发表机构
EPFL（洛桑联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明二维环面上白噪声时间流驱动的被动标量，在存在两个非平行受迫模态时，其L2耗散率一致有界；否则耗散率可任意快但最大李雅普诺夫指数下有界，方法基于弱化条件的有限模态投影。
AI 中文摘要
我们研究了二维环面上由任意有限集白噪声时间无散度傅里叶模态驱动的被动标量。我们证明，只要存在两个非平行的受迫模态，当扩散率趋于零时，对流扩散方程的指数$L^2$耗散率一致有界。若不存在这样的模态，则耗散率可依赖于初始数据而任意快，但最大李雅普诺夫指数仍保持一致下有界。证明使用了依赖于受迫集合的有限模态投影。我们论证的一部分遵循了Chemnitz和Chemnitz研究的四模态情形结构，但使用了比他们关于Frobenius范数与算子范数之间的矩阵不等式严格更弱的条件。结合可达性论证，这使我们能够考虑任意受迫模态集合和初始数据。
英文摘要
We study a passive scalar on the two-dimensional torus driven by an arbitrary finite set of white-in-time divergence-free Fourier modes. We prove that the exponential $L^2$ dissipation rate of the advection-diffusion equation is uniformly bounded as the diffusivity tends to zero as long as there exist two nonparallel forced modes. If not, then the dissipation rate can be arbitrarily fast depending on the initial data, but the top Lyapunov exponent remains uniformly bounded below. The proof uses a projection onto a finite set of modes which depends on the forcing set. Part of our argument follows the structure of the four-mode case investigated by Chemnitz and Chemnitz, but uses a strictly weaker condition than their matrix inequality between Frobenius and operator norms. Combined with an accessibility argument, this allows us to consider an arbitrary set of forcing modes and initial data.
