Long time finite dimensionality in charged fluids

Abstract

Abstract We consider long time dynamics of solutions of 2D periodic Nernst–Planck–Navier–Stokes systems forced by body charges and body forces. We show that, in the absence of body charges, but in the presence of fluid body forces, the charge density of the ions converges exponentially in time to zero, and the ion concentrations converge exponentially in time to equal time independent constants. This happens while the fluid continues to be dynamically active for all time. In the general case of body charges and body forces, the solutions converge in time to an invariant finite dimensional compact set in phase space.