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Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations

Year 2022, Volume: 26 Issue: 4, 695 - 702, 31.08.2022
https://doi.org/10.16984/saufenbilder.1097179

Abstract

In this paper, we prove the global existence and uniqueness of the weak solutions to the inviscid velocity-vorticity model of the g-Navier-Stokes equations. The system is performed by entegrating the velocity-pressure system which is involved by using the rotational formulation of the nonlinearity and the vorticity equation for the g-Navier-Stokes equations without viscosity term. In this study we particularly interest the inviscid velocity-vorticity system of the g-Navier-Stokes equations over the two dimensional periodic box Ω=(0,1)^2⊂R^2.

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References

  • [1] M. Akbas, L. G. Rebholz, C. Zerfas, “Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations,” Calcolo, Vol. 55, no. 1, pp.1-29, 2018.
  • [2] M. Gardner, A. Larios, L. G. Rebholz, D. Vargun, C. Zerfas, “Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations,” Electronic Research Archive, Vol. 29, no. 3, pp. 2223-2247, 2021.
  • [3] T. B. Gatski, “Review of incompressible fluid flow computations using the vorticity-velocity formulation,” Applied Numerical Mathematics, Vol. 7, no. 3, pp. 227-239, 1991.
  • [4] T. Heister, M. A. Olshanskii, L. G. Rebholz, “Unconditional long-time stability of a velocity- vorticity method for the 2D Navier-Stokes equations,” Numerische Mathematik, Vol. 135, no. 1, pp. 143-167, 2017.
  • [5] A. Larios, Y. Pei, L. Rebholz, “Global well-posedness of the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations,” Journal of Differential Equations, Vol. 266, no. 5, pp. 2435-2465, 2019.
  • [6] Y. Pei, “Regularity and Convergence Results of the Velocity-Vorticity-Voigt Model of the 3D Boussinesq Equations,” Acta Applicandae Mathematicae, Vol. 176, no. 1, pp. 1-25, 2021.
  • [7] Y. Cao, E. M. Lunasin, E. S. Titi, “Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models,” Communications in Mathematical Sciences, Vol. 4, no. 4, pp. 823-848, 2006.
  • [8] A. Larios, E. S. Titi, “Higher-order global regularity of an inviscid Voigt-regularization of the three-dimensional inviscid resistive magnetohydrodynamic equations,” Journal of Mathematical Fluid Mechanics, Vol. 16 no.1, pp. 59-76, 2014.
  • [9] J. Wu, “Viscous and inviscid magneto-hydrodynamics equations,” Journal d'analyse Mathematique, Vol. 73 no. 1, pp. 251-265, 1997.
  • [10] Ö. Kazar, M. Kaya, “On the weak and strong solutions of the velocity-vorticity model of the g-Navier-Stokes equations,” (to appear).
  • [11] J. Roh, “g-Navier-Stokes equations,” PhD, University of Minnesota, Minneapolis, MN, USA, 2001.
  • [12] R. Temam, “Navier-Stokes equations, theory and numerical analysis,” American Mathematical Society, Chelsea Publication, Vol.343, pp. 161-163, pp. 252-290, 2001.
  • [13] M. Schechter, “An Introduction to Nonlinear Analysis,” Cambridge University Press, 2004.
Year 2022, Volume: 26 Issue: 4, 695 - 702, 31.08.2022
https://doi.org/10.16984/saufenbilder.1097179

