May  2013, 12(3): 1349-1361. doi: 10.3934/cpaa.2013.12.1349

Spatial decay bounds in a linearized magnetohydrodynamic channel flow

1. 

Chung-Ang University, Heuksuk-Dong, Donggak-Gu, 156-756, South Korea

2. 

Hanyang University, Ansan, Gyeonggido 426-791, South Korea

Received  April 2012 Revised  July 2012 Published  September 2012

This paper establishes exponential decay bounds for a transient magnetohydrodynamic flow in a semi-infinite channel. If net entrance flows into the channel are nonzero, then the solutions will not tend to zero as the distance from the entrance end tends to infinity when homogeneous lateral surface boundary conditions and homogenous initial conditions are applied. Assuming that the entrance data are small enough so that flows converge to transient laminar flows as the distance from the entrance section tends to infinity, we linearize the magnetohydrodynamic equations and derive an integro-differential inequality that leads to an exponential decay estimate. This paper also indicates how to bound the total energy in the spirit of earlier work of Lin and Payne [11].
Citation: Julie Lee, J. C. Song. Spatial decay bounds in a linearized magnetohydrodynamic channel flow. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1349-1361. doi: 10.3934/cpaa.2013.12.1349
References:
[1]

K. A. Ames, L. E. Payne and P. W. Schaefer, Spatial decay estimates in time-dependent Stokes flow, SIAM J. Math. Anal., 24 (1993), 1395-1413.

[2]

K. A. Ames and J. C. Song, Decay bounds for magnetohydrodynamic geophysical flow, Nonlinear Analysis, 65 (2006), 1318-1333.

[3]

S. Chiriţă and M. Ciarletta, Spatial behaviour of solutions in the plane Stokes flow, J. Math. Anal. Appl., 277 (2003), 571-588.

[4]

C. O. Horgan, Plane steady flows and energy estimates for the Navier-Stokes equations, Arch. Rat. Mech. Anal., 68 (1978), 359-381.

[5]

C. O. Horgan, Recent developments concerning Saint-Venant's principle: An update, Appl. Mech. Rev., 42 (1989), 295-303.

[6]

C. O. Horgan, Recent developments concerning Saint-Venant's principle: A second update, Appl. Mech. Rev., 49 (1996), 101-111.

[7]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle, in "Advances in Applied Mechanics'' (J. W. Hutchinson ed.), Academic Press, New York, 1983, Vol. 23, pp. 179-269.

[8]

C. O. Horgan and L. T. Wheeler, Spatial decay estimates for the Navier-Stokes equations with application to the problem of entry flow, SIAM J. Appl. Math., 35 (1978), 97-116.

[9]

Y. Li, Y. Liu, S. Luo and C. Lin, Decay estimates for the Brinkman-Forchheimer equations in a semi-infinite pipe, Z. Angew. Math. Mech., 92 (2012), 160-176.

[10]

C. Lin, Spatial decay estimates and energy bounds for the Stokes flow equations, Stability Appl. Anal. Contin. Media, 2 (1992), 249-264.

[11]

C. Lin and L. E. Payne, Spatial decay bounds in the channel flow of an incompressible viscous fluid, Math. Models Methods Appl. Sci., 14 (2004), 795-818.

[12]

R. Quintanilla, Spatial decay estimate for the hyperbolic heat equation, SIAM J. Math. Anal., 27 (1996), 78-91.

[13]

J. C. Song, Decay estimates for steady magnetohydrodynamic pipe flow, Nonlinear Analysis, 54 (2003), 1029-1044.

[14]

J. C. Song, Improved decay estimates in time-dependent Stokes flow, J. Math. Anal. Appl., 288 (2003), 505-517.

[15]

J. C. Song, Improved spatial decay bounds in the plane Stokes flow, Appl. Math. Mech., 30 (2009), 833-838.

[16]

J. C. Song, Spatial decay estimates in time-dependent double-diffusive flow, J. Math. Anal. Appl., 267 (2001), 76-88.

[17]

B. Straughan, "Stability and Wave Motion in Porous Media," Springer-Verlag, New York, 2008.

[18]

P. Vafeades and C. O. Horgan, Exponential decay estimates for solutions of the van Kármán equations on a semi-infinite strip, Arch. Rat. Mech. Anal., 104 (1988), 1-25.

