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Existence and decay of solutions of the 2D QG equation in the presence of an obstacle
Stokes and Navier-Stokes equations with perfect slip on wedge type domains
1. | Heinrich-Heine-Universität Düsseldorf, Mathematisches Institut, 40204 Düsseldorf, Germany, Germany |
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. |
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