Long time finite dimensionality in charged fluids

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.

Automated summary

The study focuses on the long time dynamics of solutions of 2D periodic Nernst-Planck-Navier-Stokes systems driven by body charges and body forces. It is demonstrated that in the absence of body charges but with fluid body forces present, the charge density of the ions exponentially converges to zero, and the ion concentrations exponentially converge to time independent constants. Meanwhile, the fluid remains dynamically active throughout the process. 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.