American Institute of Mathematical Sciences

Advanced
June  2012, 5(3): 631-639. doi: 10.3934/dcdss.2012.5.631

Survey on time periodic problem for fluid flow under inhomogeneous boundary condition

Hiroko Morimoto 1,

1. 

Department of Mathematics, Meiji University, Kawasaki, 214-8571, Japan

Received  September 2010 Revised  October 2010 Published  October 2011

We consider the time dependent motion of incompressible viscous fluid with non-homogeneous boundary condition. We suppose that the bounded domain filled by the fluid has at least two boundary components, and the boundary data for the fluid velocity satisfies only the general outflow condition (GOC). The existence of solutions for the stationary problem and time periodic problem is not known in general context. We present results for the Navier-Stokes equations and the Boussinesq equations.
Keywords: general outflow condition, symmetry., Time periodic Navier-Stokes equations, time periodic Boussinesq equations, non-homogeneous boundary condition.
Mathematics Subject Classification: Primary: 35Q30, Secondary: 76D0.
Citation: Hiroko Morimoto. Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 631-639. doi: 10.3934/dcdss.2012.5.631
References:
[1]

C. J. Amick, Existence of solutions to the nonhomogeneous steady Navier-Stokes equations,, Indiana Univ. Math. J., 33 (1984), 817.  doi: 10.1512/iumj.1984.33.33043.  Google Scholar

[2]

R. Farwig and H. Morimoto, Leray's inequality for fluid flow in symmetric multi-connected doamins,, Darmstadt University of Technology, 2612 (2010).   Google Scholar

[3]

H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem,, J. Fac. Sci. Univ. Tokyo, 9 (1961), 59.   Google Scholar

[4]

H. Fujita, On stationary solutions to Navier-Stokes equations in symmetric plane domains under general outflow condition,, in, 388 (1998), 16.   Google Scholar

[5]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II, Nonlinear Steady Problems,", Springer Tracts in Natural Philosophy, 39 (1994).   Google Scholar

[6]

E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik,, Math. Ann., 117 (1941), 764.  doi: 10.1007/BF01450040.  Google Scholar

[7]

E. Hopf, "On Nonlinear Partial Differential Equations,", Lecture Series of the Symposium on Partial Differential Equations, (1957), 1.   Google Scholar

[8]

H. Inoue and M. Ôtani, Periodic Problems for Heat Convection Equations in Noncylindrical Domains,, Funkcialaj Ekvacioj, 40 (1997), 19.   Google Scholar

[9]

S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes eqautions,, Arch. Rat. Mech. Anal., 24 (1967), 363.  doi: 10.1007/BF00253153.  Google Scholar

[10]

T-P. Kobayashi, Time periodic solutions of the Navier-Stokes equations under general outflow condition,, Tokyo Journal of Mathematics, 32 (2009), 409.  doi: 10.3836/tjm/1264170239.  Google Scholar

[11]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, revised and enlarged,, Translated from the Russian by Richard A. Silverman and John Chu, (1969).   Google Scholar

[12]

J. Leray, Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique,, J. Math. Pure Appl., 12 (1933), 1.   Google Scholar

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod, (1969).   Google Scholar

[14]

H. Morimoto, On the existence of periodic weak solutions of the Navier-Stokes equations in regions with periodically moving boundaries,, J. Fac. Sci. Univ. Tokyo, 18 (): 499.   Google Scholar

[15]

H. Morimoto, Non-stationary Boussinesq equations,, J. Fac. Science Univ. Tokyo Sec. IA, 39 (1992), 61.   Google Scholar

[16]

H. Morimoto, General outflow condition for Navier-Stokes flow,, in, 16 (1998), 209.   Google Scholar

[17]

H. Morimoto, A remark on the existence of 2-D steady Navier-Stokes flow in symmetric domain under general outflow condition,, J. Math. Fluid Mech., 9 (2007), 411.  doi: 10.1007/s00021-005-0206-2.  Google Scholar

[18]

H. Morimoto, Time periodic Navier-Stokes flow with nonhomogeneous boundary condition,, Journal of Mathematical Sciences The University of Tokyo, 16 (2009), 113.   Google Scholar

[19]

H. Morimoto, Heat convection equation with nonhomogeneous boundary condition,, Funkciaj Ekvacioj, 53 (2010), 213.  doi: 10.1619/fesi.53.213.  Google Scholar

[20]

K. Ōeda, Weak and strong solutions of the heat convection equations in regions with moving boundaries,, J. Fac. Science, 36 (1989), 491.   Google Scholar

[21]

G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso di bidimensionale,, Rendi Semi. Mat. Univ. Padova, 30 (1960), 1.   Google Scholar

[22]

J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 3 (1959), 120.  doi: 10.1007/BF00284169.  Google Scholar

[23]

A. Takeshita, On the reproductive property of 2-dimensional Navier-Stokes equations,, J. Fac. Sci. Univ. Tokyo Sect. I, 16 (1970), 297.   Google Scholar

