Boundedness and stability of solutions to the non-autonomous Oseen-Navier-Stokes equation
Main Article Content
Abstract
We consider the motion of a viscous imcompressible fluid past a rotating rigid body in three-dimensional, where the translational and angular velocities of the body are prescribed but time-dependent. In a referenceframe attaches to the body, we have the non-autonomous Oseen-Navier-Stokes equations in a fixed exterior domains. We prove the existence and stability of bounded mild solutions in time t to ONSE in three-dimensional exterior domains when the coefficients are time dependent. Our method is based on the L^p-L^q-estimates of the evolution family (U(t,s)) and that of its gradient to prove boundedness of solution to linearized equations. After, we use fixed-point arguments to obtain the result on boundedness of solutions to non-linearized equations when the data belong to L^p-space and are sufficiently small. Finally, we prove existence and polynomial stability of bounded solutions to ONSE with the same condition. Our result must be useful for the study the time-periodic mild solution to the non-autonomous Oseen-Navier-Stokes equations in an exterior domains.
Keywords
boundedness and stability of solutions, exterior domains, non-autonomous equations, Oseen-Navier-Stokes flows
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
[1] W. Borchers and T. Miyakawa, L2-Decay for NavierStokes flows in unbounded domains, with applications
to exterior stationary flows, Arch. Rational Mech.
Anal., vol. 118, pp. 273-295, 1992
https://doi.org/10.1007/BF00387899
[2] T. Hishida, Large time behavior of a generalized Oseen
evolution operator, with applications to the NavierStokes flow past a rotating obstacle, Math. Ann., vol.
372, pp. 915-949, Dec. 2018.
https://doi.org/10.1007/s00208-018-1649-0
[3] T. Hishida, Decay estimates of gradient of a generalized
Oseen evolution operator arising from time-dependent
rigid motions in exterior domains, Arch. Rational
Mech. Analy., vol 238, pp. 215-254, Jun. 2020.
https://doi.org/10.1007/s00205-020-01541-3
[4] G. P. Galdi, H. Sohr, Existence and uniqueness of timeperiodic physically reasonable Navier-Stokes flows
past a body, Arch. Ration. Mech. Anal., vol. 172, pp.
363-406, Feb.2004.
https://doi.org/10.1007/s00205-004-0306-9
[5] G. P. Galdi, A. L. Silvestre, Existence of time-periodic
solutions to the Navier-Stokes equations around a
moving body, Pacific J. Math., vol. 223, pp. 251-267,
Feb. 2006.
https://doi.org/10.2140/pjm.2006.223.251
[6] T. Hansel, On the Navier-Stokes equations with rotating
effect and prescribed out-flow velocity, J.Math. Fluid
Mech., vol. 13, pp. 405-419, Jun. 2011.
https://doi.org/10.1007/s00021-010-0026-x
[7] T. Hansel and A. Rhandi, The Oseen-Navier-Stokes
flow in the exterior of a rotating obstacle: the nonautonomous case, J. Reine Angew. Math., vol. 694, pp.
1-26, Jan. 2014.
https://doi.org/10.1515/crelle-2012-0113
[8] M. Yamazaki, The Navier-Stokes equations in the
weak-Ln space with time-dependent external force,
Math. Ann., vol. 317, pp. 635-675, Aug. 2000.
https://doi.org/10.1007/PL00004418
[9] Nguyen Thieu Huy, Periodic Motions of Stokes and
Navier-Stokes Flows Around a Rotating Obstacle, Arch. Rational Mech. Analy., vol. 213, pp. 689-703,
Apr. 2014.
https://doi.org/10.1007/s00205-014-0744-y
[10] W. Borchers and T. Miyakawa, On stability of exterior
stationary Navier-Stokes flows, Acta Math., vol. 174,
pp. 311-382, Sep. 1995.
https://doi.org/10.1007/BF02392469
[11] M. Yamazaki, The Navier-Stokes equations in the
weak-Ln space with time-dependent external force,
Math. Ann., vol. 317, pp. 635-675, Aug. 2000.
https://doi.org/10.1007/PL00004418
to exterior stationary flows, Arch. Rational Mech.
Anal., vol. 118, pp. 273-295, 1992
https://doi.org/10.1007/BF00387899
[2] T. Hishida, Large time behavior of a generalized Oseen
evolution operator, with applications to the NavierStokes flow past a rotating obstacle, Math. Ann., vol.
372, pp. 915-949, Dec. 2018.
https://doi.org/10.1007/s00208-018-1649-0
[3] T. Hishida, Decay estimates of gradient of a generalized
Oseen evolution operator arising from time-dependent
rigid motions in exterior domains, Arch. Rational
Mech. Analy., vol 238, pp. 215-254, Jun. 2020.
https://doi.org/10.1007/s00205-020-01541-3
[4] G. P. Galdi, H. Sohr, Existence and uniqueness of timeperiodic physically reasonable Navier-Stokes flows
past a body, Arch. Ration. Mech. Anal., vol. 172, pp.
363-406, Feb.2004.
https://doi.org/10.1007/s00205-004-0306-9
[5] G. P. Galdi, A. L. Silvestre, Existence of time-periodic
solutions to the Navier-Stokes equations around a
moving body, Pacific J. Math., vol. 223, pp. 251-267,
Feb. 2006.
https://doi.org/10.2140/pjm.2006.223.251
[6] T. Hansel, On the Navier-Stokes equations with rotating
effect and prescribed out-flow velocity, J.Math. Fluid
Mech., vol. 13, pp. 405-419, Jun. 2011.
https://doi.org/10.1007/s00021-010-0026-x
[7] T. Hansel and A. Rhandi, The Oseen-Navier-Stokes
flow in the exterior of a rotating obstacle: the nonautonomous case, J. Reine Angew. Math., vol. 694, pp.
1-26, Jan. 2014.
https://doi.org/10.1515/crelle-2012-0113
[8] M. Yamazaki, The Navier-Stokes equations in the
weak-Ln space with time-dependent external force,
Math. Ann., vol. 317, pp. 635-675, Aug. 2000.
https://doi.org/10.1007/PL00004418
[9] Nguyen Thieu Huy, Periodic Motions of Stokes and
Navier-Stokes Flows Around a Rotating Obstacle, Arch. Rational Mech. Analy., vol. 213, pp. 689-703,
Apr. 2014.
https://doi.org/10.1007/s00205-014-0744-y
[10] W. Borchers and T. Miyakawa, On stability of exterior
stationary Navier-Stokes flows, Acta Math., vol. 174,
pp. 311-382, Sep. 1995.
https://doi.org/10.1007/BF02392469
[11] M. Yamazaki, The Navier-Stokes equations in the
weak-Ln space with time-dependent external force,
Math. Ann., vol. 317, pp. 635-675, Aug. 2000.
https://doi.org/10.1007/PL00004418