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Pointwise spatial decay of time-dependent Oseen flows: The case of data with noncompact support
1. | Univ Lille Nord de France, 59000 Lille |
References:
[1] |
R. A. Adams, "Sobolev Spaces," Academic Press, New York e.a., 1975. |
[2] |
K. I. Babenko and M. M. Vasil'ev, On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body, Prikl. Mat. Meh., 37 (1973), 690-705 (Russian); English translation, J. Appl. Math. Mech., 37 (1973), 651-665. |
[3] |
H.-O. Bae and B. J. Jin, Estimates of the wake for the 3D Oseen equations, DCDS-B, 10 (2008), 1-18.
doi: 10.3934/dcdsb.2008.10.1. |
[4] |
H.-O. Bae and J. Roh, Stability for the 3D Navier-Stokes equations with nonzero far field velocity on exterior domains, J. Math. Fluid Mech., 14 (2012), 117-139.
doi: 10.1007/s00021-010-0040-z. |
[5] |
P. Deuring, Exterior stationary Navier-Stokes flows in 3D with nonzero velocity at infinity: asymptotic behaviour of the velocity and its gradient, IASME Transactions, 6 (2005), 900-904. |
[6] |
P. Deuring, The single-layer potential associated with the time-dependent Oseen system, in "Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics" Chalkida, Greeece, (2006), 117-125. |
[7] |
P. Deuring, On volume potentials related to the time-dependent Oseen system, WSEAS Transactions on Math., 5 (2006), 252-259. |
[8] |
P. Deuring, On boundary driven time-dependent Oseen flows, Banach Center Publications, 81 (2008), 119-132.
doi: 10.4064/bc81-0-8. |
[9] |
P. Deuring, A potential theoretic approach to the time-dependent Oseen system, in "Advances in Mathematical Fluid Mechanics. Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday" (eds. R. Rannacher and A. Sequeira), Springer (2010), 191-214.
doi: 10.1007/978-3-642-04068-9_12. |
[10] |
P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922.
doi: 10.1137/080723831. |
[11] |
P. Deuring, A representation formula for the velocity part of 3D time-dependent Oseen flows,, accepted by J. Math. Fluid Mechanics., ().
|
[12] |
P. Deuring, The Cauchy problem for the homogeneous time-dependent Oseen system in $ \mathbbR^3 $: Spatial decay of the velocity,, to appear in Mathematica Bohemica., ().
|
[13] |
P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity,, submitted., ().
|
[14] |
P. Deuring and S. Kračmar, Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: Approximation by flows in bounded domains, Math. Nachr., 269/270 (2004), 86-115.
doi: 10.1002/mana.200310167. |
[15] |
P. Deuring, S. Kračmar and Š. Nečasová, On pointwise decay of linearized stationary incompressible viscous flow around rotating and tranlating bodies, SIAM J. Math. Anal., 43 (2011), 705-738.
doi: 10.1137/100786198. |
[16] |
Y. Enomoto and Y. Shibata, Local energy decay of solutions to the Oseen equation in the exterior domain, Indiana Univ. Math. J., 53 (2004), 1291-1330.
doi: 10.1512/iumj.2004.53.2463. |
[17] |
Y. Enomoto and Y. Shibata, On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation, J. Math. Fluid Mech., 7 (2005), 339-367.
doi: 10.1007/s00021-004-0132-8. |
[18] |
R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.
doi: 10.1007/BF02571437. |
[19] |
R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. |
[20] |
S. Fučik, O. John and A. Kufner, "Function Spaces," Noordhoff, Leyden, 1977. |
[21] |
G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems," (corr. 2nd print.), Springer, New York e.a., 1998.
doi: 10.1007/978-1-4612-5364-8. |
[22] |
G. P. Galdi, "An introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer, New York e.a., 1994.
doi: 10.1007/978-1-4612-5364-8. |
[23] |
J. G. Heywood, The exterior nonstationary problem for the Navier-Stokes equations, Acta Math., 129 (1972), 11-34. |
[24] |
J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681.
