Advertisement

A Stability Technique for Evolution Partial Differential Equations

A Dynamical Systems Approach

  • Victor A. Galaktionov
  • Juan Luis Vázquez

Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 56)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 1-12
  3. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 13-55
  4. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 57-79
  5. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 81-125
  6. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 127-167
  7. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 169-187
  8. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 189-215
  9. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 217-236
  10. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 237-263
  11. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 265-298
  12. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 299-325
  13. Victor A. Galaktionov, Juan Luis Vázquez
    Pages 327-357
  14. Back Matter
    Pages 359-377

About this book

Introduction

common feature is that these evolution problems can be formulated as asymptoti­ cally small perturbations of certain dynamical systems with better-known behaviour. Now, it usually happens that the perturbation is small in a very weak sense, hence the difficulty (or impossibility) of applying more classical techniques. Though the method originated with the analysis of critical behaviour for evolu­ tion PDEs, in its abstract formulation it deals with a nonautonomous abstract differ­ ential equation (NDE) (1) Ut = A(u) + C(u, t), t > 0, where u has values in a Banach space, like an LP space, A is an autonomous (time-independent) operator and C is an asymptotically small perturbation, so that C(u(t), t) ~ ° as t ~ 00 along orbits {u(t)} of the evolution in a sense to be made precise, which in practice can be quite weak. We work in a situation in which the autonomous (limit) differential equation (ADE) Ut = A(u) (2) has a well-known asymptotic behaviour, and we want to prove that for large times the orbits of the original evolution problem converge to a certain class of limits of the autonomous equation. More precisely, we want to prove that the orbits of (NDE) are attracted by a certain limit set [2* of (ADE), which may consist of equilibria of the autonomous equation, or it can be a more complicated object.

Keywords

Navier-Stokes equation continuum mechanics differential equation fluid dynamics functional analysis nonlinear analysis ordinary differential equation partial differential equation pdes

Authors and affiliations

  • Victor A. Galaktionov
    • 1
    • 2
  • Juan Luis Vázquez
    • 3
  1. 1.Department of Mathematical SciencesUniversity of BathBathUK
  2. 2.Keldysh Institute of Applied MathematicsMoscowRussia
  3. 3.Department of MathematicsUniversidad Autónoma de MadridMadridSpain

Bibliographic information