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Erschienen in: Journal of Applied Mathematics and Computing 1-2/2021

26.09.2020 | Original Research

Existence and uniqueness of solutions to the damped Navier–Stokes equations with Navier boundary conditions for three dimensional incompressible fluid

verfasst von: Subha Pal, Rajib Haloi

Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2021

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Abstract

In this article, we study the solutions of the damped Navier–Stokes equation with the Navier slip boundary condition in a bounded domain \(\Omega \) in \({\mathbb {R}}^3\) with sufficiently smooth boundary. We employ the Galerkin method to approximate the solutions of the damped Navier–Stokes equations with the Navier-slip boundary conditions. The existence of the solutions is global for \(\beta \ge 1\). We also established the regularity of the solutions for \(\beta \ge 3\), and the uniqueness of the solutions for \(\beta \ge 1\).

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Literatur
1.
Zurück zum Zitat Amrouche, C., Rejaiba, A.: Navier–Stokes equations with Navier boundary condition. Math. Methods Appl. Sci. 39(17), 5091–5112 (2016)MathSciNetMATHCrossRef Amrouche, C., Rejaiba, A.: Navier–Stokes equations with Navier boundary condition. Math. Methods Appl. Sci. 39(17), 5091–5112 (2016)MathSciNetMATHCrossRef
2.
Zurück zum Zitat Antontsev, S.N., de Oliveira, H.B.: Navier–Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time. RIMS Kôkyûroku Bessatsu, B 1, 21–41 (2007)MathSciNetMATH Antontsev, S.N., de Oliveira, H.B.: Navier–Stokes equations with absorption under slip boundary conditions: existence, uniqueness and extinction in time. RIMS Kôkyûroku Bessatsu, B 1, 21–41 (2007)MathSciNetMATH
3.
Zurück zum Zitat Antontsev, S.N., de Oliveira, H.B.: The Navier–Stokes problem modified by an absorption term. Appl. Anal. 89(12), 1805–1825 (2010)MathSciNetMATHCrossRef Antontsev, S.N., de Oliveira, H.B.: The Navier–Stokes problem modified by an absorption term. Appl. Anal. 89(12), 1805–1825 (2010)MathSciNetMATHCrossRef
4.
Zurück zum Zitat da Beirão Veiga, H., Crispo, F.: Sharp inviscid limit results under Navier type boundary conditions: an \(L^p\) theory. J. Math. Fluid Mech 12(3), 397–411 (2010)MathSciNetMATH da Beirão Veiga, H., Crispo, F.: Sharp inviscid limit results under Navier type boundary conditions: an \(L^p\) theory. J. Math. Fluid Mech 12(3), 397–411 (2010)MathSciNetMATH
5.
Zurück zum Zitat da Veiga, Beirão, Crispo, F.: The 3-D inviscid limit result under slip boundary conditions, A negative answer. J. Math. Fluid Mech. 14(1), 55–59 (2012)MathSciNetMATHCrossRef da Veiga, Beirão, Crispo, F.: The 3-D inviscid limit result under slip boundary conditions, A negative answer. J. Math. Fluid Mech. 14(1), 55–59 (2012)MathSciNetMATHCrossRef
6.
Zurück zum Zitat Cai, X., Jiu, Q.: Weak and strong solutions for the incompressible Navier–Stokes equations with damping. J. Math. Anal. Appl. 343, 799–809 (2008)MathSciNetMATHCrossRef Cai, X., Jiu, Q.: Weak and strong solutions for the incompressible Navier–Stokes equations with damping. J. Math. Anal. Appl. 343, 799–809 (2008)MathSciNetMATHCrossRef
7.
Zurück zum Zitat Clopeau, T., Mikelić, A., Robert, R.: On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions. Nonlinearity 11(6), 1625–1636 (1998)MathSciNetMATHCrossRef Clopeau, T., Mikelić, A., Robert, R.: On the vanishing viscosity limit for the 2D incompressible Navier–Stokes equations with the friction type boundary conditions. Nonlinearity 11(6), 1625–1636 (1998)MathSciNetMATHCrossRef
