Infinite Dimensional Dynamical Systems

This books sets forth a set of truly controversial and astonishing theories: First, it proposes that below the surface of the earth is a biosphere of greater mass and volume than the biosphere -- the total sum of living things -- on our planet's continents and in its oceans. Second, it proposes that the inhabitants of this subterranean biosphere are not plants or animals as we know them, but heat-loving bacteria that survive on a diet consisting solely of hydrocarbons -- that is, natural gas and petroleum. And third and perhaps most heretically, the book advances the stunning idea that most hydrocarbons on Earth are not the byproduct of biological debris ("fossil fuels"), but were a common constituent of the materials from which the Earth itself was formed some 4.5 billion years ago.
The implications are astounding. The theory proposes answers to often-asked questions: Is the deep hot biosphere where life originated, and do Mars and other seemingly barren planets contain deep biospheres? Even more provocatively, is it possible that there is an enormous store of hydrocarbons upwelling from deep within the Earth that can provide us with abundant supplies of gas and petroleum?
However far-fetched these ideas seem, they are supported by a growing body of evidence, and by the indisputable stature and seriousness Gold brings to any scientific debate. In this book we see a brilliant and boldly original thinker, increasingly a rarity in modern science, as he develops potentially revolutionary ideas about how our world works.
Thomas Gold is a member of the National Academy of Sciences, a Fellow of the Royal Society, and an Emeritus Professor of Physics at Cornell University. Regarded as one of the most creative and wide-ranging scientists of his generation, he has taught at Cambridge and Harvard, and most recently was for 20 years Director of the Cornell Center for Radiophysics and Space Research.

Contents

• Preface
• Introduction 1

Pt. I Functional Analysis 9

• 1 Banach and Hilbert Spaces 11
• 2 Ordinary Differential Equations 42
• 3 Linear Operators 62
• 4 Dual Spaces 89
• 5 Sobolev Spaces 109

Pt. II Existence and Uniqueness Theory 157

• 6 The Laplacian 159
• 7 Weak Solutions of Linear Parabolic Equations 188
• 8 Nonlinear Reaction-Diffusion Equations 213
• 9 The Navier-Stokes Equations: Existence and Uniqueness 234

Pt. III Finite-Dimensional Global Attractors 259

• 10 The Global Attractor: Existence and General Properties 261
• 11 The Global Attractor for Reaction-Diffusion Equations 285
• 12 The Global Attractor for the Navier-Stokes Equations 309
• 13 Finite-Dimensional Attractors: Theory and Examples 325

Pt. IV Finite-Dimensional Dynamics 357

• 14 The Squeezing Property: Determining Modes 359
• 15 The Strong Squeezing Property: Inertial Manifolds 385
• 16 A Direct Approach 406
• 17 The Kuramoto-Sivashinsky Equation 426

• App. A Sobolev Spaces of Periodic Functions 435
• App. B Bounding the Fractal Dimension Using the Decay of Volume Elements 439
• References 445
• Index