[24]

A. Takeshita, A remark on Leray's inequality,, Pacific J. Math., 157 (1993), 151.   Google Scholar

[25]

I. Yudovič, Periodic motions of a viscous incompressible fluid,, Doklady Acad. Nauk., 130 (1960), 1214.   Google Scholar

show all references

References:
[1]

C. J. Amick, Existence of solutions to the nonhomogeneous steady Navier-Stokes equations,, Indiana Univ. Math. J., 33 (1984), 817.  doi: 10.1512/iumj.1984.33.33043.  Google Scholar

[2]

R. Farwig and H. Morimoto, Leray's inequality for fluid flow in symmetric multi-connected doamins,, Darmstadt University of Technology, 2612 (2010).   Google Scholar

[3]

H. Fujita, On the existence and regularity of the steady-state solutions of the Navier-Stokes theorem,, J. Fac. Sci. Univ. Tokyo, 9 (1961), 59.   Google Scholar

[4]

H. Fujita, On stationary solutions to Navier-Stokes equations in symmetric plane domains under general outflow condition,, in, 388 (1998), 16.   Google Scholar

[5]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II, Nonlinear Steady Problems,", Springer Tracts in Natural Philosophy, 39 (1994).   Google Scholar

[6]

E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik,, Math. Ann., 117 (1941), 764.  doi: 10.1007/BF01450040.  Google Scholar

[7]

E. Hopf, "On Nonlinear Partial Differential Equations,", Lecture Series of the Symposium on Partial Differential Equations, (1957), 1.   Google Scholar

[8]

H. Inoue and M. Ôtani, Periodic Problems for Heat Convection Equations in Noncylindrical Domains,, Funkcialaj Ekvacioj, 40 (1997), 19.   Google Scholar

[9]

S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes eqautions,, Arch. Rat. Mech. Anal., 24 (1967), 363.  doi: 10.1007/BF00253153.  Google Scholar

[10]

T-P. Kobayashi, Time periodic solutions of the Navier-Stokes equations under general outflow condition,, Tokyo Journal of Mathematics, 32 (2009), 409.  doi: 10.3836/tjm/1264170239.  Google Scholar

[11]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow," Second English edition, revised and enlarged,, Translated from the Russian by Richard A. Silverman and John Chu, (1969).   Google Scholar

[12]

J. Leray, Etude de diverses équations intégrales nonlinéaires et de quelques problèmes que pose l'hydrodynamique,, J. Math. Pure Appl., 12 (1933), 1.   Google Scholar

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod, (1969).   Google Scholar

[14]

H. Morimoto, On the existence of periodic weak solutions of the Navier-Stokes equations in regions with periodically moving boundaries,, J. Fac. Sci. Univ. Tokyo, 18 (): 499.   Google Scholar

[15]

H. Morimoto, Non-stationary Boussinesq equations,, J. Fac. Science Univ. Tokyo Sec. IA, 39 (1992), 61.   Google Scholar

[16]

H. Morimoto, General outflow condition for Navier-Stokes flow,, in, 16 (1998), 209.   Google Scholar

[17]

H. Morimoto, A remark on the existence of 2-D steady Navier-Stokes flow in symmetric domain under general outflow condition,, J. Math. Fluid Mech., 9 (2007), 411.  doi: 10.1007/s00021-005-0206-2.  Google Scholar

[18]

H. Morimoto, Time periodic Navier-Stokes flow with nonhomogeneous boundary condition,, Journal of Mathematical Sciences The University of Tokyo, 16 (2009), 113.   Google Scholar

[19]

H. Morimoto, Heat convection equation with nonhomogeneous boundary condition,, Funkciaj Ekvacioj, 53 (2010), 213.  doi: 10.1619/fesi.53.213.  Google Scholar

[20]

K. Ōeda, Weak and strong solutions of the heat convection equations in regions with moving boundaries,, J. Fac. Science, 36 (1989), 491.   Google Scholar

[21]

G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso di bidimensionale,, Rendi Semi. Mat. Univ. Padova, 30 (1960), 1.   Google Scholar

[22]

J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations,, Arch. Rational Mech. Anal., 3 (1959), 120.  doi: 10.1007/BF00284169.  Google Scholar

[23]

A. Takeshita, On the reproductive property of 2-dimensional Navier-Stokes equations,, J. Fac. Sci. Univ. Tokyo Sect. I, 16 (1970), 297.   Google Scholar

[24]

A. Takeshita, A remark on Leray's inequality,, Pacific J. Math., 157 (1993), 151.   Google Scholar

[25]

I. Yudovič, Periodic motions of a viscous incompressible fluid,, Doklady Acad. Nauk., 130 (1960), 1214.   Google Scholar

[1]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241
[2]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[3]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340
[4]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137
[5]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336
[6]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073
[7]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318
[8]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273
[9]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
[10]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267
[11]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[12]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383
[13]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242
[14]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[15]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[16]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[17]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166
[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[19]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341
[20]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences