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Optimal algebraic decay of solutions for 2D Navier-Stokes equations with partial dissipation

  • School of Mathematical Sciences, Anhui University, Hefei 230601, China

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https://doi.org/10.3969/j.issn.0253-2778.2018.05.003
  • Received Date: 30 August 2017
  • Accepted Date: 25 January 2018
  • Rev Recd Date: 25 January 2018
  • Publish Date: 31 May 2018
    Abstract

  • Abstract

    The optimal algebraic decay of solutions for two-dimensional Navier-Stokes equation with partial dissipation was studied. By developing the classic Fourier splitting methods together with inductive methods, the higher-order derivatives of solutions in the optimal algebraic rates was obtained.

    Abstract

    The optimal algebraic decay of solutions for two-dimensional Navier-Stokes equation with partial dissipation was studied. By developing the classic Fourier splitting methods together with inductive methods, the higher-order derivatives of solutions in the optimal algebraic rates was obtained.
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    • References(18)
  • [1]
    LADYZHENSKAYA O A. The Mathematical Theory of Viscous Incompressible Fluids[M]. New York: Gordon Breach, 1969.
    [2]
    TEMAN R. The Navier-Stokes Equations[M]. Amsterdam: North-Holland, 1977.
    [3]
    LERAY J. Sur le mouvement d’un liquide visqueux remplissant l’espace[J]. Acta Math, 1934, 63: 193-248.
    [4]
    SCHONBEK M. L2 decay for weak solutions of the Navier-Stokes equations[J]. Arch Rational Mech Anal, 1985, 88: 209-222.
    [5]
    WIEGNER M. Decay results for weak solutions of the Navier-Stokes equations in Rn[J]. J London Math Soc, 1987, 35: 303-313.
    [6]
    OLIVER M, TITI E S. Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in Rn[J]. J Funct Anal, 2000, 172: 1-18.
    [7]
    KAJIKIYA R, MIYAKAWA T. On L2 decay of weak solutions of Navier-Stokes equations in Rn[J]. Math Z, 1986, 192: 135-148.
    [8]
    KATO T. Strong Lp solution of the Navier-Stokes equations in Rm, with applications to weak solutions[J]. Math Z, 1984, 187: 471-480.
    [9]
    CARPIO A. Large time behavior in incompressible Navier-Stokes equations[J]. SIAM J Math Anal, 1996, 27: 449-475.
    [10]
    CHEN Z M. A sharp decay result on strong solutions of the Navier-Stokes equations in the whole space[J]. Comm Partial Differential Equs, 1991, 16: 801-820.
    [11]
    DONG B Q, CHEN Z M. Remarks on upper and lower bounds of solutions to the Navier-Stokes equations in R2[J]. Appl Math Computations, 2006, 182: 553-558.
    [12]
    HAN P. Long-time behavior for Navier-Stokes flows in a two-dimensional exterior domain[J]. J Funct Anal, 2016, 270: 1091-1152.
    [13]
    HE C, XIN Z. On the decay properties of solutions to the nonstationary Navier-Stokes equations in R3[J]. Proc Roy Soc Edinburgh, 2001, 131: 597-619.
    [14]
    NICHE C. Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum[J]. J Differential Equations, 2016, 260: 4440-4453.
    [15]
    ZHAO J, ZHENG L. Temporal decay for the generalized Navier-Stokes equations[J]. Nonlinear Anal: Theory, Methods & Applications, 2016, 141: 191-210.
    [16]
    ZHANG L. Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equation[J]. Comm Partial Differential Equs, 1995, 20: 119-127.
    [17]
    JIA Y, ZHANG X, DONG B Q. Global well-posedness of the 2D micropolar fluid flows with mixed dissipation[J]. Electron J Differential Equations, 2016, 154: 1-10.
    [18]
    TAO Qunqun, ZHANG Fei, DONG Boqing. Rapid time decay of weak solutions to the generalized Hallmagneto-hydrodynamics equations[J]. Journal of University of Science and Technology of China, 2015, 45(9): 727-732.
    陶群群,章飞,董柏青.广义霍尔磁流体力学方程弱解的快速衰减[J]. 中国科学技术大学学报, 2015, 45(9): 727-732.)
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    [1]
    LADYZHENSKAYA O A. The Mathematical Theory of Viscous Incompressible Fluids[M]. New York: Gordon Breach, 1969.
    [2]
    TEMAN R. The Navier-Stokes Equations[M]. Amsterdam: North-Holland, 1977.
    [3]
    LERAY J. Sur le mouvement d’un liquide visqueux remplissant l’espace[J]. Acta Math, 1934, 63: 193-248.
    [4]
    SCHONBEK M. L2 decay for weak solutions of the Navier-Stokes equations[J]. Arch Rational Mech Anal, 1985, 88: 209-222.
    [5]
    WIEGNER M. Decay results for weak solutions of the Navier-Stokes equations in Rn[J]. J London Math Soc, 1987, 35: 303-313.
    [6]
    OLIVER M, TITI E S. Remark on the rate of decay of higher order derivatives for solutions to the Navier-Stokes equations in Rn[J]. J Funct Anal, 2000, 172: 1-18.
    [7]
    KAJIKIYA R, MIYAKAWA T. On L2 decay of weak solutions of Navier-Stokes equations in Rn[J]. Math Z, 1986, 192: 135-148.
    [8]
    KATO T. Strong Lp solution of the Navier-Stokes equations in Rm, with applications to weak solutions[J]. Math Z, 1984, 187: 471-480.
    [9]
    CARPIO A. Large time behavior in incompressible Navier-Stokes equations[J]. SIAM J Math Anal, 1996, 27: 449-475.
    [10]
    CHEN Z M. A sharp decay result on strong solutions of the Navier-Stokes equations in the whole space[J]. Comm Partial Differential Equs, 1991, 16: 801-820.
    [11]
    DONG B Q, CHEN Z M. Remarks on upper and lower bounds of solutions to the Navier-Stokes equations in R2[J]. Appl Math Computations, 2006, 182: 553-558.
    [12]
    HAN P. Long-time behavior for Navier-Stokes flows in a two-dimensional exterior domain[J]. J Funct Anal, 2016, 270: 1091-1152.
    [13]
    HE C, XIN Z. On the decay properties of solutions to the nonstationary Navier-Stokes equations in R3[J]. Proc Roy Soc Edinburgh, 2001, 131: 597-619.
    [14]
    NICHE C. Decay characterization of solutions to Navier-Stokes-Voigt equations in terms of the initial datum[J]. J Differential Equations, 2016, 260: 4440-4453.
    [15]
    ZHAO J, ZHENG L. Temporal decay for the generalized Navier-Stokes equations[J]. Nonlinear Anal: Theory, Methods & Applications, 2016, 141: 191-210.
    [16]
    ZHANG L. Sharp rate of decay of solutions to 2-dimensional Navier-Stokes equation[J]. Comm Partial Differential Equs, 1995, 20: 119-127.
    [17]
    JIA Y, ZHANG X, DONG B Q. Global well-posedness of the 2D micropolar fluid flows with mixed dissipation[J]. Electron J Differential Equations, 2016, 154: 1-10.
    [18]
    TAO Qunqun, ZHANG Fei, DONG Boqing. Rapid time decay of weak solutions to the generalized Hallmagneto-hydrodynamics equations[J]. Journal of University of Science and Technology of China, 2015, 45(9): 727-732.
    陶群群,章飞,董柏青.广义霍尔磁流体力学方程弱解的快速衰减[J]. 中国科学技术大学学报, 2015, 45(9): 727-732.)
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