# American Institute of Mathematical Sciences

May  2021, 20(5): 1833-1849. doi: 10.3934/cpaa.2021044

## Spatial asymptotics of mild solutions to the time-dependent Oseen system

 Univ. du Littoral Côte d'Opale, EA 2797 – LMPA – Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, F-62228 Calais, France

Received  April 2020 Revised  January 2021 Published  March 2021

We consider mild solutions to the 3D time-dependent Oseen system with homogeneous Dirichlet boundary conditions, under weak assumptions on the data. Such solutions are defined via the semigroup generated by the Oseen operator in $L^q.$ They turn out to be also $L^q$-weak solutions to the Oseen system. On the basis of known results about spatial asymptotics of the latter type of solutions, we then derive pointwise estimates of the spatial decay of mild solutions. The rate of decay depends in particular on $L^p$-integrability in time of the external force.

Citation: Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1833-1849. doi: 10.3934/cpaa.2021044
##### References:
 [1] P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922.  doi: 10.1137/080723831.  Google Scholar [2] P. Deuring, The Cauchy problem for the homogeneous time-dependent Oseen system in $\mathbb{R}^3$: spatial decay of the velocity, Math. Bohemica, 138 (2013), 299-324.   Google Scholar [3] P. Deuring, Pointwise spatial decay of time-dependent Oseen flows: the case of data with noncompact support, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2757-2776.  doi: 10.3934/dcds.2013.33.2757.  Google Scholar [4] P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity, SIAM J. Math. Anal., 45 (2013), 1388-1421.  doi: 10.1137/120872255.  Google Scholar [5] P. Deuring, Oseen resolvent estimates with small resolvent parameter, J. Diff. Equ., 265 (2018), 280-311.  doi: 10.1016/j.jde.2018.02.033.  Google Scholar [6] P. Deuring, Pointwise decay in space and in time for incompressible flow around a rigid body moving with constant velocity, J. Math. Fluid Mech., 21 (2019), article 11. doi: 10.1007/s00021-019-0414-9.  Google Scholar [7] P. Deuring, The 3D time-dependent Oseen system: link between $L^p$-integrability in time and pointwise decay in space, to appear in J. Math. Fluid Mech, available from https://hal.archives-ouvertes.fr/hal-02465649. Google Scholar [8] P. Deuring, $L^q$-weak solutions to the time-dependent Oseen system: decay estimates, to appear in Math. Nachr, available from https://hal.archives-ouvertes.fr/hal-02465651. Google Scholar [9] P. Deuring, Time-dependent incompressible viscous flows around a rigid body: estimates of spatial decay independent of boundary conditions, preprint, Available from https://hal.archives-ouvertes.fr/hal-02508815. doi: 10.1137/080723831.  Google Scholar [10] P. Deuring and W. Varnhorn, On Oseen resolvent estimates, Diff. Int. Equat., 23 (2010), 1139-1149.   Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] R. Farwig and J. Neustupa, On the spectrum of an Oseen-type operator arising from flow around a rotating body, Int. Equ. Oper. Theory, 62 (2008), 169-189.  doi: 10.1007/s00020-008-1616-3.  Google Scholar [14] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2$^nd$ edition, Springer, New York e.a., 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar [15] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Math. Soc. Colloquium Publications, American Mathematical Society, Providence R. I., 1957.  Google Scholar [16] T. Hishida, Large time behavior of a generalized Oseen evolution operator, with applications to the Navier-Stokes flow past a rotating obstacle, Math. Ann., 372 (2018), 915-949.  doi: 10.1007/s00208-018-1649-0.  Google Scholar [17] T. Hishida, Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains, Arch. Rational Mech. Anal., 238 (2020), 215-254.  doi: 10.1007/s00205-020-01541-3.  Google Scholar [18] G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations, in Approximation methods for Navier-Stokes problems (ed. R. Rautmann), Springer, 1979.  Google Scholar [19] T. Kobayashi and Y. Shibata, On the Oseen equation in three-dimensional exterior domains, Math. Ann., 310 (1998), 1-45.  doi: 10.1007/s002080050134.  Google Scholar [20] H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains, Math. Ann., 312 (1998), 319-340.  doi: 10.1007/s002080050224.  Google Scholar [21] H. Kozono, Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains, Math. Ann., 320 (2001), 709-730.  doi: 10.1007/PL00004492.  Google Scholar [22] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.   Google Scholar [23] 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.  Google Scholar [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Science, Springer, New York, 1983. doi: 10.1007/PL00004457.  Google Scholar [25] L. Weis, Operator-valued Fourier multiplier theorems and maximal regularity, Math. Ann., 319 (2001), 735-758.   Google Scholar [26] K. Yosida, Functional Analysis, 6$^{th}$ edition, Springer, Berlin, 1980.  Google Scholar

