Regularity criteria for a mathematical model for the deformation of electrolyte droplets

Abstract We prove some regularity criteria for the strong solutions to a mathematical model for the deformation of electrolyte droplets introduced by Ryham, Liu, and Zikatanov [R. J. Ryham, C. Liu, L. Zikatanov, Mathematical models for the deformation of electrolyte droplets, Discrete Contin. Dyn. Syst. Ser. B 8 (2007) 649–661]. This model is a strongly coupled system which contains the incompressible Navier–Stokes equations, the Poisson–Nernst–Planck equations and the Allen–Cahn equation, and finds many promising applications in microfluidics and materials science engineering.

[1]  J. Serrin.  On the interior regularity of weak solutions of the Navier-Stokes equations , 1962 .

[2]  Takehiko Ohyama,et al.  Interior regularity of weak solutions of the time-dependent Navier-Stokes equation , 1960 .

[3]  H.BeirāodaVeiga.  A New Regularity Class for the Navier-Stokes Equations in IR^n , 1995 .

[4]  Jihong Zhao,et al.  Well-posedness for the Navier–Stokes–Nernst–Planck–Poisson system in Triebel–Lizorkin space and Besov space with negative indices , 2011 .

[5]  Rolf J. Ryham,et al.  Existence, Uniqueness, Regularity and Long-term Behavior for Dissipative Systems Modeling Electrohydrodynamics , 2009, 0910.4973.

[6]  Markus Schmuck,et al.  ANALYSIS OF THE NAVIER–STOKES–NERNST–PLANCK–POISSON SYSTEM , 2009 .

[7]  Jie Shen,et al.  A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method , 2003 .

[8]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[9]  Luigi C. Berselli,et al.  Regularity criteria involving the pressure for the weak solutions to the Navier-Stokes equations , 2002 .

[10]  Joseph W. Jerome,et al.  ANALYTICAL APPROACHES TO CHARGE TRANSPORT IN A MOVING MEDIUM , 2002 .

[11]  Ludmil Zikatanov,et al.  Mathematical models for the deformation of electrolyte droplets , 2007 .

[13]  Fucai Li,et al.  Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics , 2009, 0905.2893.

[14]  Edriss S. Titi,et al.  Global Regularity Criterion for the 3D Navier–Stokes Equations Involving One Entry of the Velocity Gradient Tensor , 2010, 1005.4463.

[15]  Andreas Prohl,et al.  CONVERGENT FINITE ELEMENT DISCRETIZATIONS OF THE NAVIER-STOKES-NERNST-PLANCK-POISSON SYSTEM , 2010 .

[16]  Alexis Vasseur,et al.  Regularity criterion for 3d navier-stokes equations in terms of the direction of the velocity , 2007, 0705.2446.

[17]  Jishan Fan,et al.  On the Cauchy problem for a model of electro-kinetic fluid , 2012, Appl. Math. Lett..