Abstract
We establish the existence and uniqueness of solutions to stochastic Two-Dimensional Navier–Stokes equations in a time-dependent domain driven by Brownian motion. A martingale solution is constructed through domain transformation and appropriate finite-dimensional approximations on time-dependent spaces. The probabilistic strong solution follows from the pathwise uniqueness and the Yamada–Watanabe theorem. Because the state space of the solution changes with time, we need to deal with the various problems caused by the lack of appropriate chain rules/Itô’s formula, apart from the nonlinearity of the Navier–Stokes equation.
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Acknowledgements
This work is partially supported by NSFC (Nos. 12131019, 11971456, 11721101) and School Start-up Fund (USTC) KY0010000036 and Fundamental Research Funds for the Central Universities WK3470000016.
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Wang, W., Zhai, J. & Zhang, T. Stochastic Two-Dimensional Navier–Stokes Equations on Time-Dependent Domains. J Theor Probab 35, 2916–2939 (2022). https://doi.org/10.1007/s10959-021-01150-0
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DOI: https://doi.org/10.1007/s10959-021-01150-0