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Attractors for Equations of Mathematical Physics
By: Vladimir V. Chepyzhov, Mark I. Vishik
Hardcover | 1 November 2001
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363 Pages
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Introduction | p. 1 |
Attractors of Autonomous Equations | p. 15 |
Attractors of Autonomous Ordinary Differential Equations | p. 17 |
Semigroups and attractors | p. 17 |
Examples of ordinary differential equations and their attractors | p. 21 |
Attractors of Autonomous Partial Differential Equations | p. 27 |
Function spaces and embedding theorems | p. 28 |
Operator semigroups. Basic notions | p. 35 |
Attractors of semigroups | p. 37 |
Reaction-diffusion systems | p. 38 |
2D Navier-Stokes system | p. 46 |
Hyperbolic equation with dissipation | p. 49 |
Dimension of Attractors | p. 51 |
Fractal and Hausdorff dimension | p. 51 |
Dimension of invariant sets | p. 53 |
Optimization of the bound for the fractal dimension | p. 59 |
Application to semigroups | p. 62 |
Applications to evolution equations | p. 65 |
Lower bounds for the dimension of attractors | p. 73 |
Attractors of Non-autonomous Equations | p. 77 |
Processes and Attractors | p. 79 |
Symbols of non-autonomous equations | p. 80 |
Cauchy problem and processes | p. 82 |
Uniform attractors | p. 83 |
Haraux's example | p. 85 |
The reduction to a semigroup | p. 86 |
On uniform (w.r.t. [tau] [set membership] R) attractors | p. 92 |
Translation Compact Functions | p. 95 |
Almost periodic functions | p. 95 |
Translation compact functions in C(R; M) | p. 97 |
Translation compact functions in L[superscript loc subscript p] (R; [varepsilon]) | p. 101 |
Translation compact functions in L[superscript loc subscript p, w] (R; [varepsilon]) | p. 104 |
Other translation compact functions | p. 106 |
Attractors of Non-autonomous Partial Differential Equations | p. 107 |
2D Navier-Stokes system | p. 107 |
Non-autonomous reaction-diffusion systems | p. 114 |
Non-autonomous Ginzburg-Landau equation and others | p. 118 |
Non-autonomous damped hyperbolic equations | p. 119 |
Semiprocesses and Attractors | p. 129 |
Families of semiprocesses and their attractors | p. 129 |
On the reduction to the semigroup | p. 132 |
Non-autonomous equations with tr.c. on R[subscript +] symbols | p. 135 |
Prolongations of semiprocesses to processes | p. 137 |
Asymptotically almost periodic functions | p. 140 |
Non-autonomous equations with a.a.p. symbols | p. 143 |
Cascade systems and their attractors | p. 146 |
Kernels of Processes | p. 149 |
Properties of kernels | p. 149 |
On the dimension of connected sets | p. 153 |
Dimension estimates for kernel sections | p. 155 |
Applications to non-autonomous equations | p. 157 |
Kolmogorov [varepsilon]-Entropy of Attractors | p. 163 |
Estimates of the [varepsilon]-entropy | p. 163 |
Fractal dimension of attractors | p. 173 |
Functional dimension and metric order | p. 176 |
Applications to evolution equations | p. 177 |
[eta]-entropy and metric order of [Sigma] | p. 188 |
[varepsilon]-entropy in the extended phase space | p. 192 |
Trajectory Attractors | p. 197 |
Trajectory Attractors of Autonomous Ordinary Differential Equations | p. 199 |
Preliminary propositions | p. 200 |
Construction of the trajectory attractor | p. 203 |
Examples of equations | p. 205 |
Dependence on a parameter | p. 207 |
Attractors in Hausdorff Spaces | p. 211 |
Some topological preliminaries | p. 211 |
Semigroups in topological spaces and attractors | p. 214 |
Applications to (M, [xi])-attractors | p. 218 |
Trajectory Attractors of Autonomous Equations | p. 219 |
Trajectory spaces of evolution equations | p. 219 |
Existence of trajectory attractors | p. 222 |
Trajectory and global attractors | p. 224 |
Trajectory Attractors of Autonomous Partial Differential Equations | p. 229 |
Autonomous Navier-Stokes systems | p. 229 |
Autonomous hyperbolic equations | p. 242 |
Hyperbolic equations depending on a parameter | p. 251 |
Trajectory Attractors of Non-autonomous Equations | p. 259 |
Non-autonomous equations, their symbols, and trajectory spaces | p. 260 |
Existence of uniform trajectory attractors | p. 262 |
Equations with symbols on the semiaxis | p. 266 |
Trajectory Attractors of Non-autonomous Partial Differential Equations | p. 269 |
Non-autonomous Navier-Stokes systems | p. 269 |
Trajectory attractor for 2D Navier-Stokes system | p. 278 |
Reaction-diffusion systems | p. 282 |
Non-autonomous hyperbolic equations | p. 292 |
Approximation of Trajectory Attractors | p. 299 |
Trajectory attractors of non-autonomous ordinary differential equations | p. 299 |
Trajectory attractors of Galerkin systems | p. 302 |
Convergence of trajectory attractors of Galerkin systems | p. 303 |
Perturbation of Trajectory Attractors | p. 305 |
Trajectory attractors of perturbed equations | p. 305 |
Dependence of trajectory attractors on a small parameter | p. 307 |
Averaging of Attractors of Evolution Equations with Rapidly Oscillating Terms | p. 311 |
Averaging of rapidly oscillating functions | p. 311 |
Averaging of equations and systems | p. 320 |
Perturbation with rapidly oscillating terms | p. 341 |
Proofs of Theorems II.1.4 and II.1.5 | p. 345 |
Lattices and Coverings | p. 349 |
Bibliography | p. 353 |
Index | p. 361 |
Table of Contents provided by Syndetics. All Rights Reserved. |
ISBN: 9780821829509
ISBN-10: 0821829505
Series: Colloquium Publications
Published: 1st November 2001
Format: Hardcover
Language: English
Number of Pages: 363
Audience: Professional and Scholarly
Publisher: American Mathematical Society
Country of Publication: US
Dimensions (cm): 26.04 x 18.42 x 2.54
Weight (kg): 1.0
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