Bedrossian Jacob, Punshon-Smith Sam
Department of Mathematics, University of California, Los Angeles, CA 90095 USA.
Department of Mathematics, Tulane University, New Orleans, LA 70118 USA.
Commun Math Phys. 2024;405(4):107. doi: 10.1007/s00220-024-04949-0. Epub 2024 Apr 16.
We prove that all Galerkin truncations of the 2d stochastic Navier-Stokes equations in vorticity form on any rectangular torus subjected to hypoelliptic, additive stochastic forcing are chaotic at sufficiently small viscosity, provided the frequency truncation satisfies . By "chaotic" we mean having a strictly positive Lyapunov exponent, i.e. almost-sure asymptotic exponential growth of the derivative with respect to generic initial conditions. A sufficient condition for such results was derived in previous joint work with Alex Blumenthal which reduces the question to the non-degeneracy of a matrix Lie algebra implying Hörmander's condition for the Markov process lifted to the sphere bundle (projective hypoellipticity). The purpose of this work is to reformulate this condition to be more amenable for Galerkin truncations of PDEs and then to verify this condition using (a) a reduction to genericity properties of a diagonal sub-algebra inspired by the root space decomposition of semi-simple Lie algebras and (b) computational algebraic geometry executed by Maple in exact rational arithmetic. Note that even though we use a computer assisted proof, the result is valid for all aspect ratios and all sufficiently high dimensional truncations; in fact, certain steps simplify in the formal infinite dimensional limit.
我们证明，在受亚椭圆加性随机强迫作用的任何矩形环面上，二维涡度形式的随机纳维 - 斯托克斯方程的所有伽辽金截断在足够小的粘性下都是混沌的，前提是频率截断满足 。这里的“混沌”是指具有严格正的李雅普诺夫指数，即对于一般初始条件，导数几乎必然渐近指数增长。在之前与亚历克斯·布卢门撒尔的合作工作中得到了此类结果的一个充分条件，该条件将问题归结为一个矩阵李代数的非退化性，这意味着提升到球丛上的马尔可夫过程满足霍尔曼德条件（射影亚椭圆性）。这项工作的目的是重新表述这个条件，使其更适合偏微分方程的伽辽金截断，然后使用（a）受半单李代数根空间分解启发的对角子代数一般性性质的约化，以及（b）由Maple在精确有理算术下执行的计算代数几何来验证这个条件。请注意，尽管我们使用了计算机辅助证明，但该结果对所有纵横比和所有足够高维的截断都是有效的；实际上，在形式无限维极限下某些步骤会简化。
