Global Well-Posedness and Stability of Electrokinetic Flows

Abstract

Abstract. We consider a coupled system of Navier-Stokes and NernstPlanck equations, describing the evolution of the velocity and the concentration fields of dissolved constituents in an electrolyte solution. Motivated by recent applications in the field of micro-and nanofluidics, we consider the model in such generality that electrokinetic flows are included. This prohibits employing the assumption of electroneutrality of the total solution, which is a common approach in the mathematical literature in order to determine the electrical potential. Therefore we complement the system of mass and momentum balances with a Poisson equation for the electrostatic potential, with the charge density stemming from the concentrations of the ionic species. For the resulting Navier-Stokes-Nernst-Planck-Poisson system we prove the existence of unique local strong solutions in bounded domains in R n for any n ≥ 2 as well as the existence of unique global strong solutions and exponential convergence to uniquely determined steady states in two dimensions.

Short time existence for the elastic flow of clamped curves

“…We only have to take care of the first term since the others are of order ≤ 2, where by order we mean the order of derivatives with respect to the space variable. Indeed we have 4 .…”

“…are the four different roots of the equation 4 . Note that we put −λ = |λ|e iθ = 0 for some θ ∈ [ π 2 , 3π 2 ].…”

“…In [15] the short time existence for hinged curves is obtained with different methods. Maximal regularity methods for short time existence of geometric problems has been studied for instance in [9], [4], [12] or [1]. The main theorem of this paper yields a unique solution of the elastic flow to initial curves in a certain trace space which embeds merely in C 2,ε for some ε > 0, if the initial curve can be written as a normal graph over some fixed reference curve, see Assumption 1.3 and Theorem 1.2 for details.…”

On the Nernst–Planck–Navier–Stokes system

“…We sketch here for the sake of completeness our proof of Theorem 3. Local existence based on methods of maximal regularity was presented in ( [2]. )…”

Global existence for diffusion–electromigration systems in space dimension three and higher

“…The parameter τ > 0 can be viewed as the capacity of the boundary and ξ (a given datum to the problem) refers to an external potential as well as a boundary charge. For a more detailed discussion in this respect, we refer to [4]. Actually, we will assume that τ = τ (x) ≡ 0 so that pure Neumann boundary conditions may be considered on some parts of the boundary.…”