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The steady boundary value problem for charged incompressible fluids: PNP/Navier-Stokes systems. (English) Zbl 1270.35347

In this paper, the author proves the existence of a weak solution to the steady case of the Poisson-Nernst-Planck/Navier-Stokes system under Dirichlet’s boundary conditions. The weak solution is a quadruple of functions representing the velocity of the electrolyte, the electric potential and the ionic concentrations. The proof of the existence and uniqueness of the weak solution is based on the Schauder fixed point theorem under some assumptions on the kinematic viscosity and the diameter of a domain. In addition, this weak solution satisfies a weak maximum principle for the concentrations relative to their nonnegative boundary values. More precisely, if the boundary values are bounded strictly away from zero, then the same holds for the concentrations under a stricter assumption on the diameter of the domain.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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