1 Introduction

To describe electrokinetic transport occurring in micro-structures in many physical, chemical, and biological applications, a proper mathematical model adhering to the law of conservation of mass is suggested following the approach [5, 9]. The reference cross-diffusion system of elliptic-parabolic type is described by nonlinear Poisson–Nernst–Planck (PNP) equations for concentrations of charged species and overall electrostatic potential. For physical consistency, they are generalized with entropy variables associating the pressure and quasi-Fermi electro-chemical potentials.

Based on a suitable free energy (see thermodynamic principles in [24]), in [20] a variational principle was established within the Gibbs simplex, thus preserving the total mass balance and non-negative species concentrations. In [18, 19], the generalized PNP problem was stated in two-phase medium composed of pore and particle parts and taking into account for nonlinear interface reactions which are of primary importance in applications. Its rigorous asymptotic analysis was carried our in [7, 8]. For a broad class of other relevant transport equations we refer to [6, 12, 15, 22], to [13] for stochastic systems, and to [17, 25] for variational principles.

Based on the entropy variables and following the thermodynamic formalism for cross-diffusion systems introduced in [11, 16], in the current work we endow the generalized PNP problem with the structure of a gradient flow and analyze it. Within the entropy approach, the question of global solvability of related diffusion problems was investigated in [1, 2, 4]. For the general theory of linear and quasilinear parabolic equations we refer to [21]. However, the key issue of the entropy approach requires uniformly strongly elliptic property of the governing system. Unfortunately, the ellipticity fails under coupling cross-diffusive phenomena for the PNP problem, thus implying the degenerate case. For a study of degenerate elliptic operators, see [23].

In Sect. 3 we present well-posedness analysis following from the regularization approach by [26]. We set the entropy variables as independent ones. In the fully coupled case, the non-negativity of species concentrations might be lost during the time evolution. Otherwise, when the electro-chemical potentials are well-defined, then the species concentrations are expressed by a normalized canonical ensemble of Fermi–Dirac statistics, thus yielding the non-negativity and the total mass balance. Moreover, in the decoupled case, in Sect. 4 we prove directly well-posedness of the static equilibrium for the underlying problem. A rigorous derivation of energy and entropy estimates is collected in Appendix A.

2 Generalized PNP problem

We start with the geometry configuration. Let \(\varOmega \subset \mathbb {R}^d\) (with natural \(d\in \mathbb {N}\)) be a connected domain with the Lipschitz boundary \(\partial \varOmega \) and the normal vector \({\nu }=(\nu _1,\ldots ,\nu _d)^\top \) outward to \(\varOmega \). Here and in what follows the upper symbol \({}^\top \) stands for transposition swapping columns and rows. We split \(\partial \varOmega \) into two disjoint parts \(\varGamma _{\mathrm{D}}\) and \(\varGamma _{\mathrm{N}}\) corresponding to mixed Dirichlet–Neumann boundary conditions. By this consideration we associate \(\varOmega \) to a pore space with a bath boundary \(\varGamma _{\mathrm{D}}\), which is complement to a solid space (bearing in mind possibly disconnected set of micro-particles) with the boundary \(\varGamma _{\mathrm{N}}\).

For time \(t\in \mathbb {R}_+\) and spatial coordinates \(\mathbf {x} =(x_1,\ldots ,x_d)^\top \in \mathbb {R}^d\), we look for an unknown distribution over the cylinder \((0,T)\times \varOmega \) (with the final time \(T>0\)) of mass concentrations \(\varvec{\rho }(t,\mathbf {x}) =(\rho _1,\ldots ,\rho _n)^\top \) (natural \(n\ge 2\)) of charged species (ions) with electric charges \(\mathbf {z} =(z_1,\ldots ,z_n)^\top \), electro-chemical potentials \(\varvec{\mu }(t,\mathbf {x}) =(\mu _1, \ldots ,\mu _n)^\top \), the overall electrostatic potential \(\phi (t,\mathbf {x})\), and the pressure \(p(t,\mathbf {x})\) according to the generalization that was introduced in [5]. For convenience, all the physical variables and parameters of the model are gathered in Table 1.

