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Chepyzhov V.V., Vishik M.I. — Attractors for equations of mathematical physics
Chepyzhov V.V., Vishik M.I. — Attractors for equations of mathematical physics



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Название: Attractors for equations of mathematical physics

Авторы: Chepyzhov V.V., Vishik M.I.

Аннотация:

One of the major problems in the study of evolution equations of mathematical physics is the investigation of the behavior of the solutions to these equations when time is large or tends to infinity. The related important questions concern the stability of solutions or the character of the instability if a solution is unstable. In the last few decades, considerable progress in this area has been achieved in the study of autonomous evolution partial differential equations. For a number of basic evolution equations of mathematical physics, it was shown that the long time behavior of their solutions can be characterized by a very important notion of a global attractor of the equation. In this book, the authors study new problems related to the theory of infinite-dimensional dynamical systems that were intensively developed during the last 20 years. They construct the attractors and study their properties for various non-autonomous equations of mathematical physics: the 2D and 3D Navier-Stokes systems, reaction-diffusion systems, dissipative wave equations, the complex Ginzburg-Landau equation, and others. Since, as it is shown, the attractors usually have infinite dimension, the research is focused on the Kolmogorov $\varepsilon$-entropy of attractors. Upper estimates for the $\varepsilon$-entropy of uniform attractors of non-autonomous equations in terms of $\varepsilon$-entropy of time-dependent coefficients are proved. Also, the authors construct attractors for those equations of mathematical physics for which the solution of the corresponding Cauchy problem is not unique or the uniqueness is not proved. The theory of the trajectory attractors for these equations is developed, which is later used to construct global attractors for equations without uniqueness. The method of trajectory attractors is applied to the study of finite-dimensional approximations of attractors. The perturbation theory for trajectory and global attractors is developed and used in the study of the attractors of equations with terms rapidly oscillating with respect to spatial and time variables. It is shown that the attractors of these equations are contained in a thin neighborhood of the attractor of the averaged equation. The book gives systematic treatment to the theory of attractors of autonomous and non-autonomous evolution equations of mathematical physics. It can be used both by specialists and by those who want to get acquainted with this rapidly growing and important area of mathematics.