8.
Zurück zum Zitat Iftimie, D., Sueur, F.: Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199(1), 145–175 (2011)MathSciNetMATHCrossRef Iftimie, D., Sueur, F.: Viscous boundary layers for the Navier–Stokes equations with the Navier slip conditions. Arch. Ration. Mech. Anal. 199(1), 145–175 (2011)MathSciNetMATHCrossRef
9.
Zurück zum Zitat Jager, W., Mikelić, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170(1), 96–122 (2001)MathSciNetMATHCrossRef Jager, W., Mikelić, A.: On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differ. Equ. 170(1), 96–122 (2001)MathSciNetMATHCrossRef
10.
Zurück zum Zitat Kashiwabara, T.: On a strong solution of the non-stationary Navier–Stokes equations under slip or leak boundary conditions of friction type. J. Differ. Equ. 254(2), 756–778 (2013)MathSciNetMATHCrossRef Kashiwabara, T.: On a strong solution of the non-stationary Navier–Stokes equations under slip or leak boundary conditions of friction type. J. Differ. Equ. 254(2), 756–778 (2013)MathSciNetMATHCrossRef
11.
Zurück zum Zitat Kelliher, J.P.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)MathSciNetMATHCrossRef Kelliher, J.P.: Navier–Stokes equations with Navier boundary conditions for a bounded domain in the plane. SIAM J. Math. Anal. 38(1), 210–232 (2006)MathSciNetMATHCrossRef
12.
Zurück zum Zitat Lions, J.L.: Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Gauthier-Villars, Paris (1969)MATH Lions, J.L.: Quelques methodes de resolution des problemes aux limites non lineaires. Dunod, Gauthier-Villars, Paris (1969)MATH
13.
Zurück zum Zitat Lopes Filho, M.C., Nussenzveig Lopes, H.J., Planas, G.: On the inviscid limit for two-dimensional incompressible flow with Navier friction condition. SIAM J. Math. Anal. 36(4), 1130–1141 (2005)MathSciNetMATHCrossRef Lopes Filho, M.C., Nussenzveig Lopes, H.J., Planas, G.: On the inviscid limit for two-dimensional incompressible flow with Navier friction condition. SIAM J. Math. Anal. 36(4), 1130–1141 (2005)MathSciNetMATHCrossRef
14.
15.
Zurück zum Zitat Matthews, M.T., Hill, J.M.: Newtonian flow with nonlinear Navier boundary condition. Acta Mech. 191, 195–217 (2007)MATHCrossRef Matthews, M.T., Hill, J.M.: Newtonian flow with nonlinear Navier boundary condition. Acta Mech. 191, 195–217 (2007)MATHCrossRef
16.
Zurück zum Zitat Maxwell, J.C.: On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. London 170, 231–256 (1879)MATHCrossRef Maxwell, J.C.: On stresses in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. London 170, 231–256 (1879)MATHCrossRef
17.
Zurück zum Zitat Navier, C.L.M.H.: Sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Inst. Fr. 6, 389–440 (1827) Navier, C.L.M.H.: Sur les lois du mouvement des fluides. Mem. Acad. R. Sci. Inst. Fr. 6, 389–440 (1827)
18.
Zurück zum Zitat Necas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (2012)MATHCrossRef Necas, J.: Direct Methods in the Theory of Elliptic Equations. Springer, Berlin (2012)MATHCrossRef
19.
Zurück zum Zitat Pal, S., Haloi, R.: On Solution to the Navier–Stokes equations with Navier-slip boundary condition for three dimensional incompressible fluid, Acta Math. Sci. Ser. B Engl. Ed. 39(6), 1628–1638 (2019)MathSciNet Pal, S., Haloi, R.: On Solution to the Navier–Stokes equations with Navier-slip boundary condition for three dimensional incompressible fluid, Acta Math. Sci. Ser. B Engl. Ed. 39(6), 1628–1638 (2019)MathSciNet
20.
