scholarly journals Cone invariance and squeezing properties for inertial manifolds for nonautonomous evolution equations

2003 ◽  
Author(s):  
Norbert Koksch ◽  
Stefan Siegmund
Keyword(s):  
Evolution Equations ◽  
Inertial Manifolds ◽  
Cone Invariance

scholarly journals Reduction of dimension of approximate intertial manifolds by symmetry

1999 ◽  
Vol 60 (2) ◽  
pp. 319-330
Author(s):  
Anibal Rodriguez-Bernal ◽  
Bixiang Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Inertial Manifolds ◽  
Inertial Manifold ◽  
Partial Differential ◽  
Approximate Inertial Manifold ◽  
Approximate Inertial Manifolds ◽  
The Relationship

In this paper, we study approximate inertial manifolds for nonlinear evolution partial differential equations which possess symmetry. The relationship between symmetry and dimensions of approximate inertial manifolds is established. We demonstrate that symmetry can reduce the dimensions of an approximate inertial manifold. Applications for concrete evolution equations are given.

scholarly journals Approximate inertial manifolds for nonlinear parabolic equations via steady-state determining mapping

1995 ◽  
Vol 18 (1) ◽  
pp. 1-24 ◽  
Author(s):  
Yuncheng You
Keyword(s):  
Steady State ◽  
Evolution Equations ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Nonlinear Parabolic Equations ◽  
Reaction Diffusion Equations ◽  
Inertial Manifolds ◽  
Axially Symmetric ◽  
Nonlinear Parabolic ◽  
Approximate Inertial Manifolds

For nonlinear parabolic evolution equations, it is proved that, under the assumptions of local Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. It is also shown that the thickness of the exponentially attracting neighborhood of the AIM converges to zero at a fractional power rate as the dimension of the AIM increases. Applications of the obtained results to Burgers' equation, higher dimensional reaction-diffusion equations, 2D Ginzburg-Landau equations, and axially symmetric Kuramoto-Sivashinsky equations in annular domains are included.

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