In many cases of interest a nonlinear evolution partial differential equation can be written as an infinite-dimensional dynamical system $$ du/dt+A(t)u=0 $$ on a suitable functional space $X$, where $A(t)$ is a nonlinear operator, cf. e.g. the classical paper of Kato Nonlinear semigroups and evolution equations.

I would greatly appreciate references to recent surveys or books (especially those aimed at nonexperts as much as possible) of the results on nonlinear evolution PDEs obtained in this framework including the case when the original evolution PDEs are of order greater than two.

The only relatively recent reference I found so far is the survey Partial Differential Equations in the 20th Century by Brezis and Browder but there this topic is only cursorily mentioned. Perhaps I use wrong keywords for googling or something :(

Many thanks in advance for your help.

  • $\begingroup$ Infinite dimensional dynamical systems by Robinson. $\endgroup$ – Piyush Grover Nov 28 '16 at 5:53
  • $\begingroup$ @PiyushGrover: Thanks. However it does not seem to cover the systems of order greater than two. $\endgroup$ – just-someone Nov 28 '16 at 6:31
  • $\begingroup$ The book "Dynamics of Evolutionary Equations" by Sell and You is a nice introduction to the formulation of certain PDE as evolution equations. They work out many examples, including, as I recall, 3rd order (KdV) and 4th order examples. $\endgroup$ – A Blumenthal Jan 7 '17 at 19:45

Writing this as an answer since it seems to constitute one...

The book "Dynamics of Evolutionary Equations" by Sell and You provides several examples of evolution equations for third and fourth order problems. A perhaps harder and more general version of what is essentially the same theory is given in the book of Henry, "Geometric Theory of Semilinear Parabolic Equations".

These books study evolution equations of the form $$ u_t = A u + F(u, t) $$ in a Banach space $X$, where $A$ is a sectorial operator on $X$, roughly, a closed (unbounded) operator whose spectrum can be contained in a cone about the real axis. This class includes self-adjoint and skew-adjoint operators on Hilbert spaces.

This formulation is appealing to dynamicists because it provides a common framework for a large class of PDE. Typical applications include criteria for the existence of global (compact) attractors and bounding the dimension of these attractors, for which there is a considerable literature nowadays.


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