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Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

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  • Mohan, Manil T.

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.

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  • Mohan, Manil T., 2020. "Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4513-4562.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:4513-4562
    DOI: 10.1016/j.spa.2020.01.007
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    References listed on IDEAS

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    1. Röckner, Michael & Wang, Feng-Yu & Wu, Liming, 2006. "Large deviations for stochastic generalized porous media equations," Stochastic Processes and their Applications, Elsevier, vol. 116(12), pages 1677-1689, December.
    2. Sritharan, S.S. & Sundar, P., 2006. "Large deviations for the two-dimensional Navier-Stokes equations with multiplicative noise," Stochastic Processes and their Applications, Elsevier, vol. 116(11), pages 1636-1659, November.
    3. Manil T. Mohan & K. Sakthivel & Sivaguru S. Sritharan, 2019. "Ergodicity for the 3D stochastic Navier–Stokes equations perturbed by Lévy noise," Mathematische Nachrichten, Wiley Blackwell, vol. 292(5), pages 1056-1088, May.
    4. Gourcy, Mathieu, 2007. "A large deviation principle for 2D stochastic Navier-Stokes equation," Stochastic Processes and their Applications, Elsevier, vol. 117(7), pages 904-927, July.
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