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Rapid time decay of weak solutions to the generalized Hall-magneto-hydrodynamics equations

  • School of Mathematical Sciences, Anhui University, Hefei 230039, China

Funds:  Supported by the NNSF of China (11271019).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.09.004
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  • Author Bio:

    TAO Qunqun, female, born in 1990, postgraduate. Research field: differential equations. E-mail: qunquntao123@163.com

  • Corresponding author: DONG Boqing
  • Received Date: 19 May 2015
  • Accepted Date: 14 July 2015
  • Rev Recd Date: 14 July 2015
  • Publish Date: 30 September 2015
    Abstract

  • Abstract

    The rapid time decay for solutions to the generalized Hall-magneto-hydrodynamics equations was studied. By developing the classic Fourier splitting methods, the more rapid L2 decay rate of the weak solutions as (1+t)-74 was derived. The trick is mainly based on the even lower frequency effect of the nonlinear term.

    Abstract

    The rapid time decay for solutions to the generalized Hall-magneto-hydrodynamics equations was studied. By developing the classic Fourier splitting methods, the more rapid L2 decay rate of the weak solutions as (1+t)-74 was derived. The trick is mainly based on the even lower frequency effect of the nonlinear term.
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    • References(19)
  • [1]
    Sermange M, Teman R. Some mathematical questions related to the MHD equations[J]. Comm Pure Appl Math, 1983, 36: 635-664.
    [2]
    Acheritogaray M, Degond P, Frouvelle A, et al. Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system[J]. Kinet Relat Models, 2011, 4: 901-918.
    [3]
    Chae D, Degond P, Liu J. Well-posedness for Hall-magnetohydrodynamics[J]. Annales de lInstitut Henri Poincare (C) Non Linear Analysis, 2014, 31: 555-565.
    [4]
    Chae D, Schonbek M. On the temporal decay for the Hall-magnetohydrodynamic equations[J]. J Differential Equations, 2013, 255: 3 971-3 982.
    [5]
    Dong B, Li Y. Large time behavior to the system of incompressible non-Newtonian fluids in R2[J]. J Math Anal Appl, 2004, 298: 667-676.
    [6]
    Dong B, Chen Z. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid equations[J]. Discrete Contin Dyn Syst, 2009, 23: 765-784.
    [7]
    Dong B, Song J. Global regularity and asymptotic behavior of the modified Navier-Stokes equations with fractional dissipation[J]. Discrete and Continuous Dynamical Systems, 2012, 32: 57-79.
    [8]
    Guo Y, Wang Y. Decay of dissipative equations and negative Sobolev spaces[J]. Comm Partial Differential Equations, 2012, 37: 2 165-2 208.
    [9]
    Han P, He C. Decay properties of solutions to the incompressible magneto-hydrodynamics equations in a half space[J]. Math Methods Appl Sci, 2012, 35: 1 472-1 488.
    [10]
    He C, Xin Z. On the decay properties of solutions to the non-stationary Navier-Stokes equations in R3[J]. Proc Roy Soc Edinburgh Sect A, 2001, 131: 597-619.
    [11]
    Kajikiya R, Miyakawa T. On L2 decay of weak solutions of Navier-Stokes equations in Rn[J]. Math Zeit, 1986, 192: 135-148.
    [12]
    Oliver M, Titi E S. Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in Rn[J]. J Funct Anal, 2000, 172: 1-18.
    [13]
    Qin X, Wang Y. Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations[J]. SIAM J Math Anal, 2011, 43: 341-366.
    [14]
    Schonbek M E. L2 decay for weak solutions of the Navier-Stokes equations[J]. Arch Rational Mech Anal, 1985, 88: 209-222.
    [15]
    Schonbek M E, Schonbek T P, Süli E. Large-time behaviour of solutions to the magneto-hydrodynamics equations[J]. Math Ann, 1996, 304: 717-756.
    [16]
