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On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions

  • * Corresponding author

    * Corresponding author 

The first author is partially supported by NSF DMS 1522252 and ARO 65294-MA.
The second author is partially supported by NSF DMS 1522191 and ARO 65294-MA.
The third author is partially supported by NSF DMS 1418784.

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  • We study well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations, supplemented with no-slip velocity boundary conditions, a corresponding zero-normal condition for vorticity on the boundary, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier-Stokes equations.

    Mathematics Subject Classification: Primary: 35Q30, 76N10.


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  • [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
    [2] M. Akbas, L. Rebholz and C. Zerfas, Optimal vorticity accuracy in an efficient velocity-vorticity method for the 2D Navier-Stokes equations, Calcolo, 55 (2018), p3. doi: 10.1007/s10092-018-0246-7.
    [3] C. Begue, C. Conca, F. Murat and O. Pironneau, Les equations de Stokes et de Navier-Stokes avec des conditions aux limites sur la pression, in Nonlinear Partial Differential Equations and their Applications, Pitman Research Notes in Math. , (eds. H. Brezis and J. L. Lions), College de France Seminar, 181 (1988), 179–264.
    [4] M. BenziM. A. OlshanskiiL. G. Rebholz and Z. Wang, Assessment of a vorticity based solver for the Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 247 (2012), 216-225.  doi: 10.1016/j.cma.2012.07.016.
    [5] W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J, 19 (1990), 67-87.  doi: 10.14492/hokmj/1381517172.
    [6] S. CharnyiT. HeisterM. Olshanskii and L. Rebholz, On conservation laws of Navier-Stokes Galerkin discretizations, Journal of Computational Physics, 337 (2017), 289-308.  doi: 10.1016/j.jcp.2017.02.039.
    [7] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.
    [8] A. Ern and J. -L. Guermond, Theory and Practice of Finite Elements, vol. 159, Applied Mathematical Sciences, 159. Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-4355-5.
    [9] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2010, URL http://dx.doi.org/10.1090/gsm/019. doi: 10.1090/gsm/019.
    [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-state Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-0-387-09620-9.
    [11] T. B. Gatski, Review of incompressible fluid flow computations using the vorticity-velocity formulation, Appl. Numer. Math., 7 (1991), 227-239.  doi: 10.1016/0168-9274(91)90035-X.
    [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-61798-0.
    [13] V. Girault, Curl-conforming finite element methods for Navier-Stokes equations with nonstandard boundary conditions in $\textbf{R}^3$, The Navier-Stokes Equations (Oberwolfach, 1988), 201–218, Lecture Notes in Math., 1431, Springer, Berlin, 1990. doi: 10.1007/BFb0086071.
    [14] V. Girault and P. -A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5.
    [15] P. Gresho and R. Sani, Incompressible Flow and the Finite Element Method, vol. 2, Wiley, 1998.
    [16] P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972030.ch1.
    [17] G. GuevremontW. G. Habashi and M. M. Hafez, Finite element solution of the Navier-Stokes equations by a velocity-vorticity method, Int. J. Numer. Methods Fluids, 10 (1990), 461-475. 
    [18] M. D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithm, Academic Press Inc., Boston, 1989.
    [19] T. HeisterM. A. Olshanskii and L. G. Rebholz, Unconditional long-time stability of velocity-vorticity method for 2D Navier-Stokes equations, Numerische Mathematik, 135 (2017), 143-167.  doi: 10.1007/s00211-016-0794-1.
    [20] W. Layton, Introduction to Finite Element Methods for Incompressible, Viscous Flow, SIAM, Philadelphia, 2008.
    [21] H. K. LeeM. A. Olshanskii and L. G. Rebholz, On error analysis for the 3D Navier-Stokes equations in Velocity-Vorticity-Helicity form, SIAM J. Numer. Anal., 49 (2011), 711-732.  doi: 10.1137/10080124X.
    [22] D. C. LoD. L. Young and K. Murugesan, An accurate numerical solution algorithm for 3D velocity-vorticity Navier-Stokes equations by the DQ method, Commun. Numer. Meth. Engng, 22 (2006), 235-250.  doi: 10.1002/cnm.817.
    [23] A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, vol. 27, Cambridge University Press, 2002.
    [24] H. L. Meitz and H. F. Fasel, A compact-difference scheme for the Navier-Stokes equations in vorticity-velocity formulation, J. Comput. Phys., 157 (2000), 371-403.  doi: 10.1006/jcph.1999.6387.
    [25] M. A. OlshanskiiT. HeisterL. Rebholz and K. Galvin, Natural vorticity boundary conditions on solid walls, Computer Methods in Applied Mechanics and Engineering, 297 (2015), 18-37.  doi: 10.1016/j.cma.2015.08.011.
    [26] M. A. Olshanskii and L. G. Rebholz, Velocity-vorticity-helicity formulation and a solver for the Navier-Stokes equations, Journal of Computational Physics, 229 (2010), 4291-4303.  doi: 10.1016/j.jcp.2010.02.012.
    [27] A. Palha and M. Gerritsma, A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations, Journal of Computational Physics, 328 (2017), 200-220.  doi: 10.1016/j.jcp.2016.10.009.
    [28] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, North Holland Publishing Company, New York, 1977.
    [29] K. L. Wong and A. J. Baker, A 3D incompressible Navier-Stokes velocity-vorticity weak form finite element algorithm, Int. J. Numer. Meth. Fluids, 38 (2002), 99-123.  doi: 10.1002/fld.204.
    [30] X. H. WuJ. Z. Wu and J. M. Wu, Effective vorticity-velocity formulations for the three-dimensional incompressible viscous flows, J. Comput. Phys., 122 (1995), 68-82.  doi: 10.1006/jcph.1995.1197.
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