Abstract
The freezing of salt water leads to phase separation into ice and brine inclusions. The ejection of salt from the ice phase is akin to chemotaxis. We consider a regularization of a free energy for brine inclusions and address its gradient flow on a periodic domain in \({\mathbb {R}}^d(d=2,3)\). Uniqueness and global existence of classical solutions from initial data in an energy space are established for positive time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25, 1663–1763 (2015)
Fujie, K.: Boundedness in a fully parabolic chemotaxis system with singular sensitivity. J. Math. Anal. Appl. 424, 675–684 (2015)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Keller, E.F., Segel, L.A.: Traveling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235–248 (1971)
Kraitzman, N., Promislow, K., Wetton, B.: Slow migration of brine inclusions in first-year sea ice. SIAM J. Appl. Math. 82, 1470–1494 (2022)
Ladyzhenskaia, O.A., Solonnikov, V.A., Ural’tseva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type, vol. 23. AAmerican Mathematical Society, Providence (1988)
Lankeit, J.: A new approach toward boundedness in a two-dimensional parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 3, 394–404 (2016)
Lankeit, J., Winkler, M.: A generalized solution concept for the Keller–Segel system with logarithmic sensitivity: global solvability for large nonradial data. Nonlinear Differ. Equ. Appl. 24, 1–33 (2017)
Mielke, A.: Formulation of thermoplastic dissipative material behavior using generic. Contin. Mech. Thermodyn. 23, 233–256 (2011)
Nagai, T., Senba, T., Yoshida, K.: Application of the trudinger-moser inequality to a parabolic system of chemotaxis. Funkc. Ekvacioj 40, 411–433 (1997)
Wells, A., Hitchen, J., Parkinson, J.: Mushy-layer growth and convection, with application to sea ice. Philos. Trans. R. Soc. A 377, 20180165 (2019)
Winkler, M.: Aggregation vs global diffusive behavior in the higher-dimensional Keller–Segel model. J. Differ. Equ. 248, 2889–2905 (2010)
Winkler, M.: Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Math. Methods Appl. Sci. 34, 176–190 (2011)
Acknowledgements
The first author is supported by the Chinese University of Hong Kong(Shenzhen) under the start-up funding UDF01002321 and by Shenzhen government under grant C10120230023. The second author was supported by the NSF under grant DMS 2205553.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chen, Y., Promislow, K. Global existence of classical solutions to a regularized brine inclusion model. Z. Angew. Math. Phys. 74, 126 (2023). https://doi.org/10.1007/s00033-023-02019-4
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-023-02019-4