Invariant measures for a class of stochastic third-grade fluid equations in 2D and 3D bounded domains. (English) Zbl 07931661
Summary: This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third-grade fluid equations in bounded domain \(D\subset \mathbb{R}^d, d=2,3\), in the presence of a multiplicative noise. First, we show the existence of a martingale solution by coupling a stochastic compactness and monotonicity arguments. Then, we prove a stability result, which gives the pathwise uniqueness of the solution and therefore the existence of strong probabilistic solution. Secondly, we use the stability result to show that the associated semigroup is Feller and by using “Krylov-Bogoliubov Theorem,” we get the existence of an invariant probability measure. Finally, we show that all the invariant measures are concentrated on a compact subset of \(L^2\), which leads to the existence of an ergodic invariant measure.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 37L40 | Invariant measures for infinite-dimensional dissipative dynamical systems |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60J25 | Continuous-time Markov processes on general state spaces |
| 76A05 | Non-Newtonian fluids |
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