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33 results

Page 1 of 2

  • 1 - Basic notions

  • from Part I - Attractors for the Semigroups of Operators: An Abstract Framework
  • Olga A. Ladyzhenskaya
  • Foreword by Gregory A. Seregin, Varga K. Kalantarov, Koç University, Istanbul, Sergey V. Zelik, University of Surrey
  • Book:
    Attractors for Semigroups and Evolution Equations
    Published online:
    19 May 2022
    Print publication:
    09 June 2022, pp 3-5
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Attractors for Semigroups and Evolution Equations
  • Olga A. Ladyzhenskaya
  • Foreword by Gregory A. Seregin, Varga K. Kalantarov, Sergey V. Zelik
  • Published online:
    19 May 2022
    Print publication:
    09 June 2022
  • In this volume, Olga A. Ladyzhenskaya expands on her highly successful 1991 Accademia Nazionale dei Lincei lectures. The lectures were devoted to questions of the behaviour of trajectories for semigroups of nonlinear bounded continuous operators in a locally non-compact metric space and for solutions of abstract evolution equations. The latter contain many initial boundary value problems for dissipative partial differential equations. This work, for which Ladyzhenskaya was awarded the Russian Academy of Sciences' Kovalevskaya Prize, reflects the high calibre of her lectures; it is essential reading for anyone interested in her approach to partial differential equations and dynamical systems. This edition, reissued for her centenary, includes a new technical introduction, written by Gregory A. Seregin, Varga K. Kalantarov and Sergey V. Zelik, surveying Ladyzhenskaya's works in the field and subsequent developments influenced by her results.

Attractors for Semi-groups and Evolution Equations
  • Olga Ladyzhenskaya
  • Published online:
    19 January 2010
    Print publication:
    22 August 1991
  • This book presents an expansion of the highly successful lectures given by Professor Ladyzhenskaya at the University of Rome, 'La Sapienza', under the auspices of the Accademia dei Lencei. The lectures were devoted to questions of the behaviour of trajectories for semi-groups of non-linear bounded continuous operators in a locally non-compact metric space and for solutions of abstract evolution equations. The latter contain many boundaries value problems for partial differential equations of a dissipative type. Professor Ladyzhenskaya was an internationally renowned mathematician and her lectures attracted large audiences. These notes reflect the high calibre of her lectures and should prove essential reading for anyone interested in partial differential equations and dynamical systems.

  • Preface

  • Olga Ladyzhenskaya
  • Book:
    Attractors for Semi-groups and Evolution Equations
    Published online:
    19 January 2010
    Print publication:
    22 August 1991, pp ix-xii

  • Summary

    These lecture notes are devoted to questions of the behaviour, when t → ∞, of trajectories Vt (ν), t ∈ ℔+ = [0, ∞) for semigroups {Vt, t ∈ ℔+, X} of nonlinear bounded continuous operators Vt in a locally non-compact metric space X and for solutions of abstract evolution equations. The latter contain many boundary value problems for PDE (partial differential equations) of a dissipative type.

    In contrast to the traditional theory of the local stability of PDE (i.e. in the vicinity of a solution) we study the behaviour of all trajectories or solutions for the problems and give a description of the set of all limit states. We will not make assumptions either about the smallness of the parameters in the problem or on the closeness of the problem to a linear one, neither will we consider any other condition that ensures that all the solutions of the problem tend to some special solution. Our purpose is to develop a global theory of stability for problems of mathematical physics with dissipation. The principal ideas in this subject were formulated in paper [1] and I follow them here. The object of paper [1] concerns boundary value problems for Navier–Stokes equations. This object helped us to understand which properties of semigroup {Vt, t ∈ ℔+, X} imply the compactness of the set of all limit states (or, which is the same, the minimal global B-attractor), its invariance, the possibility of continuing the semigroup restricted on M. to the full group on M and a finiteness of dynamics {Vt, t ∈ ℔ = (−∞, + ∞), M}.

Page 1 of 2