Foundations of fluid dynamics

Foundations of Fluid Mechanics......Page 1
Preface......Page 3
Indice......Page 7
1.1 Continua.......Page 9
(I) Mass conservation.......Page 10
(III) Angular momentum conservation.......Page 11
(IV) Energy conservation.......Page 13
Problems.......Page 16
1.2 Equations of motion of a fluid in general. Ideal and incom-pressible cases. Incompressible Euler, Navier–Stokes and Navier–Stokes-Fourier equations.......Page 23
(A) Incompressible non viscous fluid (Euler equations).......Page 24
(B) Incompressible, non heat conducting, viscous fluid (Navier–Stokes equa-tions).......Page 25
(C) Incompressible thermoconducting viscous fluid......Page 26
(E) The case of incompressible Euler equations......Page 27
(G) The case in which heating in adiabatic compressions is not negligible.......Page 28
Note to §1.2: dimensional arguments.......Page 29
Problems: Stokes formula.......Page 30
(1) Incompressible Euler equation.......Page 33
1 Theorem (incompressible Euler limit)......Page 37
2 Theorem (incompressibility; the NS case)......Page 38
(A) Isoentropic case......Page 40
(B) Non isoentropic case.......Page 41
(C) Stability of equilibria......Page 42
(2) Hydrostatics in presence of thermoconduction.......Page 43
(3) Current lines and the Bernoulli theorem.......Page 46
Problems......Page 48
(A) General considerations on convection.......Page 51
(B) The physical assumptions of the Rayleigh’s convection model.......Page 52
(C) The Rayleigh model.......Page 55
(D) Rescalings: a systematic analysis.......Page 60
Problem......Page 61
(A) Incompressible fields in the whole space as rotations of a vector potential.......Page 62
(B) An incompressible field in a finite convex volume as a rotation: somesufficient conditions.......Page 63
(C) Ambiguities for vector potentials of incompressible fields.......Page 65
(E) The space Xrot() and its complement in L2().......Page 66
Problems and complements.......Page 69
(A) Thomson theorem:......Page 74
(B) Irrotational isoentropic flows:......Page 75
(C) Eulerian vorticity equation in general and in bidimensional flows:......Page 76
(D): Hamiltonian form of the incompressible Euler equations and Clebschpotentials. Normal velocity fields.......Page 77
Proposition......Page 79
Definition (normal solenoidal field)......Page 80
(E): Lagrangian form of the incompressible Euler equations, Hamiltonianform on the group of diffeomorphisms.......Page 81
(F): Hamiltonian form of the Eulerian potential flows; example in 2–dimensions.......Page 83
(G) Small waves.......Page 86
Problems: Sound and surface waves. Radiated energy.......Page 87
(A) Euler equation......Page 91
(C) Navier–Stokes equations and algorithmic difficulties.......Page 93
(D) Heat equation.......Page 95
(E) An empirical algorithm for NS:......Page 97
(F) Precision of the same algorithms applied to the heat equation, (Q1).......Page 99
(G) Analysis of the precision of the algorithms in the heat equation case,(Q2).......Page 100
2 Proposition: (anomaly of approximations convergence for the heat equation)......Page 101
(H) Comments:......Page 102
Problems: Well posedness of the heat equation and other remarks.......Page 104
(A) Periodic boundary conditions: spectral algorithm and “reduction” to anordinary differential equation.......Page 107
Theorem (spectral theory of the Laplace operator for divergenceless fields)......Page 110
(C) The Stokes problem.......Page 112
(E) Gyroscopic analogy in d = 2.......Page 113
(F) Gyroscopic analogy in d = 3.......Page 115
Problems: interior and boundary regularity of solutions of ellipticequations and for Stokes equation.......Page 118
2.3 Vorticity algorithms for incompressible Euler and Navier–Stokes fluids. The d = 2 case.......Page 126
Problems: Few vortices Hamiltonian motions. Periodic Green function.......Page 132
(A) Regular filaments. Divergences and infinities.......Page 134
(B) Thin filament. Smoke ring.......Page 137
Theorem (Integrability of the motion by curvature)......Page 139
(C) Irregular filaments: Brownian filaments.......Page 140
(D) Irregular filaments: quasi periodic filaments.......Page 144
Problems.......Page 146
Proposition (“perturbative local” solution of NS)......Page 149
Problems. Classical local theory for the Euler and Navier–Stokes equationswith periodic conditions.......Page 157
3.2 Weak global existence theorems for NS. Autoregularization,existence, regularity and uniqueness for d = 2......Page 164
1 Definition: (weak solutions of NS and Euler equations)......Page 166
I. Proposition (a priori bounds on solutions of regularized NS equations)......Page 168
II. Corollary (global existence of weak solutions for the NS equations)......Page 169
