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\({\mathbb {L}}^p\)-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations

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Abstract

In the first part of this work, we establish the existence and uniqueness of a local mild solution to deterministic convective Brinkman–Forchheimer (CBF) equations defined on the whole space, by using properties of the heat semigroup and fixed point arguments based on an iterative technique. Moreover, we prove that the solution exists globally. The second part is devoted for establishing the existence and uniqueness of a pathwise mild solution upto a random time to the stochastic CBF equations perturbed by Lévy noise by exploiting the contraction mapping principle. Then by using stopping time arguments, we show that the pathwise mild solution exists globally. We also discuss the local and global solvability of the stochastic CBF equations forced by fractional Brownian noise.

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Notes

  1. Let \({\mathbb {U}}\) be a real separable Hilbert space and \({\mathbb {X}}\) be a Banach space. A bounded linear operator \(R\in {\mathcal {L}}({\mathbb {U}},{\mathbb {X}})\) is \(\gamma \)-radonifying provided that there exists a centered Gaussian probability \(\nu \) on \({\mathbb {X}}\) such that \(\int _{{\mathbb {X}}}\varphi (x)\mathrm {d}\nu (x)=\Vert R^*\varphi \Vert _{{\mathbb {U}}},\ \varphi \in {\mathbb {X}}'\). Such a measure is at most one, and hence we set \(\Vert R\Vert _{\gamma ({\mathbb {U}},{\mathbb {X}})}^2:=\int _{{\mathbb {X}}}\Vert x\Vert _{{\mathbb {X}}}^2\mathrm {d}\nu (x).\) We denote \(\gamma ({\mathbb {U}};{\mathbb {X}})\) for the space of \(\gamma \)-radonifying operators, and \(\gamma ({\mathbb {U}},{\mathbb {X}})\) equipped with the norm \(\Vert \cdot \Vert _{\gamma ({\mathbb {U}},{\mathbb {X}})}\) is a separable Banach space.

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Acknowledgements

M. T. Mohan would like to thank the Department of Science and Technology (DST), India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). The author sincerely would like to thank the reviewer for his/her valuable comments and suggestions.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand, 247667, India

    Manil T. Mohan

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Mohan, M.T. \({\mathbb {L}}^p\)-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations. Anal.Math.Phys. 11, 164 (2021). https://doi.org/10.1007/s13324-021-00595-0

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