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Andrey Rekalo
Andrey Rekalo
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Finding good lower bounds of the Hausdorff dimension of the attractors to dissipative dynamical systems is a very difficult problem.

In the early 1970s Arnold explicitly stated it as an important open problem in case of the 2D Navier-Stokes equations in a bounded domain (see the compendium of Arnold's problems); in various forms the question probably goes back to Kolmogorov's seminar in the 1950s. If the atrractor's dimension can be shown to grow indefinitely withalong the Reynolds number Re, this can be interpreted as a manifestation of the turbulence of the fluid flow.

The upper estimates of attractors' Hausdorff dimension are much easier. For instance, the best known result for the 2D Navier-Stokes equations with the Dirichlet boundary conditions in the domain $\Omega$ says that

$$\dim_H {\rm Attr}\leq \frac{|\Omega|}{\pi \nu^2}\ \|f\|_{L^2}$$

(here $\nu$ is the viscosity and $f$ is the external force). There are no satisfactory lower bounds of the Hausdorff dimension in this case. The situation is slightly better in case of the Navier-Stokes equations on a torus: good lower bounds are known but only for very specific forces $f$.

Finding good lower bounds of the Hausdorff dimension of the attractors to dissipative dynamical systems is a very difficult problem.

In the early 1970s Arnold explicitly stated it as an important open problem in case of the 2D Navier-Stokes equations in a bounded domain (see the compendium of Arnold's problems); in various forms the question probably goes back to Kolmogorov's seminar in the 1950s. If the atrractor's dimension can be shown to grow indefinitely with the Reynolds number Re, this can be interpreted as a manifestation of the turbulence of the fluid flow.

The upper estimates of attractors' Hausdorff dimension are much easier. For instance, the best known result for the 2D Navier-Stokes equations with the Dirichlet boundary conditions in the domain $\Omega$ says that

$$\dim_H {\rm Attr}\leq \frac{|\Omega|}{\pi \nu^2}\ \|f\|_{L^2}$$

(here $\nu$ is the viscosity and $f$ is the external force). There are no satisfactory lower bounds of the Hausdorff dimension in this case. The situation is slightly better in case of the Navier-Stokes equations on a torus: good lower bounds are known but only for very specific forces $f$.

Finding good lower bounds of the Hausdorff dimension of the attractors to dissipative dynamical systems is a very difficult problem.

In the early 1970s Arnold explicitly stated it as an important open problem in case of the 2D Navier-Stokes equations in a bounded domain (see the compendium of Arnold's problems); in various forms the question probably goes back to Kolmogorov's seminar in the 1950s. If atrractor's dimension can be shown to grow indefinitely along the Reynolds number Re, this can be interpreted as a manifestation of the turbulence of the fluid flow.

The upper estimates of attractors' Hausdorff dimension are much easier. For instance, the best known result for the 2D Navier-Stokes equations with the Dirichlet boundary conditions in the domain $\Omega$ says that

$$\dim_H {\rm Attr}\leq \frac{|\Omega|}{\pi \nu^2}\ \|f\|_{L^2}$$

(here $\nu$ is the viscosity and $f$ is the external force). There are no satisfactory lower bounds of the Hausdorff dimension in this case. The situation is slightly better in case of the Navier-Stokes equations on a torus: good lower bounds are known but only for very specific forces $f$.

Source Link
Andrey Rekalo
Andrey Rekalo
  • 22.3k
  • 12
  • 89
  • 122

Finding good lower bounds of the Hausdorff dimension of the attractors to dissipative dynamical systems is a very difficult problem.

In the early 1970s Arnold explicitly stated it as an important open problem in case of the 2D Navier-Stokes equations in a bounded domain (see the compendium of Arnold's problems); in various forms the question probably goes back to Kolmogorov's seminar in the 1950s. If the atrractor's dimension can be shown to grow indefinitely with the Reynolds number Re, this can be interpreted as a manifestation of the turbulence of the fluid flow.

The upper estimates of attractors' Hausdorff dimension are much easier. For instance, the best known result for the 2D Navier-Stokes equations with the Dirichlet boundary conditions in the domain $\Omega$ says that

$$\dim_H {\rm Attr}\leq \frac{|\Omega|}{\pi \nu^2}\ \|f\|_{L^2}$$

(here $\nu$ is the viscosity and $f$ is the external force). There are no satisfactory lower bounds of the Hausdorff dimension in this case. The situation is slightly better in case of the Navier-Stokes equations on a torus: good lower bounds are known but only for very specific forces $f$.