{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:36:35Z","timestamp":1787236595073,"version":"build-2736575974"},"reference-count":21,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Dyn. Syst."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>The paper is devoted to the isotropic realizability of a regular electric field $\\nabla u$ 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 $\\nabla u$, the isotropic realizability of $\\nabla u$ holds either in $\\mathbb{R}^d$ if $\\nabla u$ does not vanish, or in the open sets $\\{c_j&lt;u&lt;c_{j+1}\\}$ if the $c_j$ are the critical values of $u$ (including $\\inf_{\\mathbb{R}^d}u$ and $\\sup_{\\mathbb{R}^d}u$) 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 $\\sigma$ on the torus when $\\nabla u$ 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 $b\\cdot\\nabla v=1$ in $\\mathbb{R}^d$. 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.<\/jats:p>","DOI":"10.1137\/19m1240411","type":"journal-article","created":{"date-parts":[[2019,10,17]],"date-time":"2019-10-17T12:17:28Z","timestamp":1571314648000},"page":"1846-1866","source":"Crossref","is-referenced-by-count":5,"title":["Isotropic Realizability of Fields and Reconstruction of Invariant Measures under Positivity Properties. Asymptotics of the Flow by a Non-Ergodic Approach"],"prefix":"10.1137","volume":"18","author":[{"given":"Marc","family":"Briane","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,10,17]]},"reference":[{"key":"atypb1","unstructured":"G.S. Alberti and Y. Capdeboscq,\n                      Lectures on elliptic methods for hybrid inverse problems\n                      , Specialized Courses 25, Soci\u00e9t\u00e9 Math\u00e9matique de France, Paris, 2018, 230 pp."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01790543"},{"key":"atypb3","first-page":"325","author":"Bal G.","year":"2013","journal-title":"Cambridge"},{"key":"atypb4","first-page":"116","author":"Bongiorno F.","year":"1977","journal-title":"Publ. I. A. C. III"},{"key":"atypb5","first-page":"119","author":"Brenier Y.","year":"1991","journal-title":"Lund"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2014.19.353"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2018013"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/140989121"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2013109"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"I.P. Cornfeld, S.V. Fomin, and Ya.G. Sinai,\n                      Ergodic Theory\n                      , Grundlehren Math. Wiss. 245, Springer-Verlag, Berlin, 1982.","DOI":"10.1007\/978-1-4615-6927-5"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"B. Dacorogna,\n                      Direct Methods in the Calculus of Variations\n                      , Appl. Math. Sci. 78, Springer-Verlag, Berlin, 1989.","DOI":"10.1007\/978-3-642-51440-1"},{"key":"atypb12","unstructured":"M.W. Hirsch, S. Smale, and R.L. Devaney,\n                      Differential Equations, Dynamical Systems, and an Introduction to Chaos\n                      , 2nd ed., Pure Appl. Math. 60, Elsevier Academic Press, Amsterdam, 2004."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/0152003"},{"key":"atypb14","first-page":"183","author":"Kuchment P.","year":"2012","journal-title":"Berlin"},{"key":"atypb15","first-page":"719","author":"Milton G.W.","year":"2002","journal-title":"Cambridge"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1017\/S014338570200144X"},{"key":"atypb17","unstructured":"M. Reed and B. Simon:\n                      \n                        Methods of Modern Mathematical Physics.\n                        I.\n                        Functional Analysis\n                      \n                      , Academic Press, New York, 1980."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/0141016"},{"key":"atypb19","unstructured":"Ya.G. Sinai,\n                      Introduction to Ergodic Theory\n                      , Math. Notes 18, Princeton University Press, Princeton, N.J., 1976."},{"key":"atypb20","first-page":"925","author":"Tartar L.","year":"1989","journal-title":"MA"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139996299820"}],"container-title":["SIAM Journal on Applied Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/19M1240411","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T13:55:24Z","timestamp":1787234124000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/19M1240411"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/19M1240411"],"URL":"https:\/\/doi.org\/10.1137\/19m1240411","relation":{},"ISSN":["1536-0040"],"issn-type":[{"value":"1536-0040","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}