Finite Dimensional Uniform Attractors for the Nonautonomous Camassa-Holm Equations
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چکیده
and Applied Analysis 3 and in H1 Ω , respectively, observe that H⊥, the orthogonal complement of H in L2 Ω , is {∇p : p ∈ H1 Ω } cf. 11 or 12 . ii We denote P : L2 Ω 3 → H the L2 orthogonal projection, usually referred as Helmholtz-Leray projector, and by A −PΔ the Stokes operator with domain D A H2 Ω 3 ∩ V . Notice that in the case of periodic boundary condition, A −Δ|D A is a selfadjoint positive operator with compact inverse. Hence the spaceH has an orthonormal basis {wj}j 1 of eigenfunctions of A, that is, Awj λjwj , with 0 < λ1 ≤ λ2 ≤ · · · ≤ λj ≤ · · · , λj −→ ∞, as j −→ ∞, 2.2 in fact, these eigenvalues have the form |k|4π/L2 with k ∈ Z3 \ {0}. iii We denote ·, · the L2-inner product and by | · | the corresponding L2-norm. By virtue of Poincaré inequality, one can show that there is a constant c > 0 such that c|Aw| ≤ ‖w‖H2 ≤ c−1|Aw| for every w ∈ D A 2.3
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تاریخ انتشار 2009