MEAN FIELD THEORIES AND DUAL VARIATION

1. Front Matter

   Pages i-x

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2. Duality - Sealed Variation

   • Takashi Suzuki

   Pages 1-140

3. Scaling - Revealing Hierarchy

   • Takashi Suzuki

   Pages 141-284

4. Back Matter

   Pages 271-288

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A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of “duality” according to the PDE weak solutions and “hierarchy” for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski–Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.

• differential equation
• non-linear differentail equations
• nonlinear partial differential equation
• partial differential equation
• turbulence
• ordinary differential equations
• partial differential equations

• Division of Mathematical Science Department of Systems Innovation, Graduate School of Engineering Science Osaka University, Osaka, Japan

  Takashi Suzuki

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