# On the spectrum of the stokes operator

@article{Ilyin2008OnTS, title={On the spectrum of the stokes operator}, author={Alexei A. Ilyin}, journal={Functional Analysis and Its Applications}, year={2008}, volume={43}, pages={254-263} }

We prove Li-Yau type lower bounds for the eigenvalues of the Stokes operator and give applications to the attractors of the Navier-Stokes equations.

## 35 Citations

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term for the eigenvalues of the Stokes operator and
improve the previously known estimates for the Laplace
operator. Generalizations to…

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### Upper bounds for the attractor dimension of damped Navier-Stokes equations in $\mathbb R^2$

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We consider finite energy solutions for the damped and driven two-dimensional Navier--Stokes equations in the plane and show that the corresponding dynamical system possesses a global attractor. We…

### On the Convergence Rate of Spectral Approximations for the Equations of Nonhomogeneous Incompressible Fluids

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We discuss the estimates for the L-norms of systems of functions that are orthonormal in L and H, respectively, and their essential role in deriving good or even optimal bounds for the dimension of…

### On the convergence of spectral approximations for the heat convection equations

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In this paper, we focus on the convergence rate of solutions of spectral Galerkin approximations for the heat convection equations on a bounded domain. Estimates in $$H^2$$H2-norm for velocity and…

### On the Convergence of Galerkin Spectral Methods for a Bioconvective Flow

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Convergence rates of the spectral Galerkin method are obtained for a system consisting of the Navier-Stokes equation coupled with a linear convection-diffusion equation modeling the concentration of…

## 34 References

### New bounds on the Lieb-Thirring constants

- Mathematics
- 2000

Abstract.Improved estimates on the constants Lγ,d, for 1/2<γ<3/2, d∈N, in the inequalities for the eigenvalue moments of Schrödinger operators are established.

### Lieb-Thirring integral inequalities and their applications to attractors of the Navier-Stokes equations

- Mathematics
- 2005

Integral inequalities of Lieb-Thirring type and their generalizations are proved. All the corresponding constants are given in explicit form. Special attention is devoted to applications to the…

### Dirichlet and Neumann Eigenvalue Problems on Domains in Euclidean Spaces

- Mathematics
- 1997

Abstract We obtain here some inequalities for the eigenvalues of Dirichlet and Neumann value problems for general classes of operators (or system of operators) acting in L 2 ( Ω ) (or L 2 ( Ω , C m…

### Lieb-Thirring inequalities with improved constants

- Mathematics
- 2007

Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allows us to improve on the known estimates of best constants in Lieb-Thirring…

### On the fractal dimension of invariant sets; applications to Navier-Stokes equations.

- Mathematics
- 2003

A semigroup $S_t$ of continuous operators in a Hilbert space
$H$ is considered. It is shown that the fractal dimension
of a compact strictly invariant set
$X$ ($X\subset H, S_tX=X$)
admits the…

### Infinite dimensional dynamical systems

- Mathematics
- 1983

Abstract : An approach is outlined for the discussion of the qualitative theory of infinite dimensional dynamical systems. Retarded functional differential equations are used to illustrate the…

### Navier-Stokes equations

- Mathematics, Physics
- 1992

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in…

### On characteristic exponents in turbulence

- Mathematics
- 1984

Ruelle has found upper bounds to the magnitude and to the number of non-negative characteristic exponents for the Navier-Stokes flow of an incompressible fluid in a domain Θ. The latter is…