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October  2014, 7(5): 1045-1063. doi: 10.3934/dcdss.2014.7.1045

Stokes and Navier-Stokes equations with perfect slip on wedge type domains

Siegfried Maier 1, and  Jürgen Saal 1,

1. 

Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut, 40204 Düsseldorf, Germany, Germany

Received  March 2013 Revised  June 2013 Published  May 2014

Well-posedness of the Stokes and Navier-Stokes equations subject to perfect slip boundary conditions on wedge type domains is studied. Applying the operator sum method we derive an $\mathcal{H}^\infty$-calculus for the Stokes operator in weighted $L^p_\gamma$ spaces (Kondrat'ev spaces) which yields maximal regularity for the linear Stokes system. This in turn implies mild well-posedness for the Navier-Stokes equations, locally-in-time for arbitrary and globally-in-time for small data in $L^p$.
Keywords: Kondrat'ev spaces, perfect slip, $\mathcal{H}^\infty$-calculus., Stokes equations, wedge domains.
Mathematics Subject Classification: Primary: 76D035, 35K65; Secondary: 76D0.
Citation: 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
References:
[1]

W. Borchers and T. Miyakawa, $L^2$ decay for the Navier-Stokes flow in halfspaces, Math. Ann., 282 (1988), 139-155. doi: 10.1007/BF01457017.  Google Scholar

[2]

G. Da Prato and P. Grisvard, Sommes d'oprateurs linaires et quations diffrentielles oprationelles, J. Math. Pures Appl., 54 (1975), 305-387.  Google Scholar

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R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.  Google Scholar

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R. Denk and M. Geißert, J. Saal and O. Sawada, The spin-coating process: Analysis of the free boundary value problem, Commun. Partial Differ. Equations, 36 (2011), 1145-1192. doi: 10.1080/03605302.2010.546469.  Google Scholar

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A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, 1969.  Google Scholar

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A. Friedman and J. L. Velázquez, Time-dependent coating flows in a strip. I: The linearized problem, Trans. Am. Math. Soc., 349 (1997), 2981-3074. doi: 10.1090/S0002-9947-97-01956-9.  Google Scholar

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G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, Springer Monographs in Mathematics, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

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Y. Giga, Solutions for semilinear parabolic equations in $L_p$ and regularity of weak solutions of the Navier-Stokes system, Journal of Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

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M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

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P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev Spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869.  Google Scholar

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N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345. doi: 10.1007/s002080100231.  Google Scholar

[13]

P. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, in Functional analytic methods for evolution equations, Lecture Notes in Math., 1855, Springer, Berlin, 2004, 65-311. doi: 10.1007/978-3-540-44653-8_2.  Google Scholar

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R. Labbas and B. Terreni, Somme d'opérateurs linéaires de type parabolique, Boll. Un. Mat. Ital., 7 (1987), 545-569.  Google Scholar

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V. N. Maslennikova and M. E. Bogovski, Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries, Rendiconti del Seminario Matematico e Fisico di Milano, 56 (1986), 125-138. doi: 10.1007/BF02925141.  Google Scholar

[16]

M. Mitrea and S. Monniaux, On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds, Transactions of the American Mathematical Society, 361 (2009), 3125-3157. doi: 10.1090/S0002-9947-08-04827-7.  Google Scholar

[17]

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

[18]

T. Nau and J. Saal, H-infinity-calculus for cylindrical boundary value problems, Advances in Differential Equations, 17 (2012), 767-800.  Google Scholar

[19]

A. I. Nazarov, $L_p$-estimates for a solution to the Dirichlet problem and to the Neumann problem for the heat equation in a wedge with edge of arbitrary codimension, J. Math. Sci., 106 (2001), 2989-3014. doi: 10.1023/A:1011319521775.  Google Scholar

[20]

A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on Lq-spaces, Math. Z., 244 (2003), 651-688.  Google Scholar

[21]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[22]

J. Prüss and S. Shimizu and Y. Shibata and G. Simonett, On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities, Evolution Equations and Control Theory, 1 (2012), 171-194. doi: 10.3934/eect.2012.1.171.  Google Scholar

[23]

J. Prüss and G. Simonett, $H^{\infty}$-calculus for the sum of non-commuting operators, Trans. Amer. Math. Soc., 359 (2007), 3549-3565. doi: 10.1090/S0002-9947-07-04291-2.  Google Scholar

[24]

J. Saal, Robin Boundary Conditions and Bounded $H^\infty$-Calculus for the Stokes Operator, Logos-Verlag, Ph.D thesis, Tu Darmstadt, 2003. Google Scholar

[25]

J. Saal, Stokes and Navier-Stokes equations with Robin boundary conditions in a half-space, J. Math. Fluid Mech., 8 (2006), 211-241. doi: 10.1007/s00021-004-0143-5.  Google Scholar

[26]

B. Schweizer, A well-posed model for dynamic contact angles, Nonlinear Anal. Theory Methods Appl., 43 (2001), 109-125. doi: 10.1016/S0362-546X(99)00183-2.  Google Scholar