Table 1 List of symbols
Full size table

Our modeling is based on the general law of cross-diffusion

$$\begin{aligned} {\textstyle \frac{\partial }{\partial t}} \rho _i =\mathrm{div}\, \mathbf {J}_i,\quad i=1,\dots ,n, \end{aligned}$$
(1a)

where the vector-valued diffusion fluxes \(\mathbf {J}_i(t,\mathbf {x}) =((J_i)_1,\ldots ,(J_i)_d)^\top \) are given by the constitutive law (see [5])

$$\begin{aligned} \mathbf {J}_i^\top =\sum _{j=1}^n \rho _j \nabla \mu _j^\top {\mathsf {D}}^{ij},\quad i=1,\dots ,n, \end{aligned}$$
(1b)

with coupling by means of diffusivity matrices \({\mathsf {D}}^{ij}\in \mathbb {R}^{d\times d}\), \(i,j=1,\dots ,n\). Here and in what follows \(\mathrm{div}\) stands for the divergence, and \(\nabla \) for the gradient. Inserting (1b) into (1a) implies a strongly nonlinear equation with respect to \(\varvec{\rho }\) and \(\varvec{\mu }\). The fluxes have to fulfill the mass conservation law:

$$\begin{aligned} \sum _{i=1}^n \mathbf {J}_i =0. \end{aligned}$$
(1c)

The electrostatic potential \(\phi \) is described by the Poisson equation

$$\begin{aligned} -\mathrm{div} (\nabla \phi ^\top {\mathsf {A}}) -\sum _{k=1}^n z_k \rho _k =0, \end{aligned}$$
(1d)

where the electric permittivity \({\mathsf {A}}\in \mathbb {R}^{d\times d}\). The Navier–Stokes equation (see e.g. [14]) with zero flow velocity results in the force balance

$$\begin{aligned} \mathrm{u} \nabla p =-\sum _{i=k}^n z_k \rho _k \nabla \phi . \end{aligned}$$
(1e)

The species concentrations should be physically consistent within a Gibbs simplex requiring non-negativity and preserving the total mass \(C>0\):

$$\begin{aligned} \sum _{i=1}^n \rho _i=C,\quad \rho _i\ge 0,\quad i=1,\ldots ,n. \end{aligned}$$
(1f)

Introducing the Lagrangian function of a free energy (see [20])

$$\begin{aligned} \mathcal {E}(\varvec{\rho },\phi ,p)&=\int _{\varOmega } \Bigl \{ {\sum _{i=1}^n} \bigl ( k_{\mathrm{B}} \theta \rho _i (\ln (\beta _i \rho _i) -1) +z_i \rho _i \phi \bigr ) -{\textstyle \frac{1}{2}} \nabla \phi ^\top {\mathsf {A}} \nabla \phi \nonumber \\&\quad +\,{\textstyle \frac{\mathrm{u}}{{C}}} p \bigl ( \sum _{i=1}^n \rho _i -C\bigr ) \Bigr \} \,d\mathbf {x} +\int _{\varGamma _{\mathrm{N}}} g\phi \,dS_{\mathbf {x}} \end{aligned}$$
(2a)

the governing laws (1) are completed with the thermodynamic equilibrium expressed by functional derivatives

$$\begin{aligned} \mu _i ={\textstyle \frac{\delta \mathcal {E}}{\delta \rho _i}} =k_{\mathrm{B}} \theta \ln (\beta _i \rho _i) +z_i\phi +{\textstyle \frac{\mathrm{u}}{{C}}}\, p, \quad i=1,\ldots ,n, \end{aligned}$$
(2b)

implying the Gibbs–Duhem equation for the electro-chemical potentials. It is worth noting that substitution of (2b) and (1b) into the diffusion equation (1a) leads to the gradient-flow structure

$$\begin{aligned} {\textstyle \frac{\partial }{\partial t}} \rho _i =\mathrm{div}\Bigl ( \sum _{j=1}^n \rho _j \nabla \left( {\textstyle \frac{\delta \mathcal {E}}{\delta \rho _j}} \right) ^\top {\mathsf {D}}^{ij}\Bigr ),\quad i=1,\dots ,n. \end{aligned}$$

