Topological Approximation Methods for Evolutionary Problems of Nonlinear Hydrodynamics
Bearbeitet von
Victor G. Zvyagin, Dmitry A. Vorotnikov
1. Auflage 2008. Buch. XII, 230 S. Hardcover ISBN 978 3 11 020222 9
Format (B x L): 17 x 24 cm Gewicht: 539 g
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Preface v
1 Non-Newtonianflows 1
1.1 Principles offlow description . . . 1
1.1.1 The basic characteristics of aflow . . . 1
1.1.2 Newtonianfluid . . . 2
1.1.3 Equation of motion . . . 3
1.1.4 No-slip condition . . . 5
1.2 One-dimensional models of viscoelastic media . . . 5
1.2.1 Method of mechanical models . . . 5
1.2.2 The Maxwell body . . . 6
1.2.3 The Jeffreys body . . . 7
1.3 Multidimensional models of viscoelastic media . . . 10
1.3.1 Passage to multidimensional models . . . 10
1.3.2 Partial derivative . . . 10
1.3.3 Substantial derivative . . . 11
1.3.4 Principle of material frame-indifference. Frame-indifferent functions . . . 14
1.3.5 The Zaremba–Zórawski theorem . . . .˙ 16
1.3.6 Objective derivatives . . . 17
1.3.7 Examples of objective derivatives . . . 19
1.4 Nonlinear effects in viscous media . . . 21
1.4.1 Nonlinear viscosity and viscoelasticity . . . 21
1.4.2 Noll’s theorem and the Stokes conjecture. . . 21
1.4.3 The Wang and Rivlin–Ericksen theorems . . . 23
1.4.4 Oldroyd’s method. Models of Prandtl and Eyring . . . 25
1.5 Combined models of nonlinear viscoelastic media . . . 27
1.5.1 Nonlinear differential constitutive relations . . . 27
1.5.2 Combined models . . . 29
2 Basic function spaces. Embedding and compactness theorems 31 2.1 Function spaces and embeddings . . . 31
2.1.1 Lebesgue and Sobolev spaces . . . 31
2.1.2 The spaces used in hydrodynamics . . . 39
2.2 Spaces of vector functions . . . 40
2.2.1 Preliminaries . . . 40
2.2.2 Classical criteria of compactness . . . 46
2.2.3 Compactness inLp.0; TIE/ . . . 47
2.2.4 Compactness of sets of vector functions with values in an “in- termediate” space . . . 51
2.2.5 The Aubin–Simon theorem . . . 52
2.2.6 Theorem on “partial” compactness . . . 56
2.2.7 Lemma on weak continuity of essentially bounded functions . 56 2.2.8 Lemma on differentiability of the quadrate of the norm of a vector function . . . 57
2.2.9 Two lemmas on absolutely continuous vector functions . . . . 57
3 Operator equations in Banach spaces 61 3.1 Linear equations . . . 61
3.1.1 The Lax–Milgram theorem . . . 61
3.1.2 Characterization of the gradient of a distribution . . . 61
3.1.3 An existence lemma . . . 64
3.1.4 Strongly positive operators and parabolic equations . . . 64
3.2 Nonlinear equations . . . 69
3.2.1 An existence theorem . . . 69
3.2.2 The Leray–Schauder degree . . . 74
4 Attractors of evolutionary equations in Banach spaces 77 4.1 Attractors of autonomous equations: classical approach . . . 77
4.1.1 Attractor of a semigroup . . . 77
4.1.2 Global.E; E0/-attractors of evolutionary equations . . . 78
4.2 Attractors of autonomous problems without uniqueness of the solution 79 4.2.1 Basic definitions . . . 80
4.2.2 Simple properties of attracting sets and auxiliary statements . 82 4.2.3 Existence of a minimal trajectory attractor . . . 86
4.2.4 Existence of a global attractor . . . 87
4.2.5 The case when a trajectory attractor is contained in the trajectory space . . . 91
4.2.6 Structure of the minimal trajectory attractor and of the homo- geneous trajectory quasiattractor . . . 92
4.2.7 Correspondence between two concepts of global attractor . . . 95
4.3 Attractors of non-autonomous equations . . . 97
5 Strong solutions for equations of motion of viscoelastic medium 103 5.1 The Guillopé–Saut theorem . . . 103
5.2 Initial-value problem for combined model of nonlinear viscoelastic medium . . . 112
5.2.1 Formulation of the initial-value problem . . . 112
5.2.2 The Leray projection inRnand some additional notations . . 113
5.2.3 The main existence and uniqueness theorem . . . 115
5.3 Operator treatment of the problem . . . 116
5.4 Auxiliary problem . . . 118
5.4.1 Solvability of the auxiliary problem . . . 118
5.4.2 Operator estimates . . . 119
5.4.3 Properties of the operatorA0 . . . 126
5.4.4 Uniqueness lemma . . . 128
5.4.5 A priori estimate . . . 130
5.4.6 Proof of Theorem 5.4.1 . . . 131
5.5 Proof of the main theorems . . . 133
5.5.1 Proof of Theorem 5.3.1 . . . 133
5.5.2 Proof of Theorem 5.2.1 . . . 136
5.5.3 The caser > 1 . . . 137
5.6 Continuous dependence of solutions on data . . . 137
6 Weak solutions for equations of motion of viscoelastic medium 141 6.1 Preliminaries . . . 141
6.1.1 Weak solutions for equations offluid dynamics: general scheme141 6.1.2 Integration by parts . . . 144
6.2 Initial-boundary value problem and its weak form . . . 149
6.2.1 Statement of the problem . . . 149
6.2.2 Weak formulation of the problem . . . 149
6.2.3 An existence result . . . 151
6.3 Auxiliary problem . . . 152
6.4 Passage to the limit. . . 158
6.5 Existence of a weak solution for the Jeffreys model . . . 163
6.5.1 Existence of velocity and stress . . . 163
6.5.2 Existence of pressure . . . 166
6.6 Uniqueness of the weak solution . . . 170
6.6.1 Differential energy inequality . . . 170
6.6.2 Uniqueness of the weak solution . . . 172
6.7 Minimal trajectory and global attractors for the Jeffreys model . . . . 178
6.7.1 Integral energy estimate: autonomous case . . . 178
6.7.2 Existence and structure of attractors . . . 181
6.8 Uniform attractors for the Jeffreys model . . . 184
6.8.1 Integral energy estimate: non-autonomous case . . . 184
6.8.2 Existence and structure of uniform attractors . . . 188
6.9 Stationary boundary-value problem for the Jeffreys model . . . 191
6.9.1 Strong and weak statements of the stationary problem . . . . 191
6.9.2 Auxiliary problem and a priori bound . . . 192
6.9.3 Solvability of the auxiliary problem . . . 193
6.9.4 Proof of Theorem 6.9.1. . . 194
7 The regularized Jeffreys model 197 7.1 Formulation of the problem and the main results . . . 197
7.2 Properties of the operators . . . 201
7.3 A priori estimates of solutions and solvability of the approximating equations . . . 206
7.4 A priori estimate and existence of solutions for the regularized problem 208 7.5 Another weak formulation for the regularized Jeffreys model . . . 211
7.6 Behaviour of solutions of regularized problems ası!0 . . . 216
7.7 Two constructions of regularization operator . . . 218
7.7.1 Thefirst construction . . . 218
7.7.2 The second construction . . . 220
References 223
Index 229
