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Diffusion in heterogeneous environment is Y.J. Kim's number one research interest right now.
Diffusion is a phenomenon by which a group of particles spreads in space due to random behavior of
individuals. Therefore, diffusion should be treated as a macroscopic view of microscopic scale
random movements. In the year 1855, Adolf Fick suggested his first law of diffusion and,
in the year 1905, Albert Einstein employed the diffusion theory to quantitatively model the Brownian
movement. Ever since Einstein's work, the theory of diffusion and Brownian movement turn out to be
the cornerstone of stochastic process, quantum mechanics, thermodynamics, the dynamical interpretation
of statistical physics and many more.
However, the diffusion law by Fick is for homogeneous case only. If the environment is heterogeneous, it fails. Furthermore, the diffusion becomes a lot different from the heat conduction equation. There many natural examples for heterogeneous diffusion in engineering, chemistry and, in particular, in biology. In ecology models species do not move in a homogeneous way. Depending on the availability of food, their movement strategy is changing. This kind of behavior cannot be explained homogeneous diffusion. Using Fick's law in heterogeneous environment is completely wrong. The project #1. Diffusion in Heterogeneous Environment is very active right now and YJK is inviting many young people to get involved. YJK's earlier research has been focused on convection and diffusion theory such as PME, PLE, Burger's equation, conservation laws and convection-diffusion-reaction equations. In particular, the longtime analysis and obtaining optimal convergence rate for large time is of main interests. These problems are studied within a frame work of parabolic equations. In fact, the convection equation or conservation laws should be understood as a zero viscosity limit of perturbed problem with small viscosity. In this way the convection and diffusion can be and should be understood in a single frame. The main contribution of YJK on the program #2. Second order parabolic equations will be providing a unified view for a large class of parabolic and hyperbolic problems. YJK is developing a theory for a conductivity reconstruction. The project #3. Inverse Problem for Anisotropic Electrical Conductivity is an inverse problem for anisotropic conductivity. If a conductivity body has a muscle fiber structure, then its conductivity is not isotropic anymore. So far most of inverse problem theory is for isotropic case. |
| #1. Diffusion in Heterogeneous Environment
The theory of diffusion has been developed as an analogy of heat conduction.
In fact Fick's law of diffusion has been introduced as an analogy of Fourier's law of heat
conduction from the beginning. It is true that there exists an exact analogy between them
for a homogeneous isotropic case. However, for heterogeneous cases, they are completely
different and that is why non-Fickian diffusion processes appear. In this project a new diffusion
law based on diffusion pressure is suggested. There are three directions that this project is aiming for.
#1. Developing mathematical method to handle new diffusion process. #2. Application to Non-Fickian
diffusion process. #3. Application to Math biology problems.
#1-1. Physics
related papers and preprints:
#1-2. Ecology
related papers and preprints:
#1-3. Biology related papers and preprints: #1-4. Mathematics related papers and preprints: |
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#2. Second order parabolic equations
There are three kinds of phenomena involved in second order parabolic equations. They are
the zero-th order reaction, the first order advection and the second order diffusion. Even if
mostly the word "diffusion" will be used in this note as most of others do,
it is more correct to consider it as "conduction". This parabolic problem connects hyperbolic
and elliptic problems. For example, the zero viscosity limit gives us a convection equation
and the long time limit gives us an elliptic one. There are two directions that this project
is aiming for. #1. Understand the common structure of various hyperbolic, parabolic and elliptic
problems. #2. Understand their differences.
#2-1. Long time asymptotics
#2-2. Short time dynamics in convection and diffusion related papers and preprints: #2-3. Competition between convection and diffusion
In many cases diffusion and convection exist together. In some aspects, they are different.
It needed to understand their differences, which is the goal of this part of research program.
#2-4. Common structures in convection and diffusion
In many cases diffusion and convection exist together. In some aspects, they are similar.
If one look at them from a wider view point, they have common features.
It needed to understand their common structure to really understand them,
which is the goal of this research program.
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#3. Inverse Problem for Anisotropic Electrical Conductivity.
If an electrical conductivity body has a muscle fiber structure like a human body, then an anisotropic model
is the correct one. In this project we develop a technique to recover such an anisotropic conductivity.
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| #4. Numerical Schemes for Convection.
We have been worked on the study of numerical schemes for the convection equations.
Godunov, WENO, central, upwind, positive, Lax-Wendroff schemes are examples.
These schemes are for convection equations and are widely used to solve various kinds of
problems. We found that, even if those schemes are used by lots of people,
there are many hidden properties. It is also interesting topic to work with higher order
diffusion terms that frequently appears in thin film equations. We are interested
in survey the properties of numerical schemes and develop better schemes for various
situations.
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