    Agapito R, Schonbek M E. Non-uniform decay of MHD equations with and without magnetic diffusion[J]. Comm Partial Differential Equations, 2007, 32: 1 791-1 812.
    [17]
    Wiegner M. Decay results for weak solutions of the Navier-Stokes equations in Rn[J]. J London Math Soc, 1987, 35: 303-313.
    [18]
    Zhang L. New results of general n-dimensional incompressible Navier-Stokes equations[J]. J Differential Equations, 2008, 245: 3 470-3 502.
    [19]
    Zhao C, Liang Y, Zhao M. Upper and lower bounds of time decay rate of solutions to a class of incompressible third grade fluid equations[J]. Nonlinear Anal Real World Appl, 2014, 15: 229-238.
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    [1]
    Sermange M, Teman R. Some mathematical questions related to the MHD equations[J]. Comm Pure Appl Math, 1983, 36: 635-664.
    [2]
    Acheritogaray M, Degond P, Frouvelle A, et al. Kinetic formulation and global existence for the Hall-magneto-hydrodynamics system[J]. Kinet Relat Models, 2011, 4: 901-918.
    [3]
    Chae D, Degond P, Liu J. Well-posedness for Hall-magnetohydrodynamics[J]. Annales de lInstitut Henri Poincare (C) Non Linear Analysis, 2014, 31: 555-565.
    [4]
    Chae D, Schonbek M. On the temporal decay for the Hall-magnetohydrodynamic equations[J]. J Differential Equations, 2013, 255: 3 971-3 982.
    [5]
    Dong B, Li Y. Large time behavior to the system of incompressible non-Newtonian fluids in R2[J]. J Math Anal Appl, 2004, 298: 667-676.
    [6]
    Dong B, Chen Z. Asymptotic profiles of solutions to the 2D viscous incompressible micropolar fluid equations[J]. Discrete Contin Dyn Syst, 2009, 23: 765-784.
    [7]
    Dong B, Song J. Global regularity and asymptotic behavior of the modified Navier-Stokes equations with fractional dissipation[J]. Discrete and Continuous Dynamical Systems, 2012, 32: 57-79.
    [8]
    Guo Y, Wang Y. Decay of dissipative equations and negative Sobolev spaces[J]. Comm Partial Differential Equations, 2012, 37: 2 165-2 208.
    [9]
    Han P, He C. Decay properties of solutions to the incompressible magneto-hydrodynamics equations in a half space[J]. Math Methods Appl Sci, 2012, 35: 1 472-1 488.
    [10]
    He C, Xin Z. On the decay properties of solutions to the non-stationary Navier-Stokes equations in R3[J]. Proc Roy Soc Edinburgh Sect A, 2001, 131: 597-619.
    [11]
    Kajikiya R, Miyakawa T. On L2 decay of weak solutions of Navier-Stokes equations in Rn[J]. Math Zeit, 1986, 192: 135-148.
    [12]
    Oliver M, Titi E S. Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in Rn[J]. J Funct Anal, 2000, 172: 1-18.
    [13]
    Qin X, Wang Y. Large-time behavior of solutions to the inflow problem of full compressible Navier-Stokes equations[J]. SIAM J Math Anal, 2011, 43: 341-366.
    [14]
    Schonbek M E. L2 decay for weak solutions of the Navier-Stokes equations[J]. Arch Rational Mech Anal, 1985, 88: 209-222.
    [15]
    Schonbek M E, Schonbek T P, Süli E. Large-time behaviour of solutions to the magneto-hydrodynamics equations[J]. Math Ann, 1996, 304: 717-756.
    [16]
    Agapito R, Schonbek M E. Non-uniform decay of MHD equations with and without magnetic diffusion[J]. Comm Partial Differential Equations, 2007, 32: 1 791-1 812.
    [17]
    Wiegner M. Decay results for weak solutions of the Navier-Stokes equations in Rn[J]. J London Math Soc, 1987, 35: 303-313.
    [18]
    Zhang L. New results of general n-dimensional incompressible Navier-Stokes equations[J]. J Differential Equations, 2008, 245: 3 470-3 502.
    [19]
    Zhao C, Liang Y, Zhao M. Upper and lower bounds of time decay rate of solutions to a class of incompressible third grade fluid equations[J]. Nonlinear Anal Real World Appl, 2014, 15: 229-238.
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