[19]

A. Yoshizawa, "Hydrodynamic and Magnetohydrodynamic Turbulent Flows," Kluwer Academic Publishers, Dordrecht, 1998.

[20]

C. Zhao, Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three dimensional domains, Math. Meth. Appl. Sci., 26 (2003), 759-781.

show all references

References:
[1]

K. A. Ames, L. E. Payne and P. W. Schaefer, Spatial decay estimates in time-dependent Stokes flow, SIAM J. Math. Anal., 24 (1993), 1395-1413.

[2]

K. A. Ames and J. C. Song, Decay bounds for magnetohydrodynamic geophysical flow, Nonlinear Analysis, 65 (2006), 1318-1333.

[3]

S. Chiriţă and M. Ciarletta, Spatial behaviour of solutions in the plane Stokes flow, J. Math. Anal. Appl., 277 (2003), 571-588.

[4]

C. O. Horgan, Plane steady flows and energy estimates for the Navier-Stokes equations, Arch. Rat. Mech. Anal., 68 (1978), 359-381.

[5]

C. O. Horgan, Recent developments concerning Saint-Venant's principle: An update, Appl. Mech. Rev., 42 (1989), 295-303.

[6]

C. O. Horgan, Recent developments concerning Saint-Venant's principle: A second update, Appl. Mech. Rev., 49 (1996), 101-111.

[7]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle, in "Advances in Applied Mechanics'' (J. W. Hutchinson ed.), Academic Press, New York, 1983, Vol. 23, pp. 179-269.

[8]

C. O. Horgan and L. T. Wheeler, Spatial decay estimates for the Navier-Stokes equations with application to the problem of entry flow, SIAM J. Appl. Math., 35 (1978), 97-116.

[9]

Y. Li, Y. Liu, S. Luo and C. Lin, Decay estimates for the Brinkman-Forchheimer equations in a semi-infinite pipe, Z. Angew. Math. Mech., 92 (2012), 160-176.

[10]

C. Lin, Spatial decay estimates and energy bounds for the Stokes flow equations, Stability Appl. Anal. Contin. Media, 2 (1992), 249-264.

[11]

C. Lin and L. E. Payne, Spatial decay bounds in the channel flow of an incompressible viscous fluid, Math. Models Methods Appl. Sci., 14 (2004), 795-818.

[12]

R. Quintanilla, Spatial decay estimate for the hyperbolic heat equation, SIAM J. Math. Anal., 27 (1996), 78-91.

[13]

J. C. Song, Decay estimates for steady magnetohydrodynamic pipe flow, Nonlinear Analysis, 54 (2003), 1029-1044.

[14]

J. C. Song, Improved decay estimates in time-dependent Stokes flow, J. Math. Anal. Appl., 288 (2003), 505-517.

[15]

J. C. Song, Improved spatial decay bounds in the plane Stokes flow, Appl. Math. Mech., 30 (2009), 833-838.

[16]

J. C. Song, Spatial decay estimates in time-dependent double-diffusive flow, J. Math. Anal. Appl., 267 (2001), 76-88.

[17]

B. Straughan, "Stability and Wave Motion in Porous Media," Springer-Verlag, New York, 2008.

[18]

P. Vafeades and C. O. Horgan, Exponential decay estimates for solutions of the van Kármán equations on a semi-infinite strip, Arch. Rat. Mech. Anal., 104 (1988), 1-25.

[19]

A. Yoshizawa, "Hydrodynamic and Magnetohydrodynamic Turbulent Flows," Kluwer Academic Publishers, Dordrecht, 1998.

[20]

C. Zhao, Initial boundary value problem for the evolution system of MHD type describing geophysical flow in three dimensional domains, Math. Meth. Appl. Sci., 26 (2003), 759-781.

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