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On Strong Solution to the 2D Stochastic Ericksen–Leslie System: A Ginzburg–Landau Approximation Approach

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Quantum and Stochastic Mathematical Physics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 377))

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Abstract

In this manuscript, we consider a highly nonlinear and constrained stochastic PDEs modelling the dynamics of 2-dimensional nematic liquid crystals under random perturbation. This system of SPDEs is also known as the stochastic Ericksen–Leslie equations (SELEs). We discuss the existence of local strong solution to the stochastic Ericksen–Leslie equations. In particular, we study the convergence of the stochastic Ginzburg–Landau approximation of SELEs, and prove that the SELEs with initial data in \(\textsf{H}^1\times \textsf{H}^2\) has at least a martingale, local solution which is strong in PDEs sense.

Part of this article was written when P. Razafimandimby was a Marie Skłodowska-Curie fellow at the University of York. This article is part of a project that received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 791735 “SELEs”.

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Notes

  1. 1.

    In the sense that for all \(n\in \mathbb {N}\), \(\tau _n \le \tau _{n+1}\), \(\mathbb {P}\)-a.s.

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Acknowledgements

The first named author would like to dedicate this work to Professor Sergio Albeverio, his teacher, a collaborator and a friend, on his 80th Birthday. Their collaboration on mathematical foundations of Feynmann path integrals has led him to understand the stochastic integral with respect to a cylindrical Wiener process.

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Authors and Affiliations

  1. Department of Mathematics, University of York, Heslington, York, YO10 5DD, UK

    Zdzisław Brzeźniak

  2. Department of Mathematics and Computer Science, University of Dschang, Dschang, Cameroon

    Gabriel Deugoué

  3. School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland

    Paul André Razafimandimby

Authors
  1. Zdzisław Brzeźniak
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  2. Gabriel Deugoué
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  3. Paul André Razafimandimby
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Correspondence to Zdzisław Brzeźniak .

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Editors and Affiliations

  1. Faculty of Technology, Linnaeus University, Växjo, Sweden

    Astrid Hilbert

  2. Department of Economics, Insubria University, Varese, Italy

    Elisa Mastrogiacomo

  3. Department of Mathematics, University of Trento, Povo, Trento, Italy

    Sonia Mazzucchi

  4. Applied Mathematics / Stochastic, University of Wuppertal, Wuppertal, Germany

    Barbara Rüdiger

  5. Department of Mathematics, University of MIlan, Milano, Italy

    Stefania Ugolini

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Brzeźniak, Z., Deugoué, G., Razafimandimby, P.A. (2023). On Strong Solution to the 2D Stochastic Ericksen–Leslie System: A Ginzburg–Landau Approximation Approach. In: Hilbert, A., Mastrogiacomo, E., Mazzucchi, S., Rüdiger, B., Ugolini, S. (eds) Quantum and Stochastic Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 377. Springer, Cham. https://doi.org/10.1007/978-3-031-14031-0_12

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