论文标题
关于lyapunov指数正规化线性演化方程的规范等效性
On the norm equivalence of Lyapunov exponents for regularizing linear evolution equations
论文作者
论文摘要
我们考虑与可分离的希尔伯特或巴纳克空间上的耗散线性进化方程相关的顶级lyapunov指数。在部分偏微分方程中的许多应用中，这种方程通常是在缓解的非量式空间的规模上提出的，例如集成性（$ l^p $）或可怜性（$ w^{s，p} $）。与有限的维度相反，Lyapunov指数可能取决于所使用的规范的选择。在本文中，我们表明，在相当一般的条件下，紧凑型线性算子的cocycle的Lyapunov指数与所使用的规范无关。我们将此结果应用于流体力学中的两个重要问题：对流扩散方程的增强耗散速率，具有沿阵贡速度场；以及具有随机或周期性强迫的2D Navier-Stokes方程的Lyapunov指数。
We consider the top Lyapunov exponent associated to a dissipative linear evolution equation posed on a separable Hilbert or Banach space. In many applications in partial differential equations, such equations are often posed on a scale of nonequivalent spaces mitigating, e.g., integrability ($L^p$) or differentiability ($W^{s, p}$). In contrast to finite dimensions, the Lyapunov exponent could apriori depend on the choice of norm used. In this paper we show that under quite general conditions, the Lyapunov exponent of a cocycle of compact linear operators is independent of the norm used. We apply this result to two important problems from fluid mechanics: the enhanced dissipation rate for the advection diffusion equation with ergodic velocity field; and the Lyapunov exponent for the 2d Navier-Stokes equations with stochastic or periodic forcing.
