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Well posedness,large deviations and ergodicity of the stochastic 2D Oldroyd model of order one
Institution:1. School of Mathematics and Statistics, Jiangsu Normal University, 221116, Xuzhou, China;2. Fakultät für Mathematik, Universität Bielefeld, D-33501, Bielefeld, Germany;3. Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), 100190, Beijing, China;1. Department of Mathematics and Computer Science, University of Dschang, P.O. BOX 67, Dschang, Cameroon;2. Department of Mathematics, Florida International University, DM413B University Park, Miami, FL 33199, USA;1. School of Mathematics and Statistics, Xi''an Jiaotong University, Xi''an, 710049, PR China;2. Dpto. Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, C/ Tarfia s/n, 41012-Sevilla, Spain;1. Department of Mathematics, Wenzhou University, Wenzhou, Zhejiang Province, 325035, People''s Republic of China;2. Departmento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, c/Tarfia s/n, 41012-Sevilla, Spain
Abstract:In this work, we establish the unique global solvability of the stochastic two dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows perturbed by multiplicative Gaussian noise. A local monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. The Laplace principle for the strong solution of the stochastic system is established in a suitable Polish space using a weak convergence approach. The Wentzell–Freidlin large deviation principle is proved using the well known results of Varadhan and Bryc. The large deviations for shot time are also considered. We also establish the existence of a unique ergodic and strongly mixing invariant measure for the stochastic system with additive Gaussian noise, using the exponential stability of strong solutions.
Keywords:Viscoelastic fluids  Oldroyd fluid  Large deviation principle  Gaussian noise  Invariant measure  Exponential stability
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