Attractors for Equations of Mathematical Physics
This book evolves around the notion of the global attractor, its dimension, and various generalisations of these concepts. The concrete equations studied throughout the book include general reaction-diffusion equations, the Navier-Stokes equations and hyperbolic equations of dissipative type.
The first part reviews classical material on the attractors and its dimension in the autonomous setting; several results concerning lower bounds for the dimension are also given. In the second part, these results are generalised to the non-autonomous setting. The concept of ‘process’ as a generalisation of semigroup is introduced, and a notion of Kolmogorov entropy (a generalisation of the fractal dimension) is studied. One chapter covers a lot of material concerning translation compact spaces of functions. Finally, in the third part, the new concept of ‘trajectory attractor’ is introduced in order to study evolutionary problems with possible non-uniqueness: notable examples are the three-dimensional Navier-Stokes system and the wave equation with supercritical non-linearity.