doi: 10.1512/iumj.1980.29.29048. |
[25] |
G. H. Knightly, A Cauchy problem for the Navier-Stokes equations in $ \mathbbR ^n$, SIAM J. Math. Anal., 3 (1972), 506-511. |
[26] |
G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations, in "Approximation Methods for Navier-Stokes Problems" (ed. R. Rautmann), Lecture Notes in Math., 771, Springer (1979), 287-298. |
[27] |
T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains, Math. Ann., 310 (1998), 1-45.
doi: 10.1007/s002080050134. |
[28] |
S. Kračmar, A. Novotný and M. Pokorný, Estimates of Oseen kernels in weighted $L^p$ spaces, J. Math. Soc. Japan, 53 (2001), 59-111.
doi: 10.2969/jmsj/05310059. |
[29] |
K. Masuda, On the stability of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 27 (1975), 294-327. |
[30] |
M. McCracken, The resolvent problem for the Stokes equations on halfspace in $L_p^*$, SIAM J. Math. Anal., 12 (1981), 201-228.
doi: 10.1137/0512021. |
[31] |
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. |
[32] |
R. Mizumachi, On the asymptotic behaviour of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 36 (1984), 497-522.
doi: 10.2969/jmsj/03630497. |
[33] |
Zongwei Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, American J. Math., 113 (1991), 293-373.
doi: 10.2307/2374910. |
[34] |
Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quarterly Appl. Math., 57 (1999), 117-155. |
[35] |
V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231 (Russian); English translation, J. Soviet Math., 8 (1977), 467-529. |
[36] |
S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789.
doi: 10.1016/S0362-546X(98)00070-4. |
[37] |
R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence R.I., 2001. |
[38] |
K. Yoshida, "Functional Analysis," (6th ed.), Springer, Berlin e.a., 1980. |
show all references
References:
[1] |
R. A. Adams, "Sobolev Spaces," Academic Press, New York e.a., 1975. |
[2] |
K. I. Babenko and M. M. Vasil'ev, On the asymptotic behavior of a steady flow of viscous fluid at some distance from an immersed body, Prikl. Mat. Meh., 37 (1973), 690-705 (Russian); English translation, J. Appl. Math. Mech., 37 (1973), 651-665. |
[3] |
H.-O. Bae and B. J. Jin, Estimates of the wake for the 3D Oseen equations, DCDS-B, 10 (2008), 1-18.
doi: 10.3934/dcdsb.2008.10.1. |
[4] |
H.-O. Bae and J. Roh, Stability for the 3D Navier-Stokes equations with nonzero far field velocity on exterior domains, J. Math. Fluid Mech., 14 (2012), 117-139.
doi: 10.1007/s00021-010-0040-z. |
[5] |
P. Deuring, Exterior stationary Navier-Stokes flows in 3D with nonzero velocity at infinity: asymptotic behaviour of the velocity and its gradient, IASME Transactions, 6 (2005), 900-904. |
[6] |
P. Deuring, The single-layer potential associated with the time-dependent Oseen system, in "Proceedings of the 2006 IASME/WSEAS International Conference on Continuum Mechanics" Chalkida, Greeece, (2006), 117-125. |
[7] |
P. Deuring, On volume potentials related to the time-dependent Oseen system, WSEAS Transactions on Math., 5 (2006), 252-259. |
[8] |
P. Deuring, On boundary driven time-dependent Oseen flows, Banach Center Publications, 81 (2008), 119-132.
doi: 10.4064/bc81-0-8. |
[9] |
P. Deuring, A potential theoretic approach to the time-dependent Oseen system, in "Advances in Mathematical Fluid Mechanics. Dedicated to Giovanni Paolo Galdi on the Occasion of his 60th Birthday" (eds. R. Rannacher and A. Sequeira), Springer (2010), 191-214.
doi: 10.1007/978-3-642-04068-9_12. |
[10] |
P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922.
doi: 10.1137/080723831. |
[11] |
P. Deuring, A representation formula for the velocity part of 3D time-dependent Oseen flows,, accepted by J. Math. Fluid Mechanics., ().
|
[12] |
P. Deuring, The Cauchy problem for the homogeneous time-dependent Oseen system in $ \mathbbR^3 $: Spatial decay of the velocity,, to appear in Mathematica Bohemica., ().
|
[13] |
P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity,, submitted., ().
|
[14] |
P. Deuring and S. Kračmar, Exterior stationary Navier-Stokes flows in 3D with non-zero velocity at infinity: Approximation by flows in bounded domains, Math. Nachr., 269/270 (2004), 86-115.
