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 & 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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle,, in, (1983), 179.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

B. Straughan, "Stability and Wave Motion in Porous Media,", Springer-Verlag, (2008).   Google Scholar

[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.   Google Scholar

[19]

A. Yoshizawa, "Hydrodynamic and Magnetohydrodynamic Turbulent Flows,", Kluwer Academic Publishers, (1998).   Google Scholar

[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.   Google Scholar

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.   Google Scholar

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

C. O. Horgan and J. K. Knowles, Recent developments concerning Saint-Venant's principle,, in, (1983), 179.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[10]

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

[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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

B. Straughan, "Stability and Wave Motion in Porous Media,", Springer-Verlag, (2008).   Google Scholar

[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.   Google Scholar

[19]

A. Yoshizawa, "Hydrodynamic and Magnetohydrodynamic Turbulent Flows,", Kluwer Academic Publishers, (1998).   Google Scholar

[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.   Google Scholar

[1]

José Luiz Boldrini, Jonathan Bravo-Olivares, Eduardo Notte-Cuello, Marko A. Rojas-Medar. Asymptotic behavior of weak and strong solutions of the magnetohydrodynamic equations. Electronic Research Archive, 2021, 29 (1) : 1783-1801. doi: 10.3934/era.2020091

[2]

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, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[3]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[4]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142

[5]

Yang Liu. Global existence and exponential decay of strong solutions to the cauchy problem of 3D density-dependent Navier-Stokes equations with vacuum. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1291-1303. doi: 10.3934/dcdsb.2020163

[6]

Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020362

[7]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038

[8]

Mahir Demir, Suzanne Lenhart. A spatial food chain model for the Black Sea Anchovy, and its optimal fishery. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 155-171. doi: 10.3934/dcdsb.2020373

[9]

Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150

[10]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

[11]

Xiaoping Zhai, Yongsheng Li. Global large solutions and optimal time-decay estimates to the Korteweg system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1387-1413. doi: 10.3934/dcds.2020322

[12]

Klemens Fellner, Jeff Morgan, Bao Quoc Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 635-651. doi: 10.3934/dcdss.2020334

[13]

Xiaoli Lu, Pengzhan Huang, Yinnian He. Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 815-845. doi: 10.3934/dcdsb.2020143

[14]

Xing Wu, Keqin Su. Global existence and optimal decay rate of solutions to hyperbolic chemotaxis system in Besov spaces. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021002

[15]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[16]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477

[17]

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

[18]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[19]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[20]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

2019 Impact Factor: 1.105

Metrics

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

Other articles
by authors

[Back to Top]