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Babin A.V., Vishik M.I. — Attractors of Evolution Equations
Babin A.V., Vishik M.I. — Attractors of Evolution Equations



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Название: Attractors of Evolution Equations

Авторы: Babin A.V., Vishik M.I.

Аннотация:

Problems, ideas and notions from the theory of finite-dimensional dynamical systems have penetrated deeply into the theory of infinite-dimensional systems and partial differential equations. From the standpoint of the theory of the dynamical systems, many scientists have investigated the evolutionary equations of mathematical physics. Such equations include the Navier-Stokes system, magneto-hydrodynamics equations, reaction-diffusion equations, and damped semilinear wave equations. Due to the recent efforts of many mathematicians, it has been established that the attractor of the Navier-Stokes system, which attracts (in an appropriate functional space) as t - infin; all trajectories of this system, is a compact finite-dimensional (in the sense of Hausdorff) set. Upper and lower bounds (in terms of the Reynolds number) for the dimension of the attractor were found. These results for the Navier-Stokes system have stimulated investigations of attractors of other equations of mathematical physics. For certain problems, in particular for reaction-diffusion systems and nonlinear damped wave equations, mathematicians have established the existence of the attractors and their basic properties; furthermore, they proved that, as t - +infin;, an infinite-dimensional dynamics described by these equations and systems uniformly approaches a finite-dimensional dynamics on the attractor U, which, in the case being considered, is the union of smooth manifolds. This book is devoted to these and several other topics related to the behaviour as t - infin; of solutions for evolutionary equations.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1992

