Necessary and sufficient conditions for the existence of global attractors for semigroups and applications

First we establish some necessary and suYcient con- ditions for the existence of the global attractor of an infinite dimensional dynamical system, using the measure of noncom- pactness. Then we give a new method/recipe for proving the existence of the global attractor. The main advantage of this new method/recipe is that one needs only to verify a necessary compactness condition with the same type of energy estimates as those for establishing the absorbing set. In other words, one doesn't need to obtain estimates in function spaces of higher regu- larity. In particular, this property is useful when higher regularity is not available, as demonstrated in the example on the Navier- Stokes equations on nonsmooth domains.

[1]  C. Foiaș,et al.  Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$ , 1967 .

[2]  Zhongwei Shen,et al.  On the dimension of the attractor for the non-homogeneous Navier-Stokes equations in non-smooth domains , 2000 .

[3]  K. Deimling.  Nonlinear functional analysis , 1985 .

[4]  Xiaoming Wang,et al.  An energy equation for the weakly damped driven nonlinear Schro¨dinger equations and its application to their attractors , 1995 .

[5]  R. Temam.  Navier-Stokes Equations , 1977 .

[6]  R. Temam,et al.  Modelling of the interaction of small and large eddies in two dimensional turbulent flows , 1988 .

[7]  Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbb{R}^2$ , 1996 .

[8]  M. Ziane.  Optimal bounds on the dimension of the attractor of the Navier-Stokes equations , 1997 .

[9]  R. Temam,et al.  Asymptotic analysis of the navier-stokes equations , 1983 .

[10]  Regularity of the attractor for Schrödinger equation , 1997 .

[11]  R. Temam,et al.  On the Hausdorff dimension of an attractor for the two-dimensional Navier-Stokes equations , 1983 .

[12]  Peter Constantin,et al.  Global Lyapunov Exponents, Kaplan-Yorke Formulas and the Dimension of the Attractors for 2D Navier-Stokes Equations , 1985 .

[13]  Tomasz W. Dłotko,et al.  Global Attractors in Abstract Parabolic Problems , 2000 .

[14]  J. Hale.  Asymptotic Behavior of Dissipative Systems , 1988 .

[15]  John D. Gibbon,et al.  Attractor dimension and small length scale estimates for the three-dimensional Navier - Stokes equations , 1997 .

[16]  M. Marion.  Attractors for reaction-diffusion equations: existence and estimate of their dimension , 1987 .