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A weak-Lp Prodi-Serrin type regularity criterion for electro-hydrodynamics

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

Funds:  Supported by National Natural Science Foundation of China(11301003).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.09.003
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  • Author Bio:

    SHAO Guangming, male, born in 1988, master. Research field: Infinite-dimensional dynamical systems.E-mail: guangmingshao@163.com

  • Received Date: 10 July 2016
  • Accepted Date: 18 November 2016
  • Rev Recd Date: 18 November 2016
  • Publish Date: 30 September 2017
    Abstract

  • Abstract

    Regularity criteria for weak solution of the electro-hydrodynamics was studied. It was proved that the solution (u,n,p,Ψ) remains strong on [0,T] if u∈Ls(0,T;Lr,∞(Ω)) or ‖u‖Ls,∞(0,T;Lr,∞(Ω))≤C, where (3/r)+(2/s)=1 and r∈(3,∞], C>0 depending only on r and Ω.

    Abstract

    Regularity criteria for weak solution of the electro-hydrodynamics was studied. It was proved that the solution (u,n,p,Ψ) remains strong on [0,T] if u∈Ls(0,T;Lr,∞(Ω)) or ‖u‖Ls,∞(0,T;Lr,∞(Ω))≤C, where (3/r)+(2/s)=1 and r∈(3,∞], C>0 depending only on r and Ω.
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    • References(11)
  • [1]
    RYHAM R J, LIU C, ZIKATANOV L. Mathematical models for the deformation of electrolyte droplets[J]. Discrete Contin Dyn Syst Ser B, 2007, 8(3): 649-661.
    [2]
    DENG C, ZHAO J, CUI S. Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices[J]. J Math Anal Appl, 2011, 377(1): 392-405.
    [3]
    FAN J, NAKAMURA G, ZHOU Y. On the Cauchy problem for a model of electro-kinetic fluid[J]. Appl Math Lett, 2012, 25(1): 33-37.
    [4]
    ZHAO J, BAI M. Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics[J]. Nonlinear Analysis: Real World Applications, 2016, 31: 210-226.
    [5]
    LI F. Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics[J]. J Differential Equations, 2009, 246(9): 3620-3641.
    [6]
    SHAO G, CHAI X. Approximation of the 2D incompressible electrohydrodynamics system by the artificial compressibility method[J]. Boundary Value Problems, 2017, 2017(1): 14.
    [7]
    RYHAM R J. Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics[DB/OL]. arXiv:0910.4973.
    [8]
    FAN J, LI F, NAKAMURA G. Regularity criteria for a mathematical model for the deformation of electrolyte droplets[J]. Appl Math Lett, 2013, 26: 494-499.
    [9]
    BOSIA S, PATA V, ROBINSON J. A weak-Lp Prodi-Serrin type regularity criterion for the Navier-Stokes equations[J]. J Math Fluid Mech, 2014, 16: 721-725.
    [10]
    GRAFAKOS L. Classical Fourier Analysis[M]. New York: Springer, 2008.
    [11]
    PATA V, MIRANVILLE A. On the regularity of solutions to the Navier-Stokes equations[J]. Commun Pure Appl Anal, 2012, 11: 747-761.)
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    [1]
    RYHAM R J, LIU C, ZIKATANOV L. Mathematical models for the deformation of electrolyte droplets[J]. Discrete Contin Dyn Syst Ser B, 2007, 8(3): 649-661.
    [2]
    DENG C, ZHAO J, CUI S. Well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in Triebel-Lizorkin space and Besov space with negative indices[J]. J Math Anal Appl, 2011, 377(1): 392-405.
    [3]
    FAN J, NAKAMURA G, ZHOU Y. On the Cauchy problem for a model of electro-kinetic fluid[J]. Appl Math Lett, 2012, 25(1): 33-37.
    [4]
    ZHAO J, BAI M. Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics[J]. Nonlinear Analysis: Real World Applications, 2016, 31: 210-226.
    [5]
    LI F. Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics[J]. J Differential Equations, 2009, 246(9): 3620-3641.
    [6]
    SHAO G, CHAI X. Approximation of the 2D incompressible electrohydrodynamics system by the artificial compressibility method[J]. Boundary Value Problems, 2017, 2017(1): 14.
    [7]
    RYHAM R J. Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics[DB/OL]. arXiv:0910.4973.
    [8]
    FAN J, LI F, NAKAMURA G. Regularity criteria for a mathematical model for the deformation of electrolyte droplets[J]. Appl Math Lett, 2013, 26: 494-499.
    [9]
    BOSIA S, PATA V, ROBINSON J. A weak-Lp Prodi-Serrin type regularity criterion for the Navier-Stokes equations[J]. J Math Fluid Mech, 2014, 16: 721-725.
    [10]
    GRAFAKOS L. Classical Fourier Analysis[M]. New York: Springer, 2008.
    [11]
    PATA V, MIRANVILLE A. On the regularity of solutions to the Navier-Stokes equations[J]. Commun Pure Appl Anal, 2012, 11: 747-761.)
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