Energy transfer in the Kuramoto-Sivashinskii equation on a multidimensional torus with Riemannian metric期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Energy transfer in the Kuramoto-Sivashinskii equation on a multidimensional torus with Riemannian metric

Authors:	A M Arkhipov

Abstract:	In this paper, we consider the Kuramoto-Sivashinskii equation on the multidimensional torus with a Riemannian metric: $u_t = - (P(\nabla u,\nabla u) + \Delta u + \nu \Delta ^2 u),\bar u = 0,x \in T^n ,$ where $\bar u = \frac{1}{{volT^n }}\int\limits_{T^n } {ud\mu } ,Pu = u - \bar u,\nu > 0$ . For this equation the theorem on energy transfer holds. It is formulated in the following way. Let $\sum {a_k \xi _k }$ be the Fourier expansion of the function u. Denote by P _N and P _N ^⊥ the operators of rejection of the “leading” and, respectively, “lowest” terms of the Fourier expansion (harmonics), i.e., $P_N u = \sum\limits_1^N {a_k \xi _k } ,P_N^ \bot = u - P_N u$ . For any ρ > 0,N ∈ ℕ, s ≥ n/2+5, and λ ∈ (0, 1), there exists R such that for any function. ϕ ∈ $\bar C^\infty (T^n )$ lying outside the ball $n_{C^1 } \leqslant R$ in the ball $Q = \{ n_s \leqslant \rho \left\\| \varphi \right\\|_{C^1 } \}$ , there exists an instant of time t ∈ (0, 1) such that g _KS ^t ϕ=ψ and $\left\\| {P_N^ \bot \psi } \right\\|_s^2 \geqslant \lambda \left\\| \psi \right\\|_s^2$ . Here, R is a constant depending on the metric (g), n _s is the sth Sobolev norm, and $n_{C^1 }$ is the C ¹-norm. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 24, Dynamical Systems and Optimization, 2005.

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