## Tohoku Mathematical Journal

### Time periodic solutions of the Navier-Stokes equations with the time periodic Poiseuille flow in two and three dimensional perturbed channels

Teppei Kobayashi

#### Abstract

H. Beirão da Veiga proved that, for a straight channel in $\boldsymbol{R}^n$ ($n$ arbitarily large) and for a given flux with the time periodicity, there exists a unique time periodic Poiseuille flow in a straight channel in $\boldsymbol{R}^n$. Furthermore, the existence of a time periodic solution in a perturbed channel (Leray's problem) is shown for the Stokes problem (arbitary dimension) and for the Navier-Stokes problem ($n\le4$). Concerning the Navier-Stokes case, a quatitative condition requaired to show the existence of a time periodic solution depends not just on the flux of the time periodic Poiseuille flow but also on the domain it self. In this paper, by applying the result of H. Beirão da Veiga and C. J. Amick, we succeed in proving the independence of such a condition on the particular domain.

#### Article information

Source
Tohoku Math. J. (2), Volume 66, Number 1 (2014), 119-135.

Dates
First available in Project Euclid: 7 April 2014

https://projecteuclid.org/euclid.tmj/1396875666

Digital Object Identifier
doi:10.2748/tmj/1396875666

Mathematical Reviews number (MathSciNet)
MR3189483

Zentralblatt MATH identifier
1293.35208

#### Citation

Kobayashi, Teppei. Time periodic solutions of the Navier-Stokes equations with the time periodic Poiseuille flow in two and three dimensional perturbed channels. Tohoku Math. J. (2) 66 (2014), no. 1, 119--135. doi:10.2748/tmj/1396875666. https://projecteuclid.org/euclid.tmj/1396875666

#### References

• R. A. Adamas and J. J. F. Fournier, Sobolev spaces. Second edition, Pure Appl. Math. (Amst.) 140, Elsevier/Academic Press, Amsterdam, 2003.
• C. J. Amick, Steady solutions of the Navier-Stokes equations in unbounded channels and pipes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 473–513.
• C. J. Amick, Properties of steady Navier-Stokes solutions for certain unbounded channels and pipes, Nonlinear Anal. 2 (1978), 689–720.
• H. Beirão da Veiga, Time-periodic solutions of the Navier-Stokes equations in unbounded cylindrical domains–Leray's problem for periodics flows, Arch. Ration. Mech. Anal. 178 (2005), 301–325.
• R. Finn, On the steady-state solutions of the Navier-Stokes equations, III, Acta. Math. 105 (1961), 197–244.
• H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes equation, J. Fac. Sci. Univ. Tokyo Sect. I 9 (1961), 59–102.
• G. P. Galdi, An Introduction to the mathematical theory of the Navier-Stokes equations, Vol. I, Linearized steady problems, Springer Tracts Nat. Philos. 38, Springer-Verlag, New York, 1994.
• G. P. Galdi and A. M. Robertson, The relation between flow rate and axial pressure gradient for time-periodic Poiseuille Flow in a pipe, J. Math. Fluid. Mech. 7 (2005), suppl. 2, S215-S223.
• D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Second edition, Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1983.
• D. D. Joseph and S. Carmi, Stability of Poiseuille flow in pipes, annuli, and channels, Quart. Appl. Math. 26 (1969), 575–599.
• S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes equations, Arch. Rational Mech. Anal. 24 (1967), 363–369.
• T. Kobayashi, Time periodic solutions of the Navier-Stokes equations under general outflow condition, Tokyo J. Math. 32 (2009), no. 2, 409–424.
• T. Kobayashi, The relation between stationary and periodic solutions of the Navier-Stokes equations in two or three dimensional channels, J. Math. Kyoto Univ. 49 (2009), no. 2, 307–323.
• T. Kobayashi, Time periodic solutions of the Navier-Stokes equations under general outflow condition in a two dimensional symmetric channel, Hokkaido Math. J. 39 (2010), no. 3, 291–316.
• O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Second English edition, revised and enlarged, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.
• J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaries, Dunod, Gauthier-Villars, Paris, 1969.
• K. Masuda, Weak solutions of Navier-Stokes equations, Tôhoku Math. J. 36 (1984), 623–646.
• H. Morimoto and H. Fujita, A remark on the existence of steady Navier-Stokes flow in 2D semi-infinite channel involving the general outflow condition, Math. Bohem. 126 (2001), no. 2, 457–468.
• H. Morimoto, Stationary Navier-Stokes flow in 2-D channels involving the general outflow condition, Handbook of differential equations, stationary partial differential equations. Vol. IV, 299–353, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2007.
• H. Morimoto, Time periodic Navier-Stokes flow with nonhomogeneous boundary condition, J. Math. Sci. Univ. Tokyo 16 (2009), no. 1, 113–123.
• A. Takeshita, On the reproductive property of the 2-dimensional Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sec. IA 16 (1970), 297–311.
• K. Pileckas, On nonstationary two-dimensional Leray's problem for Poiseuille flow, Adv. Math. Sci. Appl. 16 (2006), no. 1, 141–174.
• V. A. Solonnikov, Solvability of a problem of the flow of a viscous incompressible fluid into an infinite open basin, Proc. Steklov Inst. Math. (1989), no. 2, 193–225.
• R. Temam, Navier-Stokes equations, Theory and numerical analysis, With an appendix by F. Thomasset, Third edition, Stud. Math. Appl. 2, North-Holland Publishing Co., Amsterdam, 1984.
• K. Yosida, Functional analysis, Third edition, Springer-Verlag, 1980.
• V. I. Yudovič, Periodic motions of a viscous incompressible fluid, Soviet Math. Dokl. 1 (1960), 168–172.