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Analytic study of shell models of turbulence. (English) Zbl 1113.34044
The authors study analytically the viscous ‘sabra’ shell model of the energy turbulent cascade ${du_n\over dt}= i(ak_{n+1} u_{n+2} u^*_{n+1}+ bk_n u_{n+1} u^*_{n-1}- ck_{n-1} u_{n-1} u_{n-2})- \nu k^2_n u_n+ f_n,$ respectively the functional form ${du\over dt}+ \nu Au+ B(u, u)= f,\quad u(0)= u^{in}$ in the Hilbert space $$l^2$$ (sequence space over $$\mathbb{C}$$), $$A$$ is a linear and $$B$$ a bilinear suitably defined operator.
They prove by using Galerkin methods the existence of weak solutions, smooth dependence on the initial data and uniqueness, and they give appropriate assumptions for the force $$f$$ (exponential decay) that the solution $$u$$ belongs to a certain Gevrey class of regularity. In the last sections they prove (under certain assumptions) that the shell model has a finite dimensional attractor (they give an estimate for the Hausdorff and the fractal dimension) and they further show that the model possesses a finite dimensional inertial manifold (which is unknown for the Navier-Stokes equation). The authors discuss their assumptions and their results and compare the results with those of the Navier-Stokes equation and other similar problems.

MSC:
 34G20 Nonlinear differential equations in abstract spaces 76D05 Navier-Stokes equations for incompressible viscous fluids 34A45 Theoretical approximation of solutions to ordinary differential equations
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