# American Institute of Mathematical Sciences

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

 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.
Citation: Hiroko Morimoto. Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete and 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. [2] R. Farwig and H. Morimoto, Leray's inequality for fluid flow in symmetric multi-connected doamins, Darmstadt University of Technology, preprint, 2612 (2010). [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. [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. [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. [6] E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., 117 (1941), 764-775. doi: 10.1007/BF01450040. [7] E. Hopf, "On Nonlinear Partial Differential Equations," Lecture Series of the Symposium on Partial Differential Equations, Univ. Kansas, (1957), 1-31. [8] H. Inoue and M. Ôtani, Periodic Problems for Heat Convection Equations in Noncylindrical Domains, Funkcialaj Ekvacioj, 40 (1997), 19-39. [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. [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. [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. [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. [13] J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 1969. [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. [15] H. Morimoto, Non-stationary Boussinesq equations, J. Fac. Science Univ. Tokyo Sec. IA, 39 (1992), 61-75. [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. [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. [18] H. Morimoto, Time periodic Navier-Stokes flow with nonhomogeneous boundary condition, Journal of Mathematical Sciences The University of Tokyo, 16 (2009), 113-123. [19] H. Morimoto, Heat convection equation with nonhomogeneous boundary condition, Funkciaj Ekvacioj, 53 (2010), 213-229. doi: 10.1619/fesi.53.213. [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. [21] G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso di bidimensionale, Rendi Semi. Mat. Univ. Padova, 30 (1960), 1-15. [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. [23] A. Takeshita, On the reproductive property of 2-dimensional Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sect. I, 16 (1970), 297-311. [24] A. Takeshita, A remark on Leray's inequality, Pacific J. Math., 157 (1993), 151-158. [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.

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##### 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. [2] R. Farwig and H. Morimoto, Leray's inequality for fluid flow in symmetric multi-connected doamins, Darmstadt University of Technology, preprint, 2612 (2010). [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. [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. [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. [6] E. Hopf, Ein allgemeiner Endlichkeitssatz der Hydrodynamik, Math. Ann., 117 (1941), 764-775. doi: 10.1007/BF01450040. [7] E. Hopf, "On Nonlinear Partial Differential Equations," Lecture Series of the Symposium on Partial Differential Equations, Univ. Kansas, (1957), 1-31. [8] H. Inoue and M. Ôtani, Periodic Problems for Heat Convection Equations in Noncylindrical Domains, Funkcialaj Ekvacioj, 40 (1997), 19-39. [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. [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. [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. [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. [13] J. L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 1969. [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. [15] H. Morimoto, Non-stationary Boussinesq equations, J. Fac. Science Univ. Tokyo Sec. IA, 39 (1992), 61-75. [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. [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. [18] H. Morimoto, Time periodic Navier-Stokes flow with nonhomogeneous boundary condition, Journal of Mathematical Sciences The University of Tokyo, 16 (2009), 113-123. [19] H. Morimoto, Heat convection equation with nonhomogeneous boundary condition, Funkciaj Ekvacioj, 53 (2010), 213-229. doi: 10.1619/fesi.53.213. [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. [21] G. Prodi, Qualche risultato riguardo alle equazioni di Navier-Stokes nel caso di bidimensionale, Rendi Semi. Mat. Univ. Padova, 30 (1960), 1-15. [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. [23] A. Takeshita, On the reproductive property of 2-dimensional Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo Sect. I, 16 (1970), 297-311. [24] A. Takeshita, A remark on Leray's inequality, Pacific J. Math., 157 (1993), 151-158. [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.
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