2019/01/17 by Marc Briane, Briane, Marc
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1901.09675
openalex publication_date 2019/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper is devoted to the isotropic realizability of a regular gradient field u or a more general vector field b, namely the existence of a continuous positive function σ such that σb is divergence free in R d or in an open set of R d. First, we prove that under some suitable positivity condition satisfied by u, the isotropic realizability of u holds either in R d if u does not vanish, or in the open sets c j < u < c j+1 if the c j are the critical values of u (including inf R d u and sup R d u) which are assumed to be in finite number. It turns out that this positivity condition is not sufficient to ensure the existence of a continuous positive invariant measure σ on the torus when u is periodic. Then, we establish a new criterium of the existence of an invariant measure for the flow associated with a regular periodic vector field b, which is based on the equality b × v = 1 in R d. We show that this gradient invertibility is not related to the classical ergodic assumption, but it actually appears as an alternative to get the asymptotics of the flow.