Abstract

Project Number

-

References

  • [1] M. Akbas, L. G. Rebholz, C. Zerfas, “Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations,” Calcolo, Vol. 55, no. 1, pp.1-29, 2018.
  • [2] M. Gardner, A. Larios, L. G. Rebholz, D. Vargun, C. Zerfas, “Continuous data assimilation applied to a velocity-vorticity formulation of the 2D Navier-Stokes equations,” Electronic Research Archive, Vol. 29, no. 3, pp. 2223-2247, 2021.
  • [3] T. B. Gatski, “Review of incompressible fluid flow computations using the vorticity-velocity formulation,” Applied Numerical Mathematics, Vol. 7, no. 3, pp. 227-239, 1991.
  • [4] T. Heister, M. A. Olshanskii, L. G. Rebholz, “Unconditional long-time stability of a velocity- vorticity method for the 2D Navier-Stokes equations,” Numerische Mathematik, Vol. 135, no. 1, pp. 143-167, 2017.
  • [5] A. Larios, Y. Pei, L. Rebholz, “Global well-posedness of the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations,” Journal of Differential Equations, Vol. 266, no. 5, pp. 2435-2465, 2019.
  • [6] Y. Pei, “Regularity and Convergence Results of the Velocity-Vorticity-Voigt Model of the 3D Boussinesq Equations,” Acta Applicandae Mathematicae, Vol. 176, no. 1, pp. 1-25, 2021.
  • [7] Y. Cao, E. M. Lunasin, E. S. Titi, “Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models,” Communications in Mathematical Sciences, Vol. 4, no. 4, pp. 823-848, 2006.
  • [8] A. Larios, E. S. Titi, “Higher-order global regularity of an inviscid Voigt-regularization of the three-dimensional inviscid resistive magnetohydrodynamic equations,” Journal of Mathematical Fluid Mechanics, Vol. 16 no.1, pp. 59-76, 2014.
  • [9] J. Wu, “Viscous and inviscid magneto-hydrodynamics equations,” Journal d'analyse Mathematique, Vol. 73 no. 1, pp. 251-265, 1997.
  • [10] Ö. Kazar, M. Kaya, “On the weak and strong solutions of the velocity-vorticity model of the g-Navier-Stokes equations,” (to appear).
  • [11] J. Roh, “g-Navier-Stokes equations,” PhD, University of Minnesota, Minneapolis, MN, USA, 2001.
  • [12] R. Temam, “Navier-Stokes equations, theory and numerical analysis,” American Mathematical Society, Chelsea Publication, Vol.343, pp. 161-163, pp. 252-290, 2001.
  • [13] M. Schechter, “An Introduction to Nonlinear Analysis,” Cambridge University Press, 2004.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Özge Kazar 0000-0003-4876-3077

Meryem Kaya 0000-0002-5932-9105

Project Number -
Publication Date August 31, 2022
Submission Date April 1, 2022
Acceptance Date May 30, 2022
Published in Issue Year 2022 Volume: 26 Issue: 4

Cite

APA Kazar, Ö., & Kaya, M. (2022). Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations. Sakarya University Journal of Science, 26(4), 695-702. https://doi.org/10.16984/saufenbilder.1097179
AMA Kazar Ö, Kaya M. Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations. SAUJS. August 2022;26(4):695-702. doi:10.16984/saufenbilder.1097179
Chicago Kazar, Özge, and Meryem Kaya. “Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The G-Navier-Stokes Equations”. Sakarya University Journal of Science 26, no. 4 (August 2022): 695-702. https://doi.org/10.16984/saufenbilder.1097179.
EndNote Kazar Ö, Kaya M (August 1, 2022) Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations. Sakarya University Journal of Science 26 4 695–702.
IEEE Ö. Kazar and M. Kaya, “Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations”, SAUJS, vol. 26, no. 4, pp. 695–702, 2022, doi: 10.16984/saufenbilder.1097179.
ISNAD Kazar, Özge - Kaya, Meryem. “Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The G-Navier-Stokes Equations”. Sakarya University Journal of Science 26/4 (August 2022), 695-702. https://doi.org/10.16984/saufenbilder.1097179.
JAMA Kazar Ö, Kaya M. Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations. SAUJS. 2022;26:695–702.
MLA Kazar, Özge and Meryem Kaya. “Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The G-Navier-Stokes Equations”. Sakarya University Journal of Science, vol. 26, no. 4, 2022, pp. 695-02, doi:10.16984/saufenbilder.1097179.
Vancouver Kazar Ö, Kaya M. Global Existence and Uniqueness of The Inviscid Velocity-Vorticity Model of The g-Navier-Stokes Equations. SAUJS. 2022;26(4):695-702.