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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), 817-830. 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, preprint, 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, Sec. I, 9 (1961), 59-102.  Google Scholar

[4]

H. Fujita, On stationary solutions to Navier-Stokes equations in symmetric plane domains under general outflow condition, in "Navier-Stokes Equations: Theory and Numerical Methods" (Varenna, 1997), Pitman Research Notes in Mathematics Ser., 388, Longman, Harlow, (1998), 16-30.  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, Springer-Verlag, New York, 1994.  Google Scholar

[6]

E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., 117 (1941), 764-775. doi: 10.1007/BF01450040.  Google Scholar

[7]

E. Hopf, "On Nonlinear Partial Differential Equations," Lecture Series of the Symposium on Partial Differential Equations, Univ. Kansas, (1957), 1-31. Google Scholar

[8]

H. Inoue and M. Ôtani, Periodic Problems for Heat Convection Equations in Noncylindrical Domains, Funkcialaj Ekvacioj, 40 (1997), 19-39.  Google Scholar

[9]

S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes eqautions, Arch. Rat. Mech. Anal., 24 (1967), 363-369. 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), 409-424. 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, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 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), 1-82. Google Scholar

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 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), 61-75.  Google Scholar

[16]

H. Morimoto, General outflow condition for Navier-Stokes flow, in "Recent Topics on Mathematical Theory of Viscous Incompressible Fluid" (eds. H. Kozono and Y. Shibata) (Tsukuba, 1996), Lecture Note in Numerical and Applied Analysis, 16, Kinokuniya, Tokyo, (1998), 209-224.  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), 411-418. 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), 113-123.  Google Scholar

[19]

H. Morimoto, Heat convection equation with nonhomogeneous boundary condition, Funkciaj Ekvacioj, 53 (2010), 213-229. 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, Univ. Tokyo Sec. IA Math., 36 (1989), 491-536.  Google Scholar

[21]

G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso di bidimensionale, Rendi Semi. Mat. Univ. Padova, 30 (1960), 1-15.  Google Scholar

[22]

J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122. 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), 297-311.  Google Scholar

[24]

A. Takeshita, A remark on Leray's inequality, Pacific J. Math., 157 (1993), 151-158.  Google Scholar

[25]

I. Yudovič, Periodic motions of a viscous incompressible fluid, Doklady Acad. Nauk.,130 (1960), 1214-1217, translated as Soviet Math. Doklady, 1 (1960), 168-172.  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), 817-830. 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, preprint, 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, Sec. I, 9 (1961), 59-102.  Google Scholar

[4]

H. Fujita, On stationary solutions to Navier-Stokes equations in symmetric plane domains under general outflow condition, in "Navier-Stokes Equations: Theory and Numerical Methods" (Varenna, 1997), Pitman Research Notes in Mathematics Ser., 388, Longman, Harlow, (1998), 16-30.  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, Springer-Verlag, New York, 1994.  Google Scholar

[6]

E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., 117 (1941), 764-775. doi: 10.1007/BF01450040.  Google Scholar

[7]

E. Hopf, "On Nonlinear Partial Differential Equations," Lecture Series of the Symposium on Partial Differential Equations, Univ. Kansas, (1957), 1-31. Google Scholar

[8]

H. Inoue and M. Ôtani, Periodic Problems for Heat Convection Equations in Noncylindrical Domains, Funkcialaj Ekvacioj, 40 (1997), 19-39.  Google Scholar

[9]

S. Kaniel and M. Shinbrot, A reproductive property of the Navier-Stokes eqautions, Arch. Rat. Mech. Anal., 24 (1967), 363-369. 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), 409-424. 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, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 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), 1-82. Google Scholar

[13]

J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 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), 61-75.  Google Scholar

[16]

H. Morimoto, General outflow condition for Navier-Stokes flow, in "Recent Topics on Mathematical Theory of Viscous Incompressible Fluid" (eds. H. Kozono and Y. Shibata) (Tsukuba, 1996), Lecture Note in Numerical and Applied Analysis, 16, Kinokuniya, Tokyo, (1998), 209-224.  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), 411-418. 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), 113-123.  Google Scholar

[19]

H. Morimoto, Heat convection equation with nonhomogeneous boundary condition, Funkciaj Ekvacioj, 53 (2010), 213-229. 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, Univ. Tokyo Sec. IA Math., 36 (1989), 491-536.  Google Scholar

[21]

G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso di bidimensionale, Rendi Semi. Mat. Univ. Padova, 30 (1960), 1-15.  Google Scholar

[22]

J. Serrin, A note on the existence of periodic solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal., 3 (1959), 120-122. 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), 297-311.  Google Scholar

[24]

A. Takeshita, A remark on Leray's inequality, Pacific J. Math., 157 (1993), 151-158.  Google Scholar

[25]

I. Yudovič, Periodic motions of a viscous incompressible fluid, Doklady Acad. Nauk.,130 (1960), 1214-1217, translated as Soviet Math. Doklady, 1 (1960), 168-172.  Google Scholar

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