TY  - BOOK
AU  - Cholewa,Jan W.
AU  - Dlotko,Tomasz
AU  - Chafee,Nathaniel
ED  - London Mathematical Society,
TI  - Global attractors in abstract parabolic problems
T2  - London Mathematical Society lecture note series
SN  - 9780511526404 (ebook)
AV  - QA614.813 .C48 2000
U1  - 514/.74 21
PY  - 2000///
CY  - Cambridge
PB  - Cambridge University Press
KW  - Attractors (Mathematics)
KW  - Differential equations, Parabolic
N1  - Title from publisher's bibliographic system (viewed on 05 Oct 2015); Ch. 1. Preliminary Concepts -- 1.1. Elements of stability theory -- 1.2. Inequalities. Elliptic operators -- 1.3. Sectorial operators -- Ch. 2. The abstract Cauchy problem -- 2.1. Evolutionary equation with sectorial operator -- 2.2. Variation of constants formula -- 2.3. Local X[superscript [alpha]] solutions -- Ch. 3. Semigroups of global solutions -- 3.1. Generation of nonlinear semigroups -- 3.2. Smoothing properties of the semigroup -- 3.3. Compactness results -- Ch. 4. Construction of the global attractor -- 4.1. Dissipativeness of {T(t)} -- 4.2. Existence of a global attractor -- abstract setting -- 4.3. Global solvability and attractors in X[superscript [alpha]] scales -- Ch. 5. Application of abstract results to parabolic equations -- 5.1. Formulation of the problem -- 5.2. Global solutions via partial information -- 5.3. Existence of a global attractor -- Ch. 6. Examples of global attractors in parabolic problems -- 6.1. Introductory example -- 6.2. Single second order dissipative equation -- 6.3. The method of invariant regions -- 6.4. The Cahn-Hilliard pattern formation model -- 6.5. Burgers equation -- 6.6. Navier-Stokes equations in low dimension (n [less than or equal to] 2) -- 6.7. Cauchy problems in the half-space R[superscript +] x R[superscript n] -- Ch. 7. Backward uniqueness and regularity of solutions -- 7.1. Invertible processes -- 7.2. X[superscript s+[alpha]] solutions; s [greater than or equal to] 0, [alpha][Epsilon](0,1) -- Ch. 8. Extensions -- 8.1. Non-Lipschitz nonlinearities -- 8.2. Application of the principle of linearized stability -- 8.3. The n-dimensional Navier-Stokes system -- 8.4. Parabolic problems in Holder spaces -- 8.5. Dissipativeness in Holder spaces -- 8.6. Equations with monotone operators -- Ch. 9. Appendix -- 9.1. Notation, definitions and conventions -- 9.2. Abstract version of the maximum principle -- 9.3. L[superscript [infinity]]([Omega]) estimate for second order problems -- 9.4. Comparison of X[superscript [alpha]] solution with other types of solutions -- 9.5. Final remarks
N2  - The study of dissipative equations is an area that has attracted substantial attention over many years. Much progress has been achieved using a combination of both finite dimensional and infinite dimensional techniques, and in this book the authors exploit these same ideas to investigate the asymptotic behaviour of dynamical systems corresponding to parabolic equations. In particular the theory of global attractors is presented in detail. Extensive auxiliary material and rich references make this self-contained book suitable as an introduction for graduate students, and experts from other areas, who wish to enter this field
UR  - https://doi.org/10.1017/CBO9780511526404
ER  -