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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Alf Jonsson and Hans Wallin, Function spaces on subsets of $R^n$, Math. Rep., 2 (1984), xiv+221.  Google Scholar

[8]

Jacques-Louis Lions, Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Marius Mitrea and Sylvie Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential Integral Equations, 22 (2009), 339-356.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Sylvie Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Alf Jonsson and Hans Wallin, Function spaces on subsets of $R^n$, Math. Rep., 2 (1984), xiv+221.  Google Scholar

[8]

Jacques-Louis Lions, Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs, J. Math. Soc. Japan, 14 (1962), 233-241. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

Marius Mitrea and Sylvie Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential Integral Equations, 22 (2009), 339-356.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Sylvie Monniaux, Navier-Stokes equations in arbitrary domains: The Fujita-Kato scheme, Math. Res. Lett., 13 (2006), 455-461.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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