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Attractors for the Bénard problem: Existence and physical bounds on their fractal dimension. (English) Zbl 0646.76098
This article is divided into two parts. In the first one we prove the existence of attractors or functional invariant sets describing the long- time behavior (or “permanent regime”) of the solutions of the convection problems, while the second one is devoted to the estimate of the fractal and Hausdorff dimensions of these attractors in terms of natural physical quantities. Part I starts in Section 1 by recalling the convection equations and their nondimensional form. In Section 2 we recall the functional setting of the equations and the basic properties of the solution (Sections 2.1 and 2.2); in Section 2.3 we describe other natural boundary conditions to which our results apply as well. In Section 3 we study the two-dimensional case for which we prove the existence of a maximal (universal) attractor, and discuss the three- dimensional case. Part II is devoted to the estimate of the fractal and Hausdorff dimensions of the attractors. We start in Section 4 with a survey of definitions and results on the Lyapunov exponents and the fractal dimension or capacity of the attractors. Then we proceed in Section 5 with the actual estimate of the dimensions of the attractors for the two-dimensional convection; these estimates rely on the previous results and on a version of the Lieb-Thirring inequality. The original Lieb-Thirring inequality and its extension presented hear appear as an improved form of the classical Sobolev imbedding theorems. Finally, in Section 6, the three-dimensional case is treated.
76R05Forced convection (fluid mechanics)
58J65Diffusion processes and stochastic analysis on manifolds