Язык: en

Рубрика: Математика/

Серия: Сделано в холле

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 363

Добавлена в каталог: 31.10.2012

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(E\times \Sigma,E)$-continuous family      88
$(\Theta^{loc}_{+},\Sigma)$-nclosed family      262
$\varepsilon$-entropy      51 165
$\varepsilon$-period      95
Almost periodic, function      81 95
Almost periodic, function, asymptotically      133 140
Almost periodic, function, in the Stepanov sense      97
Attracting property      84
Attractor      19 217 218
Attractor, $(\mathcal{M},\mathfrak{T})$-attractor      218
Attractor, global      19 37 225 239 249
Attractor, global, uniform      265
Attractor, Lorenz      65
Attractor, non-uniform      85
Attractor, trajectory      203 223
Attractor, trajectory, uniform      262
Attractor, uniform      84 93
Average in $L_{p,w}(\Omega)$      311
Average in $L_{\infty}(\Omega)$      312
Average, time uniform      316
Averaging, spatial      311
Averaging, time      316
Backward uniqueness property      138
Belousov — Zhabotinsky equations      43
Bochner — Amerio criterion      96
Cascade system      133
Chafee — Infante equation      41 330
Closure      212
Compactness, criterion, in $C(\mathbb{R},\mathcal{M})$      98
Compactness, criterion, in $L^{loc}_{p,w}(\mathbb{R},\mathcal{E})$      105
Compactness, criterion, in $L^{loc}_{p}(\mathbb{R},\mathcal{E})$      101
Compactness, criterion, theorems      31
Compactum      214
Complete trajectory      19 38 88 218 223 263
Continuous mapping      213
Convergent sequence      212
Convergent sequence, *-weakly      32
Convergent sequence, weakly      32
Covering      212
Covering, Density      349
Covering, radius      349
d-dimensional Hausdorff measure      52
Derivative in the distribution sense      31
Differential inequality      35
Dimension, fractal      52 173
Dimension, functional      176
Dimension, Hausdorff      52
Dimension, local, fractal      175
Dimension, local, functional      176
Dimension, Lyapunov      62
Dissipative wave equation      334
Dissipativity condition      17
Douady — Oesterle theorem      55
Energy norm      50
Equilibrium point      20
First Uryson theorem      214
Fitz — Hugh — Nagumo equations      41
Frechet — Uryson space      213
Fundamental parallelepiped      349
Fundamental region      349
Gagliardo — Nirenberg inequality      30
Galerkin approximation      23 302
Galerkin method      231 284
Ginzburg — Landau equation      42 118 328
Grashof number      47 235
Gronwall's inequality      34
Group      36
Hahn — Banach theorem      32
Haraux's example      85
Hausdorff dimension      52
Hausdorff space      213
Hoelder's inequality      34
Hull      81 96 132 135
Hyperbolic equation with dissipation      49 71 159 185 292 306
Hyperbolic equation, damped      119
Hyperbolic equation, dissipative      49
Inductive limit      221
Instability index      73
Interpolation inequality      30
k-dimensional torus      82
Kernel of equation      20 223 263
Kernel of process      88 149
Kernel of semigroup      38 218
Kernel, section      20 38 88 218
Kolmogorov $\varepsilon$-entropy      164
Ladyzhenskaya's inequality      46 230 235
Lattice      349
Lattice, cube      351
Lattice, determinant      349
Lattice, enerating matrix      349
Lattice, main Voronoi      351
Lieb — Thirring inequality      69
Lipschitz condition      165
Lorenz attractor      65
Lorenz system      23
Lotka — Volterr a system      44
Lyapunov dimension      62
Lyapunov uniform exponents      61
m-dimensional trace      62
Metric order      176
Metric order, local      176
Minimality property      84
Multiplicative properties      83
Navier — Stokes system      269
Navier — Stokes system, 2D      46 68 74 107 157 177 239 278 323
Navier — Stokes system, 3D      229 305 320
Nikol'skii space      279
Periodic orbit      20
Point, adherent      211
Point, limit      212
Process      82 83
Process, bounded      83
Process, family of processes      84
Process, periodic      87
Quasidifferential      53
Quasiperiodic function      82 96
Quasiperiodic solution      20
Quasiperiodic symbol      88
Reaction-diffusion, equation      38
Reaction-diffusion, system      66 75 114 158 181 282 325
Second axiom of count ability      212
Second Uryson theorem      214
Semigroup      18 36 214
Semigroup, (E,E)-bounded      37
Semigroup, (E,E)-continuous      37
Semigroup, asymptotically compact      37
Semigroup, compact      37
Semigroup, identity      36
Semiprocess      129
Set, $(\mathcal{M},\mathfrak{T})$-attracting      218
Set, $\omega$-limit      19 38 130 215
Set, absorbing      18 37 83
Set, attracting      37 83 223
Set, count ably precompact      214
Set, local unstable      73
Set, precompact      214
Set, relatively dense      95
Set, uniformly, absorbing      84
Set, uniformly, attracting      84 92 262
Sets, closed      211
Sets, open      211
Sine — Gordon equation      49
Sobolev embedding theorem      29
Space, compact      214
Space, count ably compact      214
Space, Frechet — Uryson      213
Space, Hausdorff      213
Space, metrizable      214
Space, normal      213
Space, separable      212
Space, topological      211
Symbol of equation      79 80
Symbol of process      84
Symbol, space      80 81 84
Topology base      212
Trajectory      220 261
Trajectory, attractor      203 223
Trajectory, attractor, uniform      262
Trajectory, space      200 219 260
Trajectory, space, united      261
Translation, bounded function      105
Translation, compact function      81 105 135
Translation, group      260
Translation, identity      83 86
Translation, semigroup      200
Uniformly quasidifferentiable, map      53
Uniformly quasidifferentiable, sequence      153
Unstable trajectory      20
Volume contracting condition      165
Voronoi region      349
Weak solution      230 242 283
Young's inequality      34
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