I have heard and seen many times in informal discussions the term "dynamics" of certain PDE. For instance, "the long time dynamics of the 2D NSE is
entirely contained in the global attractor". And even in the title of a journal:
What does "dynamics" mean in such a context? Does it have a formal definition in any references?
As a PhD candidate in Dynamical Systems, I would interpret this as a discussion about the more qualitative aspects of the PDE. Example topics (that would fall under a discussion of the dynamical system generated by a PDE acting on the space of possible initial conditions) might include
Are there any steady-state solutions? Periodic solutions?
Do these steady-state/periodic solutions attract or repel nearby initial conditions?
What is the asymptotic behavior of solutions as $t \to \pm \infty$?
Is there any sensitive dependence on initial conditions? I.e., do initial conditions that are "close" drift apart or converge? How quickly does this happen--linearly/exponentially / ... ?