2 Definition (C–weak solutions)......Page 170
III. Proposition (autoregularization)......Page 171
IV. Proposition: (existence and smoothness for NS in dimension 2)......Page 173
V. Proposition (finite vorticity implies smoothness in dimension d = 2, 3)......Page 175
VI. Proposition (uniqueness of smooth solutions of NS)......Page 176
VII. Proposition (constructive approximations errors estimate for NS solutionsin 2 dimensions)......Page 178
Proposition VIII (an analytic regularity result for NS in d = 2, (Mattingly,Sinai))......Page 179
Problems......Page 181
(A) Leray’s regularization.:......Page 182
(B) Properties of the regularized equation and new weak solutions.......Page 184
(C) The local bounds of Leray. Uniformity in the regularization parameter.......Page 185
I. Proposition (regularization independent a priori bounds)......Page 186
II. Proposition (local existence, regulatity and uniqueness, (Leray))......Page 190
2. Definition (L-weak solutions)......Page 192
IV. Proposition (velocity is unbounded near singularities)......Page 193
VI. Proposition(stronger regularization induced by heat kernels)......Page 194
Proposition VII (vorticity orientation is uncertain at singularities)......Page 195
(H) Large containers.......Page 196
Problems: Further results in Leray’s theory......Page 197
1. Definition (Hausdorff α–measure)......Page 202
(B) Hausdorff dimension of singular times in the Navier–Stokes solutions(d = 3).......Page 203
(C) Hausdorff dimension in space–time of the solutions of NS, (d = 3).......Page 205
Problems.......Page 209
(A) Energy balance for weak solutions.......Page 210
(B) General Sobolev’s inequalities and further a priori bounds.......Page 212
1. Definition (pseudo NS velocity field)......Page 213
2 Definition: (dimensionless “operators” for NS)......Page 214
II Theorem (upper bound on the dimension of the sporadic set of singulartimes for NS, (Scheffer))......Page 216
III Theorem: (upper bound on the Hausdorff dimension of the sporadicsingular points in space-time (“CKN theorem”))......Page 217
(E) Proof that the renormalization map contracts.......Page 218
Problems. The CKN theory.......Page 220
4.1 Fluids theory in absence of existence and uniqueness theo-rems for the basic fluidodynamics equations. Truncated NS equa-tions. The Rayleigh’s and Lorenz’ models.......Page 231
(A) The 2–dimensional Saltzman’s equations.......Page 234
(B) A priori estimates.......Page 235
(C) The Lorenz’ model.......Page 236
(D) Truncated NS Models.......Page 238
Problems.......Page 241
4.2: Onset of chaos. Elements of bifurcation theory.......Page 243
(B) Loss of stability of the laminar motion.......Page 249
(C) The notion of genericity.......Page 251
Definition (stability and genericity)......Page 252
Theorem (genericity of hyperbolicity and transversality of periodic orbitsin C∞ maps)......Page 255
(D) Generic routes to the loss of stability of a laminar motion. Sponta-neously broken symmetry.......Page 256
Problems. Hopf bifurcation.......Page 259
(E) Stability loss of a periodic motion. Hopf bifurcation.......Page 261
1 Definition (Poincar´e map)......Page 262
(F) Loss of stability of a periodic motion. Period doubling bifurcation.......Page 266
(H1) Intermittency scenario.......Page 267
(H2) Period doubling scenario.......Page 270
(H3) The Ruelle–Takens scenario.......Page 271
(I) Conclusions......Page 272
Problems.......Page 275
4.4: Dynamical tables.......Page 277
Definition (principal series)......Page 278
(A) Bifurcations and their graphical representation.......Page 279
(B) An example: the dynamical table for the model NS5.......Page 281
(D) Remarks.......Page 284
(E) Another example: the dynamic table of the NS7 model.......Page 286
(F) Table of the 2–dimensional Navier–Stokes equation at small Reynoldsnumber. Navier–Stokes)......Page 287
Problems.......Page 288
1 Definition (smooth flows, continuous dynamical systems)......Page 289
(A) Diophantine quasi periodic spectra.......Page 290
Proposition I: (nature of quasi periodic spectra)......Page 292
3 Definition (motions with continuous spectrum):......Page 293
(C) An example: non Euclidean geometry.......Page 294
(D) A further example: the billiard.......Page 296
Problems: Ergodic theory of motions on surfaces of constant nonpositive curvature.......Page 297
5.2 Timed observations. Random data.......Page 307
1 Definition (power spectrum for discrete time evolutions)......Page 310
2 Definition: (random choice of data)......Page 313
3 Definition (approximate random number generator)......Page 314
4 Definition (non absolutely continuous random data)......Page 315
