Ir al contenido

Documat


Well-Posedness and Dependence on the Initial Value of the Time-Fractional Navier–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. 23, Nº 5, 2024
  • Idioma: inglés
  • DOI: 10.1007/s12346-024-01063-3
  • Enlaces
  • Resumen
    • In this paper, we examine the relevant properties of mild solutions to the fractional Navier–Stokes equations with respect to a time derivative of order 0 <α< 1 in a new spatial framework. On the Heisenberg group, these mild solutions are connected to the sublaplacian provided by the left-invariant vector fields. By applying appropriate conditions, we can obtain global and local mild solutions using the classical bilinear fixed point theorem. Furthermore, we have also established the dependence on the initial value of these mild solutions

  • Referencias bibliográficas
    • 1. Birindelli, I., Ferrari, F., Valdinoci, E.: Semilinear PDEs in the Heisenberg group: the role of the right invariant vector fields. Nonlinear...
    • 2. Carvalho-Neto, P.M., Planas, G.: Mild solutions to the time fractional Navier–Stokes equations in RN . J. Differ. Equ. 259, 2948–2980 (2015)
    • 3. Choe, H.J.: Boundary regularity of suitable weak solution for the Navier–Stokes equations. J. Funct. Anal. 268, 2171–2187 (2015)
    • 4. El Asraoui, H., El Mfadel, A., Hilal, K., Elomari, M.: Sufficient conditions for existence of mild solutions for nondensely defined conformable...
    • 5. Elsayed, E.M., Shah, R., Nonlaopon, K.: The analysis of the fractional-order Navier–Stokes equations by a novel approach. J. Funct. Spaces...
    • 6. Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, vol. 40. Wiley, New York (1999)
    • 7. Gu, H.B., Trujillo, J.J.: Existence of mild solution for evolution equation with Hilfer fractional derivative. Appl. Math. Comput. 257,...
    • 8. Heck, H., Kim, H., Kozono, H.: Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle....
    • 9. Hieber, M., Shibata, Y.: The Fujita–Kato approach to the Navier–Stokes equations in the rotational framework. Math. Z. 265(2), 481–491...
    • 10. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
    • 11. Hunt, R.A.: An extension of the Marcinkiewicz interpolation theorem to Lorentz spaces. Bull. Amer. Math. Soc. 70, 803–807 (1964)
    • 12. Hulanicki, A.: The distribution of energy in the Brownian motion in the Gaussian field and analytic hypoellipticity of certain subelliptic...
    • 13. Iwabuchi, T., Takada, R.: Global well-posedness and ill-posedness for the Navier–Stokes equations with the Coriolis force in function...
    • 14. Jerison, D.S., Sanchez-Calle, A.: Estimates for the heat kernel for a sum of squares of vector fields. Indiana Univ. Math. J. 35, 835–854...
    • 15. Kato, T.: Strong L p-solutions of the Navier–Stokes equation in Rm, with applications to weak solution. Math. Z. 187, 471–480 (1984)
    • 16. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam...
    • 17. Kozono, H., Okada, A., Shimizu, S.: Necessary and sufficient condition on initial data in the Besov space for solutions in the Serrin...
    • 18. Lei, Z., Lin, F.H.: Global mild solutions of Navier–Stokes equations. Commun. Pure Appl. Math. 64(9), 1297–1304 (2011)
    • 19. Lieb, E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. In: Inequalities: Selecta of Elliott H. Lieb, pp....
    • 20. Li, S., Schul, R.: The traveling salesman problem in the Heisenberg group: upper bounding curvature. Trans. Amer. Math. Soc. 368, 4585–4620...
    • 21. Mainardi, F., Paraddisi, P., Gorenflo, R.: Probability distributions generated by fractional diffusion equations. In: Kertesz, J., Kondor,...
    • 22. Oka, Y.: Local well-posedness for semilinear heat equations on H type groups. Taiwan. J. Math. 22(5), 1091–1105 (2018)
    • 23. Oka, Y.: An existence and uniqueness result for the Navier–Stokes type equations on the Heisenberg group. J. Math. Anal. Appl. 473, 382–470...
    • 24. Omrane, I.B., Slimane, M.B., Gala, S., Ragusa, M.A.: A weak-L p Prodi–Serrin type regularity criterion for the micropolar fluid equations...
    • 25. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential equations, vol. 44. Springer, Cham (2012)
    • 26. Vasseur, A.F., Yu, C.: Existence of global weak solutions for 3D degenerate compressible Navier– Stokes equations. Invent. Math. 206,...
    • 27. Wang, R.N., Chen, D.H., Xiao, T.J.: Abstract fractional Cauchy problems with almost sectorial operators. J. Differ. Equ. 252, 202–235...
    • 28. Xiao, Y.X.: An improved Hardy type inequality on Heisenberg group. J. Inequal. Appl. 1, 1–8 (2011)
    • 29. Zhai, X.P., Li, Y.S., Zhou, F.J.: Global large solutions to the three dimensional compressible Navier– Stokes equations. SIAM J. Math....
    • 30. Zheng, X.: Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems. Fract....
    • 31. Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014)
    • 32. Zhou, Y., Peng, L.: On the time-fractional Navier–Stokes equations. Comput. Math. Appl. 73(6), 874–891 (2017)
    • 33. Zhou, Y., Peng, L.: Weak solutions of the time-fractional Navier–Stokes equations and optimal control. Comput. Math. Appl. 73(6), 1016–1027...

Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno