Abstract
In this paper, we investigate the existence and uniqueness of mild solutions to the fractional Navier–Stokes equations related to time derivative of order \(\alpha \in (0,1)\). And the mild solution is associated with the sublaplacian provided by the left invariant vector fields on the Heisenberg group. We demonstrate that when the nonlinear external force term matches the applicable conditions, the global mild solution can be obtained by using improved Ascoli–Arzela theorem and Schaefer’s fixed point theorem.
Similar content being viewed by others
Availability of data and materials
Not applicable.
References
Carvalho-Neto, P.M., Planas, G.: Mild solutions to the time fractional Navier–Stokes equations in \({\mathbb{R} }^N\). J. Differ. Equ. 259, 2948–2980 (2015)
Chen, P., Zhang, X., Li, Y.: Fractional non-autonomous evolution equation with nonlocal conditions. J. Pseudo-Differ. Oper. Appl. 10, 955–973 (2019)
Chen, P., Zhang, X., Li, Y.: Cauchy problem for fractional non-autonomous evolution equations. Banach. J. Math. Anal. 14, 559–584 (2020)
Gu, H.B., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257, 344–354 (2015)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Howe, R.: On the role of the Heisenberg group in harmonic analysis. Bull. Am. Math. 162(3), 821–843 (1980)
Kato, T.: Strong \(L^p\)-solutions of the Navier–Stokes equation in \({\mathbb{R} }^m\), with applications to weak solution. Math. Z. 187, 471–480 (1984)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006)
Li, S., Schul, R.: The traveling salesman problem in the Heisenberg group: upper bounding curvature. Trans. Am. Math. Soc. 368, 4585–4620 (2016)
Liu, H., Song, M.: Strichartz inequalities for the Schrodinger equation with the full Laplacian on H-type groups, p. 12 (2015). arXiv:1402.4311v3
Liu, B.M., Liu, L.S.: The decision theorems of relative compactness for two classes of abstract function groups in an infinite interval and its applications. J. Sys. Sci. Math. Sci. 30, 1008–1019 (2010)
Mainardi, F., Paraddisi, P., Gorenflo, R.: Probability Distributions Generated by Fractional Diffusion Equations. In: Kertesz, J., Kondor, I. (eds.) Econophysics: An Emerging Science. Kluwer, Dordrecht (2000)
Oka, Y.: Local well-posedness for semilinear heat equations on H type groups. Taiwanese J. Math. 22(5), 1091–1105 (2018)
Oka, Y.: An existence and uniqueness result for the Navier–Stokes type equations on the Heisenberg group. J. Math. Anal. Appl. 473, 382–470 (2019)
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44. Springer, Berlin (2012)
Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235 (2012)
Wang, J.N., Zhou, Y., Alsaedi, A., Ahmad, B.: Well-posedness and regularity of fractional Rayleigh–Stokes problems. Z. Angew. Math. Phys. 73(4), 161 (2022)
Xi, X.X., Hou, M.M., Zhou, X.F., et al.: Approximate controllability for mild solution of time-fractional Navier–Stokes equations with delay. Z. Angew. Math. Phys. 72(3), 1–26 (2021)
Yang, Q., Zhu, F.: The heat kernel on H-type groups. Proc. Am. Math. Soc. 136(4), 1457–1464 (2008)
Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)
Zhou, Y., He, J.W.: Well-posedness and regularity for fractional damped wave equations. Monatsh. Math. 194(2), 425–458 (2021)
Zhou, Y., Peng, L.: On the time-fractional Navier–Stokes equations. Comput. Math. Appl. 73(6), 874–891 (2017)
Zhou, Y., Peng, L.: Weak solutions of the time-fractional Navier–Stokes equations and optimal control. Comput. Math. Appl. 73(6), 1016–1027 (2017)
Funding
This research was supported by the Fundo para o Desenvolvimento das Ciências e da Tecnologia of Macau (No. 0092/2022/A).
Author information
Authors and Affiliations
Contributions
Xiaolin Liu wrote the main manuscript text after discussing main technique with Yong Zhou. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Ethical approval
Not applicable.
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
Liu, X., Zhou, Y. Globally Well-Posedness Results of the Fractional Navier–Stokes Equations on the Heisenberg Group. Qual. Theory Dyn. Syst. 23, 52 (2024). https://doi.org/10.1007/s12346-023-00910-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-023-00910-z