July 2018, 38(7): 3459-3477. doi: 10.3934/dcds.2018148

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

1. 

Department of Mathematics, University of Houston, Houston, TX, 77204, USA

2. 

Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA

3. 

Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

* 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.

Received  July 2017 Revised  February 2018 Published  April 2018

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.

Citation: Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
References:
[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.

show all references

References:
[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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