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##### References:
 [1] P. Deuring, Spatial decay of time-dependent Oseen flows, SIAM J. Math. Anal., 41 (2009), 886-922.  doi: 10.1137/080723831.  Google Scholar [2] P. Deuring, The Cauchy problem for the homogeneous time-dependent Oseen system in $\mathbb{R}^3$: spatial decay of the velocity, Math. Bohemica, 138 (2013), 299-324.   Google Scholar [3] P. Deuring, Pointwise spatial decay of time-dependent Oseen flows: the case of data with noncompact support, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013), 2757-2776.  doi: 10.3934/dcds.2013.33.2757.  Google Scholar [4] P. Deuring, Spatial decay of time-dependent incompressible Navier-Stokes flows with nonzero velocity at infinity, SIAM J. Math. Anal., 45 (2013), 1388-1421.  doi: 10.1137/120872255.  Google Scholar [5] P. Deuring, Oseen resolvent estimates with small resolvent parameter, J. Diff. Equ., 265 (2018), 280-311.  doi: 10.1016/j.jde.2018.02.033.  Google Scholar [6] P. Deuring, Pointwise decay in space and in time for incompressible flow around a rigid body moving with constant velocity, J. Math. Fluid Mech., 21 (2019), article 11. doi: 10.1007/s00021-019-0414-9.  Google Scholar [7] P. Deuring, The 3D time-dependent Oseen system: link between $L^p$-integrability in time and pointwise decay in space, to appear in J. Math. Fluid Mech, available from https://hal.archives-ouvertes.fr/hal-02465649. Google Scholar [8] P. Deuring, $L^q$-weak solutions to the time-dependent Oseen system: decay estimates, to appear in Math. Nachr, available from https://hal.archives-ouvertes.fr/hal-02465651. Google Scholar [9] P. Deuring, Time-dependent incompressible viscous flows around a rigid body: estimates of spatial decay independent of boundary conditions, preprint, Available from https://hal.archives-ouvertes.fr/hal-02508815. doi: 10.1137/080723831.  Google Scholar [10] P. Deuring and W. Varnhorn, On Oseen resolvent estimates, Diff. Int. Equat., 23 (2010), 1139-1149.   Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] R. Farwig and J. Neustupa, On the spectrum of an Oseen-type operator arising from flow around a rotating body, Int. Equ. Oper. Theory, 62 (2008), 169-189.  doi: 10.1007/s00020-008-1616-3.  Google Scholar [14] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2$^nd$ edition, Springer, New York e.a., 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar [15] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Math. Soc. Colloquium Publications, American Mathematical Society, Providence R. I., 1957.  Google Scholar [16] T. Hishida, Large time behavior of a generalized Oseen evolution operator, with applications to the Navier-Stokes flow past a rotating obstacle, Math. Ann., 372 (2018), 915-949.  doi: 10.1007/s00208-018-1649-0.  Google Scholar [17] T. Hishida, Decay estimates of gradient of a generalized Oseen evolution operator arising from time-dependent rigid motions in exterior domains, Arch. Rational Mech. Anal., 238 (2020), 215-254.  doi: 10.1007/s00205-020-01541-3.  Google Scholar [18] G. H. Knightly, Some decay properties of solutions of the Navier-Stokes equations, in Approximation methods for Navier-Stokes problems (ed. R. Rautmann), Springer, 1979.  Google Scholar [19] T. Kobayashi and Y. Shibata, On the Oseen equation in three-dimensional exterior domains, Math. Ann., 310 (1998), 1-45.  doi: 10.1007/s002080050134.  Google Scholar [20] H. Kozono, $L^1$-solutions of the Navier-Stokes equations in exterior domains, Math. Ann., 312 (1998), 319-340.  doi: 10.1007/s002080050224.  Google Scholar [21] H. Kozono, Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains, Math. Ann., 320 (2001), 709-730.  doi: 10.1007/PL00004492.  Google Scholar [22] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.   Google Scholar [23] 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.  Google Scholar [24] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Science, Springer, New York, 1983. doi: 10.1007/PL00004457.  Google Scholar [25] L. Weis, Operator-valued Fourier multiplier theorems and maximal regularity, Math. Ann., 319 (2001), 735-758.   Google Scholar [26] K. Yosida, Functional Analysis, 6$^{th}$ edition, Springer, Berlin, 1980.  Google Scholar
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