5 Definition (ideal random number generator)......Page 316
Problems.......Page 317
5.3 Dynamical systems types. Statistics on attracting sets.......Page 319
2 Definition: (topological and differentiable dynamical systems):......Page 321
3 Definition (statistics of a random motion):......Page 322
4 Definition (metric dynamical systems, discrete and continuous):......Page 324
5 Definition: (ergodicity)......Page 325
7 Definition (continuous spectrum for a metric dynamical system)......Page 326
Problems.......Page 328
5.4 Dynamical bases and Lyapunov exponents.......Page 330
2 Definition (hyperbolic, Anosov and axiom A attracting sets)......Page 331
3 Definition (Axiom A and Anosov systems)......Page 332
4 Definition (axiom B systems)......Page 334
5 Definition: (regular singularities, regular metric dynamical systems)......Page 335
I Theorem (existence of Lyapunov exponents)......Page 337
Philosophical problems.......Page 342
(A) “Physical” (i.e. SRB) probability distributions.......Page 349
I Theorem (existence of stable and unstable manifolds and their absolutecontinuity)......Page 352
(B) Structure of axiom A attractors. Heuristic considerations.......Page 353
1 Definition (absolute continuity along the unstable manifold)......Page 356
II Theorem (periodic orbit representation of SRB distributions)......Page 359
2 Definition (attractors and information dimension)......Page 360
I’ Theorem (existence of stability manifolds and Lyapunov exponents forgeneral dynamical systems)......Page 362
Problems......Page 363
(A) Complete observations and formal symbolic dynamics.......Page 365
1 Definition (complete observation, generating pavement)......Page 366
2 Definition (frequencies of strings in a sequence of symbols)......Page 367
3 Definition (symbolic dynamics)......Page 369
4 Definition (complexity of a sequence)......Page 370
(C) Entropy of dynamical a system and the Kolmogorov–Sinai theory.......Page 373
II Theorem (average entropy)......Page 374
Problems.......Page 375
(A) Expansive maps on [0, 1]. Infinity of the number of invariant distributions.)......Page 377
(B) Application to the Lorenz model.......Page 381
(C) Hyperbolic maps and markovian pavements.......Page 382
2 Definition (compatibility matrix of symbolic dynamics)......Page 385
(D) Arnold cat map as paradigm for the properties of Markov pavements.......Page 386
(E) More general hyperbolic maps and their Markov pavements.......Page 389
I Theorem (variational principle for SRB distribution)......Page 390
(F) Representation of the SRB distribution via markovian pavements.......Page 391
II Theorem (SRB and volume measures)......Page 392
Problems.......Page 393
6.1 Functional integral representation of stationary distributions.......Page 399
Problems.......Page 407
(A): Energy dissipation: inertial and viscous scales.......Page 408
(B): Digression on the physical meaning of ”v → 0”.......Page 409
(C): The K41 tridimensional theory.......Page 410
(D): Bidimensional theory.......Page 417
(E): Remarks on the K41 theory.......Page 419
(F): The dissipative Euler equation.......Page 421
(G) The Ruelle–Lieb bounds and K41 theory......Page 423
Problems: Dissipation and attractor dimension as v → 0. Kolmogorov’sskew correlations.......Page 424
6.3 The shell model. Multifractal statistics.......Page 427
Problems.......Page 436
(A): Reversible equations for dissipative fluids.......Page 437
(B) Microscopic reversibility and macroscopic irreversibility.......Page 442
Definition: (reminder of the notion of attractor)......Page 445
Problems......Page 447
(A) The SRB distribution, and other invariant distributions.......Page 448
(B) Attractors and reversibility. Unbreakability of the time reversal symmetry.......Page 450
(C) An example.......Page 451
(D) The axiom C.......Page 453
Definition (axiom C)......Page 454
Theorem (axiom C and time reversal stability)......Page 456
(E) The chaotic hypothesis.......Page 457
(A) The fluctuation theorem.......Page 459
I Theorem (fluctuation theorem)......Page 462
II Theorem: (extended fluctuation theorem)......Page 464
(B) Onsager’s reciprocity and the chaotic hypothesis.......Page 465
(C) A fluidodynamic application.......Page 467
Theorem III (extension of Onsager–Machlup fluctuations theory)......Page 468
Appendix: Onsager reciprocity as a consequence of the fluctuationtheorem.......Page 470
(A) Reversible and irreversible equations for a real fluid.......Page 473
(B) Axiom C and the pairing rule.......Page 478
(C) Relation between the NS and ED equations: the barometric formula.......Page 484
Problems.......Page 486
Bibliography.......Page 487
Name index......Page 495
Subject index......Page 496
Citations index......Page 502