Attractors for Semigroups and Evolution Equations

Olga A. Ladyzhenskaya's work expands on her 1991 Accademia Nazionale dei Lincei lectures, covering topics such as semigroups of nonlinear bounded continuous operators and abstract evolution equations. This edition, reissued for her centenary, includes a new technical introduction by Gregory A. Seregin, Varga K. Kalantarov, and Sergey V. Zelik, surveying Ladyzhenskaya's works and their influence.

Format: Paperback / softback
Length: 115 pages
Publication date: 09 June 2022
Publisher: Cambridge University Press

In this comprehensive volume, Olga A. Ladyzhenskaya delves deeper into her renowned 1991 Accademia Nazionale dei Lincei lectures, which were dedicated to exploring the behavior of trajectories for semigroups of nonlinear bounded continuous operators in locally non-compact metric spaces, as well as the solutions of abstract evolution equations. These equations encompass numerous initial boundary value problems for dissipative partial differential equations. Recognizing the significance of her work, Ladyzhenskaya was honored with the Russian Academy of Sciences' Kovalevskaya Prize. This publication, reissued in celebration of her centenary, features a new technical introduction authored by Gregory A. Seregin, Varga K. Kalantarov, and Sergey V. Zelik, providing a comprehensive survey of Ladyzhenskaya's contributions to the field and the subsequent developments influenced by her results.

The lectures, which were delivered at the prestigious Accademia Nazionale dei Lincei, focused on the behavior of trajectories for semigroups of nonlinear bounded continuous operators in locally non-compact metric spaces. These semigroups represent mathematical models for a wide range of physical systems, including fluids, elasticity, and chemical reactions. The lectures also addressed the solutions of abstract evolution equations, which are mathematical models used to describe the evolution of complex systems over time. These equations often involve partial differential equations, which are mathematical equations that describe the rate of change of a quantity as it varies in space and time.

One of the key themes of Ladyzhenskaya's lectures was the study of the stability and ergodicity of solutions to abstract evolution equations. She showed that certain types of solutions can exhibit chaotic behavior, which is characterized by unpredictable and complex patterns of change. However, she also demonstrated that certain conditions can be imposed on the equations to ensure the stability of solutions, allowing for the prediction of long-term behavior.

Another important aspect of Ladyzhenskaya's work was her investigation of the behavior of solutions to initial boundary value problems for dissipative partial differential equations. These equations are used to model a wide range of physical phenomena, including heat transfer, fluid flow, and electromagnetic waves. Ladyzhenskaya showed that certain types of solutions to these equations can exhibit singular behavior, which is characterized by sudden and dramatic changes in the solution. She also developed methods for analyzing these solutions and predicting their long-term behavior.

In addition to her theoretical contributions, Ladyzhenskaya was also a prominent figure in the mathematical community. She served as the president of the International Commission on Mathematical Physics from 1988 to 1992 and was a member of the Russian Academy of Sciences. She was also the recipient of numerous awards and honors, including the Lomonosov Gold Medal from the Russian Academy of Sciences and the Leroy P. Steele Prize from the American Mathematical Society.

Ladyzhenskaya's work has had a significant impact on the field of mathematics and has inspired many researchers to pursue their studies in the area of partial differential equations and dynamical systems. Her lectures, which were published in this volume, are essential reading for anyone interested in her approach to these subjects and the broader field of mathematics.

In conclusion, Olga A. Ladyzhenskaya's work in the field of partial differential equations and dynamical systems is a testament to her exceptional talent and dedication. Her lectures, which were published in this volume, are essential reading for anyone interested in her approach to these subjects and the broader field of mathematics. Her contributions have had a significant impact on the field and have inspired many researchers to pursue their studies in the area.

ISBN-13: 9781009229821
Edition number: Revised ed