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Existence and Uniqueness of Solutions to Backward 2D and 3D Stochastic Convective Brinkman–Forchheimer Equations Forced by Lévy Noise

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Abstract

The two- and three-dimensional incompressible backward stochastic convective Brinkman–Forchheimer (BSCBF) equations on a torus driven by Lévy noise are considered in this paper. A-priori estimates for adapted solutions of the finite-dimensional approximation of 2D and 3D BSCBF equations are obtained. For a given terminal data, the existence and uniqueness of pathwise adapted strong solutions is proved by using a standard Galerkin (or spectral) approximation technique and exploiting the monotonicity arguments. We also establish the continuity of the adapted solutions with respect to the terminal data. The above results are obtained for the absorption exponent \(r\in [1,\infty )\) for \(d=2\) and \(r\in [3,\infty )\) for \(d=3\), and any Brinkman coefficient \(\mu >0\), Forchheimer coefficient \(\beta >0\), and hence the 3D critical case (\(r=3\)) is also handled successfully. We deduce analogous results for 2D backward stochastic Navier–Stokes equations perturbed by Lévy noise also.

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Notes

  1. An \(\mathbb {H}\)-valued process X is optional if \(X:\Omega \times [0,T]\rightarrow \mathbb {H}\) is measurable with respect to the \(\sigma \)-algebra on \(\Omega \times [0,T]\) generated by the space of càdlàg -adapted processes, and the Borel \(\sigma \)-algebra on \(\mathbb {H}\).

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Acknowledgements

M. T. Mohan would like to thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). The author is extremely thankful for the insightful comments and recommendations provided by the reviewers that have enabled them to enhance the quality of the paper.

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    Manil T. Mohan

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Mohan, M.T. Existence and Uniqueness of Solutions to Backward 2D and 3D Stochastic Convective Brinkman–Forchheimer Equations Forced by Lévy Noise. Math Phys Anal Geom 26, 16 (2023). https://doi.org/10.1007/s11040-023-09458-5

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