Zurück zum Zitat Sohr, H.: The Navier–Stokes Equations An Elementary Functional Analytic Approach Modern Birkhäuser Classics. Springer, Basel (2001)MATH Sohr, H.: The Navier–Stokes Equations An Elementary Functional Analytic Approach Modern Birkhäuser Classics. Springer, Basel (2001)MATH
21.
Zurück zum Zitat Song, X., Hou, Y.: Attractors for the three dimensional incompressible Navier–Stokes equations with damping. Discret. Contin. Dyn. Syst. 31, 239–252 (2012)MathSciNetMATHCrossRef Song, X., Hou, Y.: Attractors for the three dimensional incompressible Navier–Stokes equations with damping. Discret. Contin. Dyn. Syst. 31, 239–252 (2012)MathSciNetMATHCrossRef
22.
Zurück zum Zitat Song, X., Hou, Y.: Uniform attractors for three dimensional incompressible Navier–Stokes equation with nonlinear damping. J. Math. Anal. Appl. 422, 337–351 (2015)MathSciNetMATHCrossRef Song, X., Hou, Y.: Uniform attractors for three dimensional incompressible Navier–Stokes equation with nonlinear damping. J. Math. Anal. Appl. 422, 337–351 (2015)MathSciNetMATHCrossRef
23.
Zurück zum Zitat Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979)MATH Temam, R.: Navier–Stokes Equations. North-Holland, Amsterdam (1979)MATH
24.
Zurück zum Zitat Xiao, Y.L., Xin, Z.P.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60(7), 1027–1055 (2007)MathSciNetMATHCrossRef Xiao, Y.L., Xin, Z.P.: On the vanishing viscosity limit for the 3D Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Math. 60(7), 1027–1055 (2007)MathSciNetMATHCrossRef
25.
Zurück zum Zitat Yamazaki, K.: Stochastic Lagrangian formulations for damped Navier–Stokes equations and boussinesq system, with applications. Commun. Stoch. Anal 12, 447–471 (2018)MathSciNet Yamazaki, K.: Stochastic Lagrangian formulations for damped Navier–Stokes equations and boussinesq system, with applications. Commun. Stoch. Anal 12, 447–471 (2018)MathSciNet
26.
Zurück zum Zitat Zajaczkowski, W.M.: Global special regular solutions to the Navier–Stokes equations in a cylindrical domain without the axis of symmetry. Topol. Methods Nonlinear Anal. 24(1), 69–105 (2004)MathSciNetMATHCrossRef Zajaczkowski, W.M.: Global special regular solutions to the Navier–Stokes equations in a cylindrical domain without the axis of symmetry. Topol. Methods Nonlinear Anal. 24(1), 69–105 (2004)MathSciNetMATHCrossRef
27.
Zurück zum Zitat Zhang, Z., Wu, X., Lu, M.: On the uniqueness of strong solution to the incompressible Navier–Stokes equation with damping. J. Math. Anal. Appl. 377, 414–419 (2011)MathSciNetMATHCrossRef Zhang, Z., Wu, X., Lu, M.: On the uniqueness of strong solution to the incompressible Navier–Stokes equation with damping. J. Math. Anal. Appl. 377, 414–419 (2011)MathSciNetMATHCrossRef
28.
Zurück zum Zitat Zhou, Y.: Regularity and uniqueness for the 3D incompressible Navier–Stokes equations with damping. Appl. Math. Lett. 25, 1822–1825 (2012)MathSciNetMATHCrossRef Zhou, Y.: Regularity and uniqueness for the 3D incompressible Navier–Stokes equations with damping. Appl. Math. Lett. 25, 1822–1825 (2012)MathSciNetMATHCrossRef
Metadaten
Titel
Existence and uniqueness of solutions to the damped Navier–Stokes equations with Navier boundary conditions for three dimensional incompressible fluid
verfasst von
Subha Pal
Rajib Haloi
Publikationsdatum
26.09.2020
Verlag
Springer Berlin Heidelberg
Erschienen in
Journal of Applied Mathematics and Computing / Ausgabe 1-2/2021
Print ISSN: 1598-5865
Elektronische ISSN: 1865-2085
DOI
https://doi.org/10.1007/s12190-020-01437-1

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