Methods of Geometry in the Theory of Partial Differential Equations: Principle of the Cancellation of Singularities (414 Pages)

Mathematical models are used to describe the essence of the real world, and their analysis induces new predictions filled with unexpected phenomena. In spite of a huge number of insights derived from a variety of scientific fields in these five hundred years of the theory of differential equations, and its extensive developments in these one hundred years, several principles that ensure these successes are discovered very recently. This monograph focuses on one of them: cancellation of singularities derived from interactions of multiple species, which is described by the language of geometry, in particular, that of global analysis. Five objects of inquiry, scattered across different disciplines, are selected in this monograph: evolution of geometric quantities, models of multi-species in biology, interface vanishing of d – δ systems, the fundamental equation of electro-magnetic theory, and free boundaries arising in engineering. The relaxation of internal tensions in these systems, however, is described commonly by differential forms, and the reader will be convinced of further applications of this principle to other areas. Contents Preface Evolution of Geometric Objects 1. Curves and Surfaces in R3 1.1 Surfaces 1.2 Curves 1.3 Curves on Surfaces 1.4 Curvatures 1.5 Differential Forms 1.6 Tangent and Cotangent Spaces 1.7 Frames and Covariant Derivatives 1.8 Connections 1.9 Fundamental Equation of Surfaces 1.10 Conformal Geometry 2. Static Recursive Hierarchy 2.1 Point Vortices 2.2 Mean Field Limit 2.3 Liouville Integral 2.4 Coverings of the Sphere 2.5 Boltzmann Poisson Equation 2.6 Method of Scaling 3. Kinetic Recursive Hierarchy 3.1 Smoluchowski Poisson Equation 3.2 Trudinger Moser Inequality 3.3 Quantized Blowup Mechanism 3.4 Blowup in Finite Time 3.5 Blowup in Infinite Time 3.6 Formation of Collapses 3.7 Improved ε-Regularity 3.8 Scaling Limit 3.9 Residual Vanishing 3.10 Bounded Domains 3.11 Exclusion of Boundary Blowup Points 3.12 Global-in-Time Solution 3.13 Initial Mass Quantization 3.14 Collapse Dynamics 3.15 Simplified System of Chemotaxis 4. Diffusion Geometry 4.1 2D Normalized Ricci Flow 4.2 Analytic Approach 4.3 Geometric Argument 4.4 Logarithmic Diffusion 4.5 Benilan’s Inequality 4.6 Concentration of Probability Measures 4.7 Pre-Compactness of the Orbit 4.8 Steady States 4.9 Critical Manifolds 4.10 Łojasiewicz Simon Inequality 4.11 Convergence to the Steady State 4.12 Non-Degeneracy 4.13 Exponential Rate of Convergence 4.14 Vanishing of the Center Manifold Differential Forms and Singularities 5. Systems of Multiple Components 5.1 Languages of Geometry 5.2 Analytic Mechanics 5.3 Symplectic and Poisson Manifolds 5.4 Hamilton Jacobi Theory 5.5 Symplectic and Poisson Structures of Biological Models 5.6 Reaction Diffusion Systems 5.7 Lotka Volterra Systems with Skew-Symmetry 5.8 The Case without Linear Term 5.9 The Case with Linear Term 5.10 Integrable Systems and Differential Forms 5.11 Integrable Systems of Order 1 5.12 Integrable Systems of Order 2 5.13 Integrable Systems of Order (N − 1) 6. Interface Vanishing 6.1 Geselowitz Equation 6.2 Maxwell Equation 6.3 Differential Forms in Higher Dimension 6.4 2-Forms on Euclidean Spaces 6.5 2-Forms on Minkowski Spaces 6.6 1-Forms 6.7 d − δ Systems Theory of Transformations 7. Non-Standard Elliptic Regularity 7.1 Lipschitz Domains 7.2 H1-Solution 7.3 Sectional Curvatures 7.4 Convex Domains 8. Liouville’s Formulae 8.1 Transformation of Variables 8.2 Variational Formulae of Jacobian 8.3 Volume Derivatives 8.4 Filtration 8.5 Flux of the Flow 8.6 Stefan Condition as Heat Transfer 8.7 Area Derivatives 9. Hadamard’s Variational Formula 9.1 An Abstract Theorem 9.2 Green’s Function 9.3 Lagrange Derivatives 9.4 Euler Derivatives 9.5 C1,1 Domains 9.6 C2,θ Domains 9.7 C2,1 Domains 9.8 Neumann Problems 9.9 First Variational Formula 9.10 Second Variational Formula 9.11 Second Fundamental Form on ∂Ω 10. Perturbation of Eigenvalues 10.1 Eigenvalue Problems 10.2 Unilateral Derivatives and Rearrangements 10.3 Characterization of Derivatives 10.4 Reduction to the Abstract Theory 10.5 Continuity of Eigenvalues and Eigenspaces 10.6 First Derivatives 10.7 Rearrangement of Eigenvalues 10.8 Second Derivatives Bibliography Index