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

Özge Kazar Meryem Kaya

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.

Keywords

Existence and uniqueness, g-Navier-Stokes equations, inviscid velocity-vorticity model

Supporting Institution

-

Project Number

-

Thanks

-

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.