Abstract
This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness of the local (in time) solution, which corresponds to an addapted stochastic process with sample paths defined up to a certain positive stopping time, with values in the Sobolev space \(H^3\). Our approach combines a cut-off approximation scheme, a stochastic compactness arguments and a general version of Yamada–Watanabe theorem. This leads to the existence of a local strong pathwise solution.
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Notes
\({\mathcal {P}}_{T}:=\sigma (\{ ]s,t]\times F_s \vert 0\le s < t \le T,F_s\in {\mathcal {F}}_s \} \cup \{\{0\}\times F_0 \vert F_0\in {\mathcal {F}}_0 \})\) (see [23, p. 33]). Then, a process defined on \(\Omega _T\) with values in a given space X is predictable if it is \({\mathcal {P}}_{T}\)-measurable.
Note that the same can be reproduced with: \(\displaystyle \sum _{{\textbf{k}}\ge 1} \sigma _{\textbf{k}}^2(t,0)< \infty \).
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Acknowledgements
This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications). The authors would like also to thank the anonymous referees for their comments and help in improving our work.
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Tahraoui, Y., Cipriano, F. Local strong solutions to the stochastic third grade fluid equations with Navier boundary conditions. Stoch PDE: Anal Comp (2023). https://doi.org/10.1007/s40072-023-00314-9
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DOI: https://doi.org/10.1007/s40072-023-00314-9