New Research


2008/12/01

Study of the nonlinear principle and estimate of nonlinear phenomena

Self-organizations caused by the hierarchy and variational structure

Professor Takashi Suzuki, Associate Professor Masato Okado, Assistant Professor Atsushi Nobe

 In the Suzuki laboratory, several problems in natural science, engineering, medicine, and economics are studied from the viewpoint of mathematical analysis. Concretely, their formulation to equations based on mathematical penomena and physical principles, development of numerical schemes with numerical simulation, and theoretical study in nonlinear partial differential equations and nonlinear functional analysis are made. We also develop algorithms to solve non-appropriate problems in which the solutions are not determined uniquely. The main research objects are

1. self-interaction fluids such as nebulas, fixed stars, and plasma,
2. transportation phenomena of electrons and chemicals,
3. superconducting, condensation, and turbulent flow which are microscopically generated in extreme states,
4. macroscopically critical states such as phase transitions, phase separations, and hysteresis,
5. system of chemotaxis and tumor formation, etc.

 Through the formulation of these objects, we study stability of stationary solutions, explosion of solutions, and transitional order formations of nonlinear partial differential equations. In these problems, nonlinear partial differential equations of parabolic or elliptic type with non-local terms are arising in common through mathematical formulations such as the dual variational principle. This series of equations are newly introduced and are not included in the frame of the usual theory of nonlinear partial differential equations.
 Our main theme is to extract \"closed dissipative systems\" from biological objects in mathematical medicine and mathematical biology, from chemical objects in chemical kinetics and nonequilibrium thermodynamics, and from physical objects in gage theory and statistical physics; and to clarify the \"aspects of sealed self-organizations\". Aiming application to engineering and medicine, we also study the problems on MEG (magnetoencephalography) analysis and sound localization by using \"parallel optimization\" - a method to make a computer retrieve \"preferable solutions\" heuristically.


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