On the Space Analyticity of the Nernst–Planck–Navier–Stokes system

Abstract

We consider an electrodiffusion model that describes the intricate interplay of multiple ionic species with a twodimensional, incompressible, viscous fluid subjected to stochastic additive noise. This system involves nonlocal nonlinear drift-diffusion Nernst-Planck equations for ionic species and stochastic Navier-Stokes equations for fluid motion under the influence of electric and time-independent forces. Under the selective boundary conditions imposed on the concentrations, we establish the existence and uniqueness of global pathwise solutions to this system on smooth bounded domains. Our study also investigates long-time ionic concentration dynamics and explores Feller properties of the associated Markovian semigroup. In the context of equal diffusive species and under appropriate conditions, we demonstrate the existence of invariant ergodic measures supported on H 2 . We then enhance the ergodicity results on periodic tori and obtain smooth invariant measures under a constraint on the initial spatial averages of the concentrations. The uniqueness of the invariant measures on periodic boxes and smooth bounded domains is further established when the noise forces sufficient modes, and the diffusivities of the species are large. Finally, in the case of two ionic species with equal diffusivities and valences of 1 and −1, we study the rate of convergence of the Markov transition kernels to the invariant measure and obtain unconditional, unique exponential ergodicity for the model.

On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions

“…The forcing term f in (1) is a function of both space and time. Several approaches were developed over the years to study the analyticity of nonlinear parabolic equations on domains with boundaries ( [8,9]) based on successive applications of the L 2 norms of derivatives, and without boundaries based on Fourier series techniques ( [2,5,6,11] and references therein), a mild formulation of the complexified problem ( [1], [7]), etc. Recently, Kukavica and Vicol established in [10] a derivative reduction proof, based on classical energy inequalities, to study the analyticity up to the boundary of the d-dimensional inhomogeneous heat equation on the half-space with homogeneous Dirichlet Boundary conditions.…”

“…As for the sum which does not depend on the normal derivatives of solutions, we decompose it into three sub-sums, S 1 , S 2 , and S 3 , where S 1 includes all terms with at least two tangential derivatives, S 2 depends on exactly one tangential derivative, and S 3 is the sum of the remaining time derivative terms. The estimation of S 1 uses the structure of the diffusion driven by ∆q, which, by making use of the heat equation (1), allows us to reduce the number of tangential derivatives by increasing the number of normal derivatives. As for the sum S 2 , we interpolate in the tangential variable to have an additional tangential derivative and hence have good control of S 2 by S 1 .…”

Well-posedness of the generalized Navier–Stokes equations with damping

“…They achieved global regularity of solutions for W 2,p initial data and demonstrated their convergence to stable steady states on 2D bounded smooth domains, incorporating selective boundary conditions. In the periodic setting, the global well-posedness of the NPNS system as well as the exponential stability of solutions were established in [2,3]. See also [4,24,25] for global well-posedness results of the Nernst-Planck-Euler and Nernst-Planck-Darcy systems, which are two other closely related models.…”

“…Although the deterministic NPNS system has been extensively studied, there is a notable scarcity of research about the stochastic version of NPNS system. In a recent work [1], the authors studied the 2D stochastic NPNS system with equation (1.2) being replaced by the stochastic NSE in the Itô sense:…”

“…Our main results are the well-posedness and enhanced dissipation for the solutions to this NPNS system with Stratonovich-type transport noise. When examining the global well-posedness, the presence of transport noise renders the approach in [1] and the entropy estimates in the deterministic case [14] inapplicable. To address this challenge, we focus on the scenario involving two ionic species with identical diffusivity and opposite valences.…”