[27]

V. A. Solonnikov, On some free boundary problems for the Navier-Stokes equations with moving contact points and lines, Math. Ann., 302 (1995), 743-772. doi: 10.1007/BF01444515.  Google Scholar

show all references

References:
[1]

W. Borchers and T. Miyakawa, $L^2$ decay for the Navier-Stokes flow in halfspaces, Math. Ann., 282 (1988), 139-155. doi: 10.1007/BF01457017.  Google Scholar

[2]

G. Da Prato and P. Grisvard, Sommes d'oprateurs linaires et quations diffrentielles oprationelles, J. Math. Pures Appl., 54 (1975), 305-387.  Google Scholar

[3]

R. Denk, M. Hieber and J. Prüss, $\mathcalR$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Am. Math. Soc., 166 (2003). doi: 10.1090/memo/0788.  Google Scholar

[4]

R. Denk and M. Geißert, J. Saal and O. Sawada, The spin-coating process: Analysis of the free boundary value problem, Commun. Partial Differ. Equations, 36 (2011), 1145-1192. doi: 10.1080/03605302.2010.546469.  Google Scholar

[5]

G. Dore and A. Venni, On the closedness of the sum of two operators, Math. Z., 196 (1987), 189-201. doi: 10.1007/BF01163654.  Google Scholar

[6]

A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, 1969.  Google Scholar

[7]

A. Friedman and J. L. Velázquez, Time-dependent coating flows in a strip. I: The linearized problem, Trans. Am. Math. Soc., 349 (1997), 2981-3074. doi: 10.1090/S0002-9947-97-01956-9.  Google Scholar

[8]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, Springer Monographs in Mathematics, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[9]

Y. Giga, Solutions for semilinear parabolic equations in $L_p$ and regularity of weak solutions of the Navier-Stokes system, Journal of Differential Equations, 62 (1986), 186-212. doi: 10.1016/0022-0396(86)90096-3.  Google Scholar

[10]

M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[11]

P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev Spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869.  Google Scholar

[12]

N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345. doi: 10.1007/s002080100231.  Google Scholar

[13]

P. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, in Functional analytic methods for evolution equations, Lecture Notes in Math., 1855, Springer, Berlin, 2004, 65-311. doi: 10.1007/978-3-540-44653-8_2.  Google Scholar

[14]

R. Labbas and B. Terreni, Somme d'opérateurs linéaires de type parabolique, Boll. Un. Mat. Ital., 7 (1987), 545-569.  Google Scholar

[15]

V. N. Maslennikova and M. E. Bogovski, Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries, Rendiconti del Seminario Matematico e Fisico di Milano, 56 (1986), 125-138. doi: 10.1007/BF02925141.  Google Scholar

[16]

M. Mitrea and S. Monniaux, On the analyticity of the semigroup generated by the Stokes operator with Neumann-type boundary conditions on Lipschitz subdomains of Riemannian manifolds, Transactions of the American Mathematical Society, 361 (2009), 3125-3157. doi: 10.1090/S0002-9947-08-04827-7.  Google Scholar

[17]

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

[18]

T. Nau and J. Saal, H-infinity-calculus for cylindrical boundary value problems, Advances in Differential Equations, 17 (2012), 767-800.  Google Scholar

[19]

A. I. Nazarov, $L_p$-estimates for a solution to the Dirichlet problem and to the Neumann problem for the heat equation in a wedge with edge of arbitrary codimension, J. Math. Sci., 106 (2001), 2989-3014. doi: 10.1023/A:1011319521775.  Google Scholar

[20]

A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on Lq-spaces, Math. Z., 244 (2003), 651-688.  Google Scholar

[21]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[22]

J. Prüss and S. Shimizu and Y. Shibata and G. Simonett, On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities, Evolution Equations and Control Theory, 1 (2012), 171-194. doi: 10.3934/eect.2012.1.171.  Google Scholar

[23]

J. Prüss and G. Simonett, $H^{\infty}$-calculus for the sum of non-commuting operators, Trans. Amer. Math. Soc., 359 (2007), 3549-3565. doi: 10.1090/S0002-9947-07-04291-2.  Google Scholar

[24]

J. Saal, Robin Boundary Conditions and Bounded $H^\infty$-Calculus for the Stokes Operator, Logos-Verlag, Ph.D thesis, Tu Darmstadt, 2003. Google Scholar

[25]

J. Saal, Stokes and Navier-Stokes equations with Robin boundary conditions in a half-space, J. Math. Fluid Mech., 8 (2006), 211-241. doi: 10.1007/s00021-004-0143-5.  Google Scholar

[26]

B. Schweizer, A well-posed model for dynamic contact angles, Nonlinear Anal. Theory Methods Appl., 43 (2001), 109-125. doi: 10.1016/S0362-546X(99)00183-2.  Google Scholar

[27]

V. A. Solonnikov, On some free boundary problems for the Navier-Stokes equations with moving contact points and lines, Math. Ann., 302 (1995), 743-772. doi: 10.1007/BF01444515.  Google Scholar

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