dissipative dynamical system and it possesses the global attractor. But for largeL > 0, we do not know whether or not an inertial manifold exists. Here we introduce a nonlinear change of variables so that we transform the above equation to a reaction diffusion system which possesses exactly the same asymptotic dynamics. We then prove the existence of an inertial manifold for the transformed equation; thereby we find the ordinary differential equation which describes completely the long-time dynamics of the orginal equation.

A semilinear heat equation with singular initial data

We first prove existence and uniqueness of non-negative solutions of the equation in in the range 1 p N , when initial data u ( x , 0) = a | x | −2( p −1) , x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the form where g = g a satisfies After uniqueness is proved, the asymptotic behaviour of solutions of is studied. In particular, we show that The case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

GLOBAL EXISTENCE FOR 3D NAVIER-STOKES EQUATIONS IN A LONG PERIODIC DOMAIN

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a long periodic domain. We show by a simple argument that a strong solution exists globally in time when the initial velocity in and the forcing function in ([0; T);), T > 0, satisfy a certain condition. This condition common appears for the global existence in thin non-periodic domains. Larger and larger initial data and forcing functions satisfy this condition as the thickness of the domain tends to zero.

NEGATIVELY BOUNDED SOLUTIONS FOR A PARABOLIC PARTIAL DIFFERENTIAL EQUATION

In this note, we introduce a new proof of the unique- ness and existence of a negatively bounded solution for a parabolic partial difierential equation. The uniqueness in particular implies the flniteness of the Fourier spanning dimension of the global attrac- tor and the existence allows a construction of an inertial manifold. The long-time behaviour of solutions of a parabolic partial difierential equation is often-times investigated in the framework of an inflnite di- mensional dynamical system. The wide class of partial difierential equa- tions including reaction difiusion system, Kuramoto-Sivashinsky equa- tion, and 2D Navier-Stokes equations has the dissipative structure and possesses the global attractor. Henceforth the general understanding of the dynamics of the underlying partial difierential equations is reduced to the study of the geometric structure and dynamical property of the global attractor. See (3) and (4) for details.

REGULARITY OF SOLUTIONS OF 3D NAVIER-STOKES EQUATIONS IN A LIPSCHITZ DOMAIN FOR SMALL DATA

We consider the global existence of strong solutions of the 3D incompressible Navier-Stokes equations in a bounded Lipschitz do- main under Dirichlet boundary condition. We present by a very simple argument that a strong solution exists globally when the product of L2 norms of the initial velocity and the gradient of the initial velocity and Lp,2, p � 4 norm of the forcing function are small enough. Our condition is scale invariant and implies many typical known global existence results for small initial data including the sharp dependence of the bound on the volumn of the domain and viscosity. We also present a similar result in the whole domain with slightly stronger condition for the forcing.

THE PROCESS OF THE DEVELOPMENT OF HYPOXIA IN AN ABNORMAL BLOOD FLOW

Interrupted blood flow diminishes the concentration of oxygen in tissues. Hypoxia first appears in the region distal to the capillaries and grows throughout the entire t issue. However, the time-wise evolution of hypoxic area is diverse when some of capillaries are blocked in a multi-capillary domain with different oxygen squirt. The process of the development of hypoxia through time course is analyzed mathematically in the domain. Each source in steady state is controlled by a time sensitive function to simulate the occlusion.