# American Institute of Mathematical Sciences

October  2013, 6(5): 1355-1369. doi: 10.3934/dcdss.2013.6.1355

## Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains

 1 Sylvie Monniaux - LATP CMI Université Aix-Marseille, 39 rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France

Received  November 2011 Revised  December 2011 Published  March 2013

We present here different boundary conditions for the Navier-Stokes equations in bounded Lipschitz domains in $\mathbb{R}^3$, such as Dirichlet, Neumann or Hodge boundary conditions. We first study the linear Stokes operator associated to the boundary conditions. Then we show how the properties of the operator lead to local solutions or global solutions for small initial data.
Citation: Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
##### References:
 [1] Paul Deuring, The Stokes resolvent in 3D domains with conical boundary points: Nonregularity in $L^p$-spaces, Adv. Differential Equations, 6 (2001), 175-228. [2] Paul Deuring and Wolf von Wahl, Strong solutions of the Navier-Stokes system in Lipschitz bounded domains, Math. Nachr., 171 (1995), 111-148. doi: 10.1002/mana.19951710108. [3] Eugene Fabes, Osvaldo Mendez and Marius Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), 323-368. doi: 10.1006/jfan.1998.3316. [4] Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. [5] Yoshikazu Giga, Analyticity of the semigroup generated by the Stokes operator in $L_r$ spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869. [6] Yoshikazu Giga and Tetsuro Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875. [7] Alf Jonsson and Hans Wallin, Function spaces on subsets of $R^n$, Math. Rep., 2 (1984), xiv+221. [8] Jacques-Louis Lions, Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241. [9] Dorina Mitrea, Marius Mitrea and Sylvie Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal., 7 (2008), 1295-1333. doi: 10.3934/cpaa.2008.7.1295. [10] Dorina Mitrea, Marius Mitrea and Michael Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Mem. Amer. Math. Soc., 150 (2001), x+120. [11] Marius Mitrea and Sylvie Monniaux, The regularity of the Stokes operator and the Fujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains, J. Funct. Anal., 254 (2008), 1522-1574. doi: 10.1016/j.jfa.2007.11.021. [12] Marius Mitrea and Sylvie Monniaux, On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 3125-3157. doi: 10.1090/S0002-9947-08-04827-7. [13] Marius Mitrea and Sylvie Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential Integral Equations, 22 (2009), 339-356. [14] Marius Mitrea, Sylvie Monniaux and Matthew Wright, The Stokes operator with Neumann boundary conditions in Lipschitz domains, J. Math. Sci. (N. Y.), 176 (2011), 409-457. doi: 10.1007/s10958-011-0400-0. [15] Sylvie Monniaux, On uniqueness for the Navier-Stokes system in 3D-bounded Lipschitz domains, J. Funct. Anal., 195 (2002), 1-11. doi: 10.1006/jfan.2002.3902. [16] Sylvie Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461. [17] Sylvie Monniaux, Maximal regularity and applications to PDEs, in "Analytical and Numerical Aspects of Partial Differential Equations," Walter de Gruyter, Berlin, (2009), 247-287. [18] Michael E. Taylor, Incompressible fluid flows on rough domains, in "Semigroups of Operators: Theory and Applications" (Newport Beach, CA, 1998), Progr. Nonlinear Differential Equations Appl., 42, Birkhäuser, Basel, 2000, 320-334.

show all references

##### References:
 [1] Paul Deuring, The Stokes resolvent in 3D domains with conical boundary points: Nonregularity in $L^p$-spaces, Adv. Differential Equations, 6 (2001), 175-228. [2] Paul Deuring and Wolf von Wahl, Strong solutions of the Navier-Stokes system in Lipschitz bounded domains, Math. Nachr., 171 (1995), 111-148. doi: 10.1002/mana.19951710108. [3] Eugene Fabes, Osvaldo Mendez and Marius Mitrea, Boundary layers on Sobolev-Besov spaces and Poisson's equation for the Laplacian in Lipschitz domains, J. Funct. Anal., 159 (1998), 323-368. doi: 10.1006/jfan.1998.3316. [4] Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315. [5] Yoshikazu Giga, Analyticity of the semigroup generated by the Stokes operator in $L_r$ spaces, Math. Z., 178 (1981), 297-329. doi: 10.1007/BF01214869. [6] Yoshikazu Giga and Tetsuro Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267-281. doi: 10.1007/BF00276875. [7] Alf Jonsson and Hans Wallin, Function spaces on subsets of $R^n$, Math. Rep., 2 (1984), xiv+221. [8] Jacques-Louis Lions, Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241. [9] Dorina Mitrea, Marius Mitrea and Sylvie Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition in nonsmooth domains, Commun. Pure Appl. Anal., 7 (2008), 1295-1333. doi: 10.3934/cpaa.2008.7.1295. [10] Dorina Mitrea, Marius Mitrea and Michael Taylor, Layer potentials, the Hodge Laplacian, and global boundary problems in nonsmooth Riemannian manifolds, Mem. Amer. Math. Soc., 150 (2001), x+120. [11] Marius Mitrea and Sylvie Monniaux, The regularity of the Stokes operator and the Fujita-Kato approach to the Navier-Stokes initial value problem in Lipschitz domains, J. Funct. Anal., 254 (2008), 1522-1574. doi: 10.1016/j.jfa.2007.11.021. [12] Marius Mitrea and Sylvie Monniaux, On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds, Trans. Amer. Math. Soc., 361 (2009), 3125-3157. doi: 10.1090/S0002-9947-08-04827-7. [13] Marius Mitrea and Sylvie Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential Integral Equations, 22 (2009), 339-356. [14] Marius Mitrea, Sylvie Monniaux and Matthew Wright, The Stokes operator with Neumann boundary conditions in Lipschitz domains, J. Math. Sci. (N. Y.), 176 (2011), 409-457. doi: 10.1007/s10958-011-0400-0. [15] Sylvie Monniaux, On uniqueness for the Navier-Stokes system in 3D-bounded Lipschitz domains, J. Funct. Anal., 195 (2002), 1-11. doi: 10.1006/jfan.2002.3902. [16] Sylvie Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461. [17] Sylvie Monniaux, Maximal regularity and applications to PDEs, in "Analytical and Numerical Aspects of Partial Differential Equations," Walter de Gruyter, Berlin, (2009), 247-287. [18] Michael E. Taylor, Incompressible fluid flows on rough domains, in "Semigroups of Operators: Theory and Applications" (Newport Beach, CA, 1998), Progr. Nonlinear Differential Equations Appl., 42, Birkhäuser, Basel, 2000, 320-334.
 [1] Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151 [2] Chérif Amrouche, Nour El Houda Seloula. $L^p$-theory for the Navier-Stokes equations with pressure boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1113-1137. doi: 10.3934/dcdss.2013.6.1113 [3] Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408 [4] Siegfried Maier, Jürgen Saal. Stokes and Navier-Stokes equations with perfect slip on wedge type domains. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 1045-1063. doi: 10.3934/dcdss.2014.7.1045 [5] Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete and Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783 [6] Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169 [7] Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043 [8] Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673 [9] Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121 [10] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277 [11] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135 [12] 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 and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [13] Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325 [14] Franck Boyer, Pierre Fabrie. Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 219-250. doi: 10.3934/dcdsb.2007.7.219 [15] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 [16] Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2811-2838. doi: 10.3934/cpaa.2021080 [17] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [18] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [19] Hongjie Dong, Kunrui Wang. Interior and boundary regularity for the Navier-Stokes equations in the critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5289-5323. doi: 10.3934/dcds.2020228 [20] Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

2020 Impact Factor: 2.425