2016/12/22 by Wolf‐Jürgen Beyn, Beyn, Wolf-Jürgen, Denny Otten +1
Computer Science · Mathematics · #35B40 #35K57 (Primary) #35Pxx #35Q56 #47A55 #47N40 (Secondary) #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1612.07535
openalex publication_date 2016/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators Lv(x)=A\triangle v(x)+⟨ Sx,∇ v(x)⟩+Df(v⋆(x))v(x), x∈ℝd, d\geqslant 2 where v⋆:ℝd→ℝm is a rotating wave profile with v⋆(x)→ v∞∈ℝm as |x|→∞, f:ℝm→ℝm is smooth, A∈ℝm,m has eigenvalues with positive real parts and commutes with the limit matrix Df(v∞). The matrix S∈ℝd,d is assumed to be skew-symmetric with eigenvalues (λ1,…,λd)=(± iσ1,…,± i σk,0,…,0). The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction diffusion systems. We prove under suitable conditions that every λ∈ℂ satisfying the dispersion relation det(λIm + η2 A - Df(v∞) + i⟨ n,σ⟩ Im)=0\quadfor some η∈ℝ and n∈ℤk belongs to the essential spectrum σess(L) in Lp. For values Re λ to the right of the spectral bound for Df(v∞) we show that the operator λI-L is Fredholm of index 0, solve the identification problem for the adjoint operator (λI-L)^*, and formulate the Fredholm alternative. Moreover, we show that the set σ(S)∪\λi+λj: λi,λj∈σ(S), 1\leqslant i