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

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 ^t _KS 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.