Isotropic Realizability of Fields and Reconstruction of Invariant Measures under Positivity Properties. Asymptotics of the Flow by a Non-Ergodic Approach
Marc Briane · SIAM Journal on Applied Dynamical Systems · 2019
The paper is devoted to the isotropic realizability of a regular electric field $ abla u$ or a more general vector field $b$, namely, the existence of a continuous positive function $\sigma$ such that $\sigma b$ is divergence free in $\mathbb{R}^d$ or in an open set of $\mathbb{R}^d$. First, we prove that under some suitable positivity condition satisfied by $ abla u$, the isotropic realizability of $ abla u$ holds either in $\mathbb{R}^d$ if $ abla u$ does not vanish, or in the open sets $\{c_j