Количество страниц: 532

Добавлена в каталог: 13.02.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\langle\;,\;\rangle$      15
$\overline{u}(t)$      251 282 436
${\alpha}$-limit set      160
${\delta}$ u      13
${\delta}^{\alpha}$      14
${\Gamma}^{+}$      157
${\Gamma}^{-}$      157
${\mathfrak B}({\rm F})$      31
${\mathfrak B}_{\rm E}({\rm F})$      31
${\mathfrak N}(X)$      160
${\mathfrak N}(\Phi\;\le\;\xi)$      163
${\mathfrak R}$      343
${\mathfrak U}(\Phi\;\le\;\xi)$      162
${\mathfrak U}^{+}$      186
${\omega}$-limit set      124
${\Pi}_{+}\;{\rm and}\;{\Pi}_{-}$      201 202 221
${\rm E}_{+}$ and ${\rm E}_{-}$      202
${\rm G}_{\delta}$      (type of set) 301
${\rm M}^{+}$      281
${\rm M}^{\rm S}(z,\omega)$      220
${\rm M}^{\rm S}(\rm Y,X,{\rm D}_{0})$      158
${\rm M}^{\rm u}(z)$      217
${\rm M}^{\rm u}(z,\omega)$      217
${\rm M}^{\rm u}(\rm Y,X,{\rm D}_{0})$      157
${\rm M}_{\rm f}(\infty)$      147
${\rm N}(\lambda,-\Delta)$      319 322
${\rm r}_{0}$      201
${\rm Tr}_{\rm d}(\mathscr A)$      480
${\rm T}^{\rm n}$      13
${\rm V}_{+}$      233 242
${\rm V}_{-}$      230 242
${\sigma}_{+}(-{\rm A'}_{1}({\rm z}))$      318
*-weakly convergent sequence      20
Absorbing set      32
Absorbing set, uniformly      400
Almost stable semigroup      201
Asymptotics, of ${\rm N}({\lambda},{-\Delta})$      322
Asymptotics, of ${\rm N}({\lambda},{-\Delta}),$ spectral      251 258 259 260 392
Asymptotics, of ${\rm N}({\lambda},{-\Delta}),$ stabilized      434 see
Asymptotics, of ${\rm N}({\lambda},{-\Delta}),$ uniform      279 281 375 382 391
Attracting set      118
Attracting set, uniformly      424
Attraction      117
Attraction, exponential      4 05
Attractor      21 119
Attractor, point-attracting      163
Attractor, regular      270
Bair's Theorem      301
Boussinesg equation      502
c.l.t.      436 452 453 457 460 470 474
Class ${\rm C}^{1\;+\;\alpha}$      223
Class ${\rm C}^{1\;+\;\alpha},\; {\rm G}_{\rm q}$      223 233
Class ${\rm C}^{1\;+\;\alpha},\; {\rm G}_{\rm q}^{+,\alpha}$      241
Class ${\rm C}^{1\;+\;\alpha},\; {\rm G}_{\rm q}^{+}$      241
Class ${\rm C}^{1\;+\;\alpha},\; {\rm G}_{\rm q}^{-,\alpha}$      241
Class ${\rm C}^{1\;+\;\alpha},\; {\rm G}_{\rm q}^{-}$      241
Class ${\rm C}^{1\;+\;\alpha},\; {\rm G}_{\rm q}^{\alpha}$      223 233
Closure      21
Combined limit trajectory      436 see
Compact set      21
Continuous operator      33
Continuous operator, uniformly      33
Courant's Minimax, Principle      3 20
Critical point of a curve      335
Critical value      301
Damped hyperbolic equation      103 131 165 171 214 259 262 378 416 460 500
Damped hyperbolic system      148 170 216
Damped wave equation      see “Damped hyperbolic equation”
Derivative in the sense of distributions      19
Differential inequalities      23
Dimension (Hausdorff)      479
Dirichlet boundary condition      64
Dissipation integral      172 180
Dist      137
Distance      137
Distance, symmetric      137
Ellipticity condition      90
Embedding      17
Equilibrium point      160
F.-d.c.t.      282
Finite-dimensional, combined trajectory      28 2
Fourier coefficients      15
Frechet differentiable operators      341
Frechet differential      341
Fredholm operator, differentiable      301 313
Fredholm operator, linear      300
Gagliardo — Nirenberg inequality      18
Galerkin system      39
Gel'fand theorem      204
Green function      3 20
Gronwall inequality      23
Group      29 148
Hartman — Grobman theorem      261
Hausdorff dimension      479
Hausdorff measure      479
Hoelder condition      222
Hoelder continuous function      14
Hoelder's inequality      23
Hyperbolic equation      see “Damped hyperbolic equation”
Hyperbolic point      240 245
Implicit function theorem      346
Index, of a Fredgolm operator      300
Index, of a linear semigroup      205
Index, of instability      318 337
Inflection, point of      337
Inhomogenious variation equation      345
Injectivity      148
Instability index      337
Interpolation inequality      18
Invariance, inverse      118
Invariance, strict      119
Invariant, manifold, analytic      245
Invariant, manifold, of class ${\rm C}^{1\;+\;\alpha}$      223
Invariant, set      159
Invariant, trajectory      157
Linearization, ${\rm C}^{1}$      262
Linearization, analytical      266 268
Linearization, continuous      261
Linearization, global      262 268
Linearization, smooth      265
Liouville formula      486
Local unstable set      217
Locally invariant set      241
Lyapunov function      159
Lyapunov stability modulo attractor      120
Magnetohydrodynamics equations      503
Manifold, ${\rm M}_{+}$      223 241 249
Manifold, ${\rm M}_{+}(\rm z,r)$      249
Manifold, ${\rm M}_{-}$      233 241
Manifold, local invariant      249
Manifold, of class ${\rm C}^{1\;+\;\alpha}$      223
Massive      301
Maximal attractor      119
