Mathematical aspects of fluid mechanics / edited by James C. Robinson, José L. Rodrigo and Witold Sadowski.

Includes bibliographical references.

Print version record.

Cover; LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES; Title; Copyright; Dedication; Contents; Contents; Preface; Preface; List of Contributors; List of Contributors; 1 Towards fluid equations by approximate deconvolution models; 1.1 Introduction; 1.2 The approximate deconvolution; 1.3 High accuracy deconvolution alpha-models; 1.4 Energy spectrum; 1.5 Limiting behaviour in terms of the deconvolution parameter; References; 2 On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph; 2.1 Introduction; 2.2 Orlicz spaces; 2.3 Selections.

2.4 Convergence tools2.5 Steady flows without convection; 2.6 Steady flows with convection; 2.7 Unsteady flows without convection; 2.8 Full problem; Acknowledgments; References; 3 A continuous model for turbulent energy cascade; 3.1 Motivation for the model; 3.1.1 Onsager and Kolmogorov; 3.1.2 Onsager's Conjecture and Besov spaces; 3.1.3 Littlewood-Paley framework for intermittency; 3.2 A continuous model for the energy flux; 3.3 Inviscid case; 3.4 Viscous case; References; 4 Remarks on complex fluid models; 4.1 Introduction; 4.2 Energetics; 4.3 Global existence issues; 4.4 Uniqueness issues.

7.2.3 Stabilization via a control supported on part of the boundary7.3 Construction of a stabilizing control for the Oseen equations; 7.3.1 Reduction to the linear case; 7.3.2 Description of the "correct" initial conditions; 7.3.3 Theorem on the stabilization of the Oseen equations; 7.4 Stabilization for the Navier-Stokes equations; 7.4.1 Definition of the stable invariant manifold; 7.4.2 Feedback operator and stabilization; 7.5 Feedback property for a control; 7.5.1 Definitions. The case of initial control; 7.5.2 The case of distributed control supported in a subdomain; 7.5.3 Real processes.

7.6 Description of numerical algorithms7.6.1 General definitions; 7.6.2 Stable invariant manifold for a fixed point; 7.6.3 Projection onto the stable invariant manifold; 7.6.4 The stable manifold corresponding to a trajectory; 7.6.5 Projection onto the stable manifold; 7.6.6 Calculations with control in the right-hand side; 7.7 Results of numerical calculations; 7.7.1 The physical model and its mathematical setting; 7.7.2 The structure of the phase portrait; 7.7.3 Stabilization by control of the initial condition; 7.7.4 Stabilization by control of the right-hand side.

The rigorous mathematical theory of the equations of fluid dynamics has been a focus of intense activity in recent years. This volume is the product of a workshop held at the University of Warwick to consolidate, survey and further advance the subject. The Navier-Stokes equations feature prominently: the reader will find new results concerning feedback stabilisation, stretching and folding, and decay in norm of solutions to these fundamental equations of fluid motion. Other topics covered include new models for turbulent energy cascade, existence and uniqueness results for complex fluids and certain interesting solutions of the SQG equation. The result is an accessible collection of survey articles and more traditional research papers that will serve both as a helpful overview for graduate students new to the area and as a useful resource for more established researchers.

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