Navier-Stokes Equations and Turbulence
This book aims to bridge the gap between practicing mathematicians and the practitioners of turbulence theory. It presents the mathematical theory of turbulence to engineers and physicists, and the physical theory of turbulence to mathematicians. The book is the result of many years of research by the authors to analyze turbulence using Sobolev spaces and functional analysis.The mathematical technicalities are kept to a minimum within the book, enabling the language to be at a level understood by a borad audience. Each chapter is the accompanied by appendices giving full details of the mathematical proofs and subtleties. This unique presentation should ensure a volume of interest to mathematicians, engineers, and physicists.
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II Elements of the Mathematical Theory of the NavierStokes Equations
III Finite Dimensionality of Flows
IV Stationary Statistical Solutions of the NavierStokes Equations Time Averages and Attractors
V TimeDependent Statistical Solutions of the NavierStokes Equations and Fully Developed Turbulence
2-dimensional analyticity Appendix assume belongs bilinear operator Borel bounded in H cascade Chapter compact consider constant convergence corresponding deduce deﬁned deﬁnition denote derivative dissipation domain dµ(u dµt(u energy equation energy inequality ensemble average enstrophy exists ﬁeld ﬁnd finite ﬁrst fluid Foias Foias and Temam follows forcing term f Fourier function spaces Galerkin given global attractor Grashof Grashof number Hence homogeneous statistical solution inertial range initial condition inner product integrable kinetic energy Kolmogorov Lebesgue Lebesgue point Lemma linear mathematical Moreover Navier–Stokes equations no-slip boundary conditions norm obtain priori estimates probability distribution probability measure real-valued result Reynolds number satisfy Section self-similar sense Sobolev spaces space average space dimension space H space-periodic stationary statistical solutions Stokes operator strong solutions test functional time-average measure tion trilinear turbulent flows vector fields velocity field viscosity wavenumber weak solution weak topology zero