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

Previous Article
Non-floquet invariant tori in reversible systems
DCDS Home
This Issue
Next Article
Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

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

Maxim A. Olshanskii ^1, , Leo G. Rebholz ^2,, and Abner J. Salgado ^3,

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.

Keywords: Navier-Stokes equations, velocity-vorticity formulation, well-posedness, no-slip boundary conditions.

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

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. Benzi, M. A. Olshanskii, L. 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. Charnyi, T. Heister, M. 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 {M}ethod, vol. 2, Wiley, 1998.
[16]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972030.ch1.
[17]	G. Guevremont, W. 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. Heister, M. 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. Lee, M. 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. Lo, D. 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. Olshanskii, T. Heister, L. 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 {N}avier-{S}tokes 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. Wu, J. 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. Benzi, M. A. Olshanskii, L. 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. Charnyi, T. Heister, M. 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 {M}ethod, vol. 2, Wiley, 1998.
[16]	P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011. doi: 10.1137/1.9781611972030.ch1.
[17]	G. Guevremont, W. 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. Heister, M. 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. Lee, M. 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. Lo, D. 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. Olshanskii, T. Heister, L. 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 {N}avier-{S}tokes 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. Wu, J. 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.

[1]	Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
[2]	Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
[3]	Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143
[4]	Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
[5]	Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[6]	Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845
[7]	Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315
[8]	Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783
[9]	Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic & Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021
[10]	Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437
[11]	Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045
[12]	Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113
[13]	Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[14]	Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673
[15]	Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629
[16]	Barbara Kaltenbacher, Irena Lasiecka. Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions. Conference Publications, 2011, 2011 (Special) : 763-773. doi: 10.3934/proc.2011.2011.763
[17]	Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325
[18]	Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219
[19]	Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631
[20]	Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277

American Institute of Mathematical Sciences