Navier-Stokes Equations and Turbulence
This book aims to bridge the gap between practising 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 analyse turbulence using Sobolev spaces and functional analysis. In this way the authors have recovered parts of the conventional theory of turbulence, deriving rigorously from the Navier–Stokes equations what had been arrived at earlier by phenomenological arguments. The mathematical technicalities are kept to a minimum within the book, enabling the language to be at a level understood by a broad audience. Each chapter is 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
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2-dimensional analyticity Appendix approximation assume belongs bilinear operator Borel bounded in H cascade Chapter compact consider constant convergence corresponding deduce deﬁned deﬁnition denote derivative dissipation domain dÁ(u dÁt(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 turbulent flows vector fields velocity field viscosity wavenumber weak solution weak topology zero