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Stochastic Forcing in Hydrodynamic Models with Non-local Interactions

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Abstract

The hydrodynamical model of the collective behavior of animals consists of the Euler equation with additional non-local forcing terms representing the repulsive and attractive forces among individuals. This paper deals with the system endowed with an additional white-noise forcing and an artificial viscous term. We provide a proof of the existence of a dissipative martingale solution—a cornerstone for a subsequent analysis of the system with stochastic forcing.

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Notes

  1. For a function \(f:{\mathbb {T}}\mapsto {\mathbb {R}}\), we define its average as \((f)_{\mathbb {T}}:= \int _{\mathbb {T}}f \ \mathrm{d}x\).

  2. Here we adopt the following notation. For a general function \(f(\varrho _\varepsilon ,\mathbf{u}_\varepsilon )\), we use \(\overline{f(\varrho ,\mathbf{u})}\) to denote its weak limit (which is generaly not equal to \(f(\varrho ,\mathbf{u})\)).

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Acknowledgements

The work of Václav Mácha was supported by the Czech Science Foundation (GAČR), Grant Agreement GA18-05974S in the framework of RVO:67985840. The work of Pavel Ludvík was supported by the Grant IGA_PrF_2021_008 “Mathematical Models” of the Internal Grant Agency of Palacký University in Olomouc.

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Authors and Affiliations

  1. Department of Mathematical Analysis and Applications of Mathematics, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 77146, Olomouc, Czech Republic

    Pavel Ludvík

  2. Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Václav Mácha

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  1. Pavel Ludvík
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  2. Václav Mácha
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Correspondence to Václav Mácha.

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Ludvík, P., Mácha, V. Stochastic Forcing in Hydrodynamic Models with Non-local Interactions. J Theor Probab 35, 2806–2852 (2022). https://doi.org/10.1007/s10959-021-01137-x

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