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On the Existence, Uniqueness and Regularity of Solutions of a Viscoelastic Stokes Problem Modelling Salt Rocks

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Abstract

A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of \(L^2\)-based Sobolev spaces. The system is transformed into a generalized Stokes problem, and the proper conditions on the parameters of the model that guarantee that the system is uniformly elliptic are given. Under those conditions, existence, uniqueness and low-order regularity are obtained under classical regularity conditions on the data, while higher-order regularity is proved under less stringent conditions than classical ones. Explicit estimates for the solution in terms of the data are given accordingly.

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Notes

  1. If \(\varphi \in W^{1,\infty }(\Omega )\), let \(\varphi _\varepsilon \) be a mollification of \(\varphi \), which converges to \(\varphi \) in any \(W^{1,p}_{\mathrm{loc}}(\Omega )\), for \(1\le p < \infty \). Thus, a subsequence of \(\varphi _\varepsilon \) converges to \(\varphi \) and is such that \(\varvec{\nabla }\varphi _\varepsilon \) converges to \(\varvec{\nabla }\varphi \), almost everywhere in \(\Omega \). Since \(|\delta _i^h\varphi _\varepsilon |\le \sup _\Omega |\varvec{\nabla }\varphi _\varepsilon |\), we find at the limit that \(\Vert \delta _i^h\varphi \Vert _{L^\infty (\Omega _0)} \le \Vert \partial _{x_i}\varphi \Vert _{L^\infty (\Omega )}\), for every subdomain \(\Omega _0\) compactly included in \(\Omega \) and for every \(h\ne 0\) sufficiently small such that \(|h|\le \text {dist}(\Omega _0,\partial \Omega )\).

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Acknowledgements

All the authors acknowledge the financial support of CENPES/PETROBRÁS. R.M.S. Rosa was also partly supported by CNPq, Brasília, Brazil.

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Cipolatti, R.A., Liu, IS., Palermo, L.A. et al. On the Existence, Uniqueness and Regularity of Solutions of a Viscoelastic Stokes Problem Modelling Salt Rocks. Appl Math Optim 78, 403–456 (2018). https://doi.org/10.1007/s00245-017-9411-7

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