Well-Posedness of the Nonlinear Fractional Rayleigh-Stokes Equations on the Heisenberg Group

• Xiaolin Liu ^[1] ; Yong Zhou ^[1]
  1. [1] Macau University of Science and Technology

     Macau University of Science and Technology

     RAE de Macao (China)

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper, we investigate the existence and uniqueness of mild solutions for a class of nonlinear Rayleigh-Stokes equations on the Heisenberg group in the fractional Sobolev space. Based on the expression of the mild solutions, we study the relevant properties of the solution operator. By utilizing the contraction mapping principle, we have successfully established the well-posedness of the mild solutions and the dependence on initial value.

• Referencias bibliográficas
  • 1. Bazhlekova, E., Jin, B., Lazarov, R., Zhou, Z.: An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid. Numer....
  • 2. Bazhlekova, E.: Subordination principle for a class of fractional order dierential equations. Math. 2, 412–427 (2015)
  • 3. Binh, T.T., Nashine, H.K., Le, D.L., Nguyen, H.L., Nguyen, C.: Identification of source term for the ill-posed Rayleigh-Stokes problem...
  • 4. Birindelli, I., Ferrari, F., Valdinoci, E.: Semilinear PDEs in the Heisenberg group: the role of the right invariant vector fields. Nonlinear...
  • 5. Fetecau, C., Jamil, M., Fetecau, C., Vieru, D.: The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid. Z. Angew. Math....
  • 6. Folland, G.B.: Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13(1–2), 161–207 (1975)
  • 7. Gupta, M., Tyagi, J.: Fractional Adams-Moser-Trudinger type inequality on Heisenberg group. Nonlinear Anal. 195, 111747 (2020)
  • 8. He, J.W., Zhou, Y., Peng, L., Ahmad, B.: On well-posedness of semilinear Rayleigh-Stokes problem with fractional derivative on RN , Adv....
  • 9. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam...
  • 10. Liu,X.L., Zhou,Y.: Globally Well-Posedness Results of the Fractional Navier–Stokes Equations on the Heisenberg Group. Qualitative Theory...
  • 11. Li, S., Schul, R.: The traveling salesman problem in the Heisenberg group: upper bounding curvature. Trans. Amer. Math. Soc. 368, 4585–4620...
  • 12. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Springer Science and Business Media, Vol....
  • 13. Peng, L., Zhou, Y., Ahmad, B.: The well-posedness for fractional nonlinear Schrödinger equations. Comput. Math. Appl. 77, 1998–2005 (2019)
  • 14. J. Prüss, Evolutionary integral equations and applications, Birkhäuser, Vol. 87(2013)
  • 15. Safari, F., Sun, H.: Improved singular boundary method and dual reciprocity method for fractional derivative Rayleigh-Stokes problem....
  • 16. Saka, K.: Besov spaces and Sobolev spaces on a nilpotent Lie group. Tohoku Mathematical Journal, Second Series 31(4), 383–437 (1979)
  • 17. Salah, F., Aziz, Z.A., Ching, D.L.C.: New exact solution for Rayleigh-Stokes problem of Maxwell fluid in a porous medium and rotating...
  • 18. Samet, B.: Blow-up phenomena for a nonlinear time fractional heat equation in an exterior domain. Comput. Math. Appl. 78, 1380–1385 (2019)
  • 19. Wang, J.N., Zhou, Y., He, J.W.: Existence and regularization of solutions for nonlinear fractional Rayleigh-Stokes problem with final...
  • 20. Wang, J.N., Alsaedi, A., Ahmad, B., Zhou, Y.: Well-posedness and blow-up results for a class of nonlinear fractional Rayleigh-Stokes problem....
  • 21. Xiao, Y.X.: An improved Hardy type inequality on Heisenberg group. J. Inequal. Appl. 1, 1–8 (2011)
  • 22. Zaky, M.A.: An improved tau method for the multi-dimensional fractional Rayleigh-Stokes problem for a heated generalized second grade...
  • 23. Zhou, Y.: Basic theory of fractional differential equations. World Scientific, Singapore (2014)
  • 24. Y. Zhou, Infinite interval problems for fractional evolution equations, Mathematics, 10(6)(2022), 900