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

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

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

Sylvie Monniaux ^1,

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

Abstract
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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.

Keywords: Stokes operator, Neumann, Navier-Stokes equations., Dirichlet, Lipschitz domains, Hodge, boundary conditions.

Mathematics Subject Classification: Primary: 35Q30, 35Q35, 35J2.

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.

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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.

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