Abstract
This paper is a development of the results and techniques of the two papers (Carvalho-Neto and Planas in J Differ Equ 259:2948–2980, 2015; Oka in J Math Anal Appl 473:382–407, 2019) for the aim of addressing the existence and uniqueness of local and global mild solutions, on the Heisenberg group \(\mathbb {H}^d\), of the time-fractional Navier–Stokes system with time derivative of order \(\alpha \in (0, 1)\). The proof relies on Schaefer’s fixed point theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Carvalho-Neto, P.M.: Fractional differential equations: a novel study of local and global solutions in Banach spaces. Ph.D. Thesis, Universidade de São Paulo, São Carlos (2013)
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)
Garofalo, N., Lanconelli, E.: Existence and nonexistence results for semilinear equations on the Heisenberg group. Indiana Univ. Math. J. 41, 71–98 (1992)
Howe, R.: On the role of the Heisenberg group in harmonic analysis. Bull. Am. Math. Soc. 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, Amsterdam (2006)
Oka, Y.: An existence and uniqueness result for the Navier–Stokes type equations on the Heisenberg group. J. Math. Anal. Appl. 473, 382–407 (2019)
Planchon, F.: Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier–Stokes equations in \(\mathbb{R}^3\). Ann. Inst. H. Poincaré Anal. Non Linéaire 13, 319–336 (1996)
Tao, T.: Finite time blowup for an averaged three-dimensional Navier–Stokes equation. J. Am. Math. Soc. 29, 601–674 (2016)
Tolksdorf, P.: On the \(L^p\)-theory of the Navier–Stokes equations on three-dimensional bounded Lipschitz domains. Math. Ann. 371, 445–460 (2018)
Wang, Y., Liang, T.: Mild solutions to the time fractional Navier–Stokes delay differential inclusions. Discrete Contin. Dyn. Syst. Ser. B 24, 3713–3740 (2019)
Zhou, Y., Wang, J.R., Zhang, L.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2016)
Zhou, Y., Peng, L.: On the time-fractional Navier–Stokes equations. Comput. Math. Appl. 73, 874–891 (2017)
Zhou, Y., Peng, L.: Weak solutions of the time-fractional Navier–Stokes equations and optimal control. Comput. Math. Appl. 73, 1016–1027 (2017)
Zou, G., Lv, G., Wu, J.: Stochastic Navier–Stokes equations with Caputo derivative driven by fractional noises. J. Math. Anal. Appl. 461, 595–609 (2018)
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
About this article
Cite this article
Kirane, M., Aimene, D. & Seba, D. Local and global existence of mild solutions of time-fractional Navier–Stokes system posed on the Heisenberg group. Z. Angew. Math. Phys. 72, 116 (2021). https://doi.org/10.1007/s00033-021-01499-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-021-01499-6