Minimax principle      320
Monotone parabolic equation      34 47 127 181 189 410
Monotonicity condition      179 182
Monotonicity condition, strong      179
Morse — Smale system      295
Navier — Stokes system, three-dimensional      392
Navier — Stokes system, two-dimensional      76 129 189 260 330 383 412 413 414 492
Neighbourhood, ${\rm O}_{\rho}(\rm z)$      241
Neighbourhood, ${\rm O}_{\rho}^{+}(0)$      241
Neighbourhood, ${\rm O}_{\rho}^{-}(0)$      241
Neighbourhood, ${\rm O}_{\varepsilon}(\rm Y)$      117
Neighbourhood, of a set      117
Neumann boundary condition      64
Nonlinear theory of shells, system      170 176
Norm, ${\|\rm . \|}_{\rm C}$      14
Norm, ${\|\rm . \|}_{{\rm C}^{\alpha}}$      14
Norm, ${\|\rm . \|}_{{\rm C}^{\rm k}}$      14
Norm, ${\|\rm u\|}_{0,p}$      15
Norm, ${\|\rm u\|}_{\rm 0,\infty}$      15
Norm, ${\|\rm u\|}_{\rm l,p}$      15
Norm, ${\|\rm u\|}_{{\rm L}_{\infty}}$      15
Norm, ${\|\rm u\|}_{{\rm L}_{\rm p}}$      15
Norm, in: ${\rm C}^{\rho}(\Omega)$      14
Norm, in: ${\rm C}^{\rho}({\rm T}^{0})$      15
Norm, in: ${\rm E}_{+}\;+\;{\rm E}_{-}$      222
Norm, in: ${\rm W}^{\rm l}_{\rm p}(\Omega)$      15
Norm, in: ${\rm W}^{\rm l}_{\rm p}({\rm T}^{\rm n})$      15
Operators ${\rm L}_{+},\;{\rm L}_{-}$      222
Orbit      157
Passing neighbourhoods, in order reverse to the numeration      435
Poincare theorem      266
Precompact      21
Pressing exponential      218
Proper mapping      298
Quasidifferentiable, mapping      479
Quasidifferentiable, mapping, uniformly      479
Quasidifferential      480
Quasilinear operators      299 309
Quasilinear parabolic, equations      90 130 153 163 386 452 498
Reaction-diffusion system      63 168 208 258 368 456 487 498
Reaction-diffusion system, steady-state solutions      327
Regular value      301
Resonances      266
Rest-untending points      186
Riesz projections      326
Sard — Smale theorem      302
Scalar product in ${\rm H}^{t}(\Omega)$      15
Semigroup      29
Semigroup identity      29
Semigroup, bounded      31
Semigroup, bounded, for $t\;>\;0$      31
Semigroup, bounded, for $t\;\le\;0$      31
Semigroup, bounded, uniformly for finite t      32
Semigroup, bounded, uniformly in t      31
Semigroup, close to linear      248 354
Semigroup, continuous      3 3
Semigroup, continuous, uniformly      33
Semigroup, linear      194 201 205 206 209
Semilinear operators      299
Semiorbit      157
Semitrajectory      157
Semitrajectory, negative      147
Shmuljan — Eberlane theorem      21
Singularly perturbed equations      416 465 474
Sobolev's embedding theorems      17
Sobolev's space, ${\rm H}^{\rho}(\Omega)\;(\rho\;\le\;0)$      16
Sobolev's space, ${\rm H}^{\rm l}(\Omega)$ (integer 1)      15
Sobolev's space, ${\rm W}_{\rm p}^{\rm l}(\Omega)$      15
Spaces, ${\rm C}(\Omega)$      14
Spaces, ${\rm C}^{\alpha}(\Omega)$      14
Spaces, ${\rm C}^{\infty}(\overline{\Omega})$      16
Spaces, ${\rm C}^{\infty}_{0}(\Omega)$      16
Spaces, ${\rm C}^{\rho}(\Omega)$      14
Spaces, ${\rm C}^{\rho}({\rm T}^{\rm n})$      15
Spaces, ${\rm C}^{\rm k}([0,\;T];\;E)$      19
Spaces, ${\rm C}^{\rm k}(\Omega)$      14
Spaces, ${\rm C}^{\rm k}(\rm E)$      19
Spaces, ${\rm C}^{\rm t,2t}([0,T]\;\times\;{\overline{\Omega}})$      22
Spaces, ${\rm H}^{\rm -l}(\Omega)$      16
Spaces, ${\rm H}^{\rm l}(\Omega)$      15
Spaces, ${\rm H}^{\rm l}_{0}(\Omega)$      16
Spaces, ${\rm H}^{\rm t,2t}([0,T]\;\times\;{\overline{\Omega}})$      22
Spaces, ${\rm L}^{\infty}(\Omega)$      15
Spaces, ${\rm L}^{\rm p}(\Omega)$      14 15
Spaces, ${\rm L}^{\rm p}(\rm E)$      19
Spaces, ${\rm L}_{\rm p}([0,\;T];\;E)$      19
Spaces, ${\rm W}^{\rm l}_{\rm p,0}(\Omega)$      16
Spaces, ${\rm W}^{\rm l}_{\rm p}(\Omega)$      15
Spaces, ${\rm W}^{\rm l}_{\rm p}({\rm T}^{\rm n})$      15
Spaces, L      342
Spaces, L'      343
Stability      120
Stability, Lyapunov modulo attractor      120
Stable set      158
Steady-state solutions      297
Stretch exponential      219
Subspace, corresponding to ${\sigma}_{0}$      202
Subspace, invariant      203
Subspace, stable      204
Subspace, unstable      204
Tikhonov's lemma      154
Time of arrival      283
Torus ${\rm T}^{\rm n}$      13
Trace      480
Trajectory, bounded      157
Trajectory, invariant      157
Turning point      336
Uniformly absorbing set      400
Uniformly continuous semigroup      33
Uniformly differentiable operator      341
Uniformly quasidifferentiable mapping      479
Unstable set      157 217
Upper semicontinuous dependence      402
Value, critical      301
Value, regular      301
Variation equation inhomogenious      345
Viscoelasticity, equation      169
Viscoelasticity, linearized equation      2
Weak topology      21
Weakly continuous trajectory      21
weakly convergent sequence      20
Young's inequality      22
[L.S.U.]      91
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