doi: 10.1002/mana.200310167. |
[15] |
P. Deuring, S. Kračmar and Š. Nečasová, On pointwise decay of linearized stationary incompressible viscous flow around rotating and tranlating bodies, SIAM J. Math. Anal., 43 (2011), 705-738.
doi: 10.1137/100786198. |
[16] |
Y. Enomoto and Y. Shibata, Local energy decay of solutions to the Oseen equation in the exterior domain, Indiana Univ. Math. J., 53 (2004), 1291-1330.
doi: 10.1512/iumj.2004.53.2463. |
[17] |
Y. Enomoto and Y. Shibata, On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier-Stokes equation, J. Math. Fluid Mech., 7 (2005), 339-367.
doi: 10.1007/s00021-004-0132-8. |
[18] |
R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.
doi: 10.1007/BF02571437. |
[19] |
R. Finn, On the exterior stationary problem for the Navier-Stokes equations, and associated perturbation problems, Arch. Rational Mech. Anal., 19 (1965), 363-406. |
[20] |
S. Fučik, O. John and A. Kufner, "Function Spaces," Noordhoff, Leyden, 1977. |
[21] |
G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearised Steady Problems," (corr. 2nd print.), Springer, New York e.a., 1998.
doi: 10.1007/978-1-4612-5364-8. |
[22] |
G. P. Galdi, "An introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems," Springer, New York e.a., 1994.
doi: 10.1007/978-1-4612-5364-8. |
[23] |
J. G. Heywood, The exterior nonstationary problem for the Navier-Stokes equations, Acta Math., 129 (1972), 11-34. |
[24] |
J. G. Heywood, The Navier-Stokes equations. On the existence, regularity and decay of solutions, Indiana Univ. Math. J., 29 (1980), 639-681.
doi: 10.1512/iumj.1980.29.29048. |
[25] |
G. H. Knightly, A Cauchy problem for the Navier-Stokes equations in $ \mathbbR ^n$, SIAM J. Math. Anal., 3 (1972), 506-511. |
[26] |
G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations, in "Approximation Methods for Navier-Stokes Problems" (ed. R. Rautmann), Lecture Notes in Math., 771, Springer (1979), 287-298. |
[27] |
T. Kobayashi and Y. Shibata, On the Oseen equation in three dimensional exterior domains, Math. Ann., 310 (1998), 1-45.
doi: 10.1007/s002080050134. |
[28] |
S. Kračmar, A. Novotný and M. Pokorný, Estimates of Oseen kernels in weighted $L^p$ spaces, J. Math. Soc. Japan, 53 (2001), 59-111.
doi: 10.2969/jmsj/05310059. |
[29] |
K. Masuda, On the stability of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 27 (1975), 294-327. |
[30] |
M. McCracken, The resolvent problem for the Stokes equations on halfspace in $L_p^*$, SIAM J. Math. Anal., 12 (1981), 201-228.
doi: 10.1137/0512021. |
[31] |
T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140. |
[32] |
R. Mizumachi, On the asymptotic behaviour of incompressible viscous fluid motions past bodies, J. Math. Soc. Japan, 36 (1984), 497-522.
doi: 10.2969/jmsj/03630497. |
[33] |
Zongwei Shen, Boundary value problems for parabolic Lamé systems and a nonstationary linearized system of Navier-Stokes equations in Lipschitz cylinders, American J. Math., 113 (1991), 293-373.
doi: 10.2307/2374910. |
[34] |
Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quarterly Appl. Math., 57 (1999), 117-155. |
[35] |
V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 38 (1973), 153-231 (Russian); English translation, J. Soviet Math., 8 (1977), 467-529. |
[36] |
S. Takahashi, A weighted equation approach to decay rate estimates for the Navier-Stokes equations, Nonlinear Anal., 37 (1999), 751-789.
doi: 10.1016/S0362-546X(98)00070-4. |
[37] |
R. Teman, "Navier-Stokes Equations. Theory and Numerical Analysis," AMS Chelsea Publishing, Providence R.I., 2001. |
[38] |
K. Yoshida, "Functional Analysis," (6th ed.), Springer, Berlin e.a., 1980. |
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