Dynamics of Partial Differential Equations

Volume 18 (2021)

Number 4

Observable measures in partial differential equations

Pages: 279 – 292

DOI:  https://dx.doi.org/10.4310/DPDE.2021.v18.n4.a2

Authors

M. Dong (Department of Mathematics, College of Science, Yanbian University, Yanji City, Jilin Province, China)

W. Jung (Healthcare Datascience Center, Konyang University Hospital, Daejeon, South Korea)

C. A. Morales (Instituto de Matemática, Universidade Federal do Rio de Janeiro, Brazil)

Abstract

We prove that a broad class of PDE’s including the reaction-diffusion and 2D Navier–Stokes equations have observable measures. Moreover, these measures form the minimal weak* compact subset of Borel probability measures whose basins of attraction has total Gaussian measure. To prove these results we extend the theory of observable measures [8] from continuous maps on compact manifolds to dissipative semiflows on Polish spaces.

Keywords

SRB measure, observable measure, reaction-diffusion, Navier–Stokes, dissipative semiflow

2010 Mathematics Subject Classification

Primary 37L30. Secondary 37L40.

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Received 17 February 2020

Published 2 December 2021