• Corpus ID: 235489927

Inertial manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions

@inproceedings{Kostianko2021InertialMF,
  title={Inertial manifolds for 3D complex Ginzburg-Landau equations with periodic boundary conditions},
  author={Anna Kostianko and Chunyou Sun and Sergey Zelik},
  year={2021}
}
We prove the existence of an Inertial Manifold for 3D complex Ginzburg-Landau equation with periodic boundary conditions as well as for more general cross-diffusion system assuming that the dispersive exponent is not vanishing. The result is obtained under the assumption that the parameters of the equation is chosen in such a way that the finite-time blow up of smooth solutions does not take place. For the proof of this result we utilize the method of spatio-temporal averaging recently… 
Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces
We study asymptotically compact nonautonomous dynamical systems given by abstract cocycles in Banach spaces. Our main assumptions are given by a squeezing property in a quadratic cone field (given by

References

SHOWING 1-10 OF 47 REFERENCES
Inertial Manifolds for the 3D Cahn-Hilliard Equations with Periodic Boundary Conditions
The existence of an inertial manifold for the 3D Cahn-Hilliard equation with periodic boundary conditions is verified using a proper extension of the so-called spatial averaging principle introduced
Inertial manifolds for 1D reaction-diffusion-advection systems. Part I: Dirichlet and Neumann boundary conditions
This is the first part of our study of inertial manifolds for the system of 1D reaction-diffusion-advection equations which is devoted to the case of Dirichlet or Neumann boundary conditions.
Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg–Landau equation
  • A. Kostianko
  • Mathematics
    Proceedings of the Royal Society A
  • 2020
We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mané projectors) for semilinear parabolic systems with cross diffusion terms and
Inertial manifolds for 1D reaction-diffusion-advection systems. Part II: periodic boundary conditions
This is the second part of our study of the Inertial Manifolds for 1D systems of reaction-diffusion-advection equations initiated in [ 6 ] and it is devoted to the case of periodic boundary
Inertial manifolds for the hyperviscous Navier–Stokes equations
Inertial manifolds and finite-dimensional reduction for dissipative PDEs*
  • S. Zelik
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2014
This paper is devoted to the problem of finite-dimensional reduction for parabolic partial differential equations. We give a detailed exposition of the classical theory of inertial manifolds as well
Inertial Manifolds via Spatial Averaging Revisited
The paper gives a comprehensive study of inertial manifolds for semilinear parabolic equations and their smoothness using the spatial averaging method suggested by G. Sell and J. Mallet-Paret. We
Inertial Manifolds for Certain Subgrid-Scale α-Models of Turbulence
TLDR
It is shown that the existence of an inertial manifold, i.e., a globally invariant, exponentially attracting, finite-dimensional smooth manifold, for two different subgrid-scale α-models of turbu- lence, the simplified Bardina model and the modified Leray-α model, in two-dimensional space implies that the long-time dynamics of these turbulence models is equivalent to that of a finite- dimensional system of ordinary differential equations.
Three counterexamples in the theory of inertial manifolds
AbstractAn example of a dissipative semilinear parabolic equation in a Hilbert space without smooth inertial manifolds is constructed. Moreover, the attractor of this equation can be embedded in no
Multibump, Blow-Up, Self-Similar Solutions of the Complex Ginzburg-Landau Equation
TLDR
This article constructs a branch of solutions that are not perturbations of solutions to the nonlinear Schrodinger equation (NLS); moreover, these axisymmetric ring-like solutions exist over a broader parameter regime than the monotone profile.
...
...