Mathematical Aspects of Fluid Mechanics

doi:10.1017/CBO9781139235792

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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Contents

Frontmatter
pp i-vi
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Contents
pp vii-viii
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Preface
pp ix-x
- By James C. Robinson, Mathematics Institute, University of Warwick, José L. Rodrigo, Mathematics Institute, University of Warwick, Witold Sadowski, University of Warsaw
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List of Contributors
pp xi-xvi
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1 - Towards fluid equations by approximate deconvolution models
pp 1-22
- By L.C. Berselli, Dipartimento di Matematica Applicata “U. Dini”, Università di Pisa
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2 - On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph
pp 23-51
- By M. Bulíčcek, Charles University, P. Gwiazda, University of Warsaw, J. Málek, Charles University, K.R. Rajagopal, Texas A&M University, A. Świerczewska-Gwiazda, University of Warsaw
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3 - A continuous model for turbulent energy cascade
pp 52-69
- By A. Cheskidov, University of Illinois, R. Shvydkoy, University of Chicago, S. Friedlander, University of Southern California
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4 - Remarks on complex fluid models
pp 70-87
- By P. Constantin, Princeton University
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5 - A naive parametrization for the vortex-sheet problem
pp 88-115
- By A. Castro, Ecole Normale Superieure, D. Córdoba, Instituto de Ciencias Matemáticas, F. Gancedo, Universidad de Sevilla
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6 - Sharp and almost-sharp fronts for the SQG equation
pp 116-129
- By C.L. Fefferman, Princeton University
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7 - Feedback stabilization for the Navier–Stokes equations: theory and calculations
pp 130-172
- By A.V. Fursikov, Moscow State University, A.A. Kornev, Moscow State University
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8 - Interacting vortex pairs in inviscid and viscous planar flows
pp 173-200
- By T. Gallay, Université de Grenoble I
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9 - Stretching and folding diagnostics in solutions three-dimensional Euler and Navier–Stokes equations
pp 201-220
- By J.D. Gibbon, Imperial College London, D.D. Holm, Imperial College London
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10 - Exploring symmetry plane conditions in numerical Euler solutions
pp 221-234
- By R.M. Kerr, University of Warwick, M.D. Bustamante, University College Dublin
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11 - On the decay of solutions of the Navier–Stokes system with potential forces
pp 235-250
- By I. Kukavica, University of Southern California
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12 - Leray–Hopf solutions to Navier–Stokes equations with weakly converging initial data
pp 251-258
- By G. Seregin, Mathematical Institute, Oxford University
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Mathematical Aspects of Fluid Mechanics

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Contents

Frontmatter
pp i-vi

Contents
pp vii-viii

Preface
pp ix-x

List of Contributors
pp xi-xvi

1 - Towards fluid equations by approximate deconvolution models
pp 1-22

2 - On flows of fluids described by an implicit constitutive equation characterized by a maximal monotone graph
pp 23-51

3 - A continuous model for turbulent energy cascade
pp 52-69

4 - Remarks on complex fluid models
pp 70-87

5 - A naive parametrization for the vortex-sheet problem
pp 88-115

6 - Sharp and almost-sharp fronts for the SQG equation
pp 116-129

7 - Feedback stabilization for the Navier–Stokes equations: theory and calculations
pp 130-172

8 - Interacting vortex pairs in inviscid and viscous planar flows
pp 173-200

9 - Stretching and folding diagnostics in solutions three-dimensional Euler and Navier–Stokes equations
pp 201-220

10 - Exploring symmetry plane conditions in numerical Euler solutions
pp 221-234

11 - On the decay of solutions of the Navier–Stokes system with potential forces
pp 235-250

12 - Leray–Hopf solutions to Navier–Stokes equations with weakly converging initial data
pp 251-258

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