# American Institute of Mathematical Sciences

• Previous Article
A nonlinear effective slip interface law for transport phenomena between a fracture flow and a porous medium
• DCDS-S Home
• This Issue
• Next Article
Existence and decay of solutions of the 2D QG equation in the presence of an obstacle
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

 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 $L^p$.
Citation: 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
##### 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. [2] G. Da Prato and P. Grisvard, Sommes d'oprateurs linaires et quations diffrentielles oprationelles, J. Math. Pures Appl., 54 (1975), 305-387. [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. [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. [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. [6] A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, 1969. [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. [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. [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. [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. [11] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev Spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869. [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. [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. [14] R. Labbas and B. Terreni, Somme d'opérateurs linéaires de type parabolique, Boll. Un. Mat. Ital., 7 (1987), 545-569. [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. [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. [17] M. Mitrea and S. Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential and Integral Equations, 22 (2009), 339-356. [18] T. Nau and J. Saal, H-infinity-calculus for cylindrical boundary value problems, Advances in Differential Equations, 17 (2012), 767-800. [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. [20] A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on Lq-spaces, Math. Z., 244 (2003), 651-688. [21] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [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. [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. [24] J. Saal, Robin Boundary Conditions and Bounded $H^\infty$-Calculus for the Stokes Operator, Logos-Verlag, Ph.D thesis, Tu Darmstadt, 2003. [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. [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. [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.

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. [2] G. Da Prato and P. Grisvard, Sommes d'oprateurs linaires et quations diffrentielles oprationelles, J. Math. Pures Appl., 54 (1975), 305-387. [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. [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. [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. [6] A. Friedman, Partial Differential Equations, Holt, Rinehard and Winston, 1969. [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. [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. [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. [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. [11] P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev Spaces, Acta Math., 147 (1981), 71-88. doi: 10.1007/BF02392869. [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. [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. [14] R. Labbas and B. Terreni, Somme d'opérateurs linéaires de type parabolique, Boll. Un. Mat. Ital., 7 (1987), 545-569. [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. [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. [17] M. Mitrea and S. Monniaux, The nonlinear Hodge-Navier-Stokes equations in Lipschitz domains, Differential and Integral Equations, 22 (2009), 339-356. [18] T. Nau and J. Saal, H-infinity-calculus for cylindrical boundary value problems, Advances in Differential Equations, 17 (2012), 767-800. [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. [20] A. Noll and J. Saal, $H^\infty$-calculus for the Stokes operator on Lq-spaces, Math. Z., 244 (2003), 651-688. [21] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6. [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. [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. [24] J. Saal, Robin Boundary Conditions and Bounded $H^\infty$-Calculus for the Stokes Operator, Logos-Verlag, Ph.D thesis, Tu Darmstadt, 2003. [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. [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. [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.
 [1] Matthias Geissert, Horst Heck, Christof Trunk. $H^{\infty}$-calculus for a system of Laplace operators with mixed order boundary conditions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1259-1275. doi: 10.3934/dcdss.2013.6.1259 [2] Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315 [3] Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2155-2185. doi: 10.3934/cpaa.2021062 [4] Xingyue Liang, Jianwei Xia, Guoliang Chen, Huasheng Zhang, Zhen Wang. $\mathcal{H}_{\infty}$ control for fuzzy markovian jump systems based on sampled-data control method. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1329-1343. doi: 10.3934/dcdss.2020368 [5] Boris Muha, Zvonimir Tutek. Note on evolutionary free piston problem for Stokes equations with slip boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1629-1639. doi: 10.3934/cpaa.2014.13.1629 [6] 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 [7] 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 [8] Linjie Xiong. Incompressible Limit of isentropic Navier-Stokes equations with Navier-slip boundary. Kinetic and Related Models, 2018, 11 (3) : 469-490. doi: 10.3934/krm.2018021 [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] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [11] 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 [12] Gilberto M. Kremer, Filipe Oliveira, Ana Jacinta Soares. $\mathcal H$-Theorem and trend to equilibrium of chemically reacting mixtures of gases. Kinetic and Related Models, 2009, 2 (2) : 333-343. doi: 10.3934/krm.2009.2.333 [13] Ming Wang, Yanbin Tang. Attractors in $H^2$ and $L^{2p-2}$ for reaction diffusion equations on unbounded domains. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1111-1121. doi: 10.3934/cpaa.2013.12.1111 [14] Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^∞$ via the singular value problem. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266 [15] 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 [16] Minghua Yang, Zunwei Fu, Jinyi Sun. Global solutions to Chemotaxis-Navier-Stokes equations in critical Besov spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3427-3460. doi: 10.3934/dcdsb.2018284 [17] Eduard Feireisl, Josef Málek, Antonín Novotný. Navier's slip and incompressible limits in domains with variable bottoms. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 427-460. doi: 10.3934/dcdss.2008.1.427 [18] M. S. Mahmoud, P. Shi, Y. Shi. $H_\infty$ and robust control of interconnected systems with Markovian jump parameters. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 365-384. doi: 10.3934/dcdsb.2005.5.365 [19] Amol Sasane. Extension of the $\nu$-metric for stabilizable plants over $H^\infty$. Mathematical Control and Related Fields, 2012, 2 (1) : 29-44. doi: 10.3934/mcrf.2012.2.29 [20] Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $H_{\infty}$ control design for active suspension systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 197-212. doi: 10.3934/dcdss.2021036

2021 Impact Factor: 1.865