Since \(p+K\) is defined by (1e) up to an additive constant K, all the \(\mu _i +{\textstyle \frac{\mathrm{u}}{C}}\, K\) are determined by (2b) up to the same constant. Taking the gradient of (2b) and using the force balance (1e) leads to formulas [which will be useful later on to calculate the flux in (1b)] for \(i=1,\ldots ,n\)

$$\begin{aligned} \rho _i \nabla \mu _i =k_{\mathrm{B}} \theta \nabla \rho _i +\varUpsilon _i(\varvec{\rho }) \nabla \phi ,\quad \varUpsilon _i(\varvec{\rho }) :=\rho _i \left( z_i -{\textstyle \frac{1}{C}} \sum _{k=1}^n z_k \rho _k \right) , \end{aligned}$$
(2c)

where the functions \(\varUpsilon _1,\ldots ,\varUpsilon _n\) are uniformly bounded within the Gibbs simplex

$$\begin{aligned} |\varUpsilon _i(\varvec{\rho })|\le \rho _i (|z_i| +Z),\quad Z :=\sum _{k=1}^n |z_k|,\quad \text {for }\varvec{\rho }~\text {satisfying}~(1\hbox {f}). \end{aligned}$$
(2d)

Moreover, equating the variation \({\textstyle \frac{\delta \mathcal {E}}{\delta \phi }}\) of the function \(\mathcal {E}\) in (2a) to zero leads to the Gauss law in the form of Poisson equation (1d) and the inhomogeneous Neumann boundary condition below in (3c) for \(\phi \). From the optimization viewpoint, the pressure p enters (2a) as a Lagrange multiplier to the equality constraint in (1f) implying \({\textstyle \frac{\delta \mathcal {E}}{\delta p}} =0\).

The elliptic-parabolic system of nonlinear equations in (1)–(2) is endowed with the standard initial condition

$$\begin{aligned} \varvec{\rho } =\varvec{\rho }^0\quad \text {as}~t=0 \end{aligned}$$
(3a)

and mixed Dirichlet–Neumann boundary conditions

$$\begin{aligned}&\varvec{\rho } =\varvec{\rho }^0,\quad \phi =\phi ^0\quad \text {at}~\varGamma _{\mathrm{D}} \end{aligned}$$
(3b)
$$\begin{aligned}&\nabla \phi ^\top {\mathsf {A}} \varvec{\nu } =g,\quad \mathbf {J}_i^\top \varvec{\nu } =0, \quad i=1,\ldots ,n,\quad \text {at}~\varGamma _{\mathrm{N}}, \end{aligned}$$
(3c)

for given data functions \(g\in L^\infty (0,T; L^2(\varGamma _{\mathrm{N}}))\), \(\phi ^0\in L^\infty (0,T; H^1(\varOmega ))\), and \(\varvec{\rho }^0 =(\rho ^0_1,\ldots ,\rho ^0_n)^\top \in H^1(0,T; L^2(\varOmega ))^n \cap C([0,T]; H^1(\varOmega ))^n\) such that

$$\begin{aligned} \sum _{i=1}^n \rho ^0_i=C,\quad \rho ^0_i >0,\quad i=1,\ldots ,n. \end{aligned}$$
(3d)

It is worth remarking that an inhomogeneous condition for the normal diffusion flux in (3c) would be well-posed only when it depends nonlinearly on \(\varvec{\rho }\), this case was investigated in [18,19,20].

In order to guarantee the flux balance identity (1c), it suffices to assume

$$\begin{aligned} \sum _{i=1}^n {\mathsf {D}}^{ij} ={\mathsf {D}},\quad j=1,\ldots ,n, \end{aligned}$$
(4a)

with an elliptic matrix \({\mathsf {D}} \in \mathbb {R}^{d\times d}\). Indeed, substituting into (1c) the constitutive equations (1b) together with the expression for \(\rho _i\nabla \mu _i\) from (2c) and using the assumption (4a), after summation of the fluxes \(\mathbf {J}_i\) over \(i =1,\ldots ,n\) we have

$$\begin{aligned} \sum _{i=1}^n \mathbf {J}_i^\top =\sum _{i,j=1}^n \rho _j \nabla \mu _j^\top {\mathsf {D}}^{ij} =\sum _{j=1}^n \rho _j \nabla \mu _j^\top {\mathsf {D}} =\sum _{j=1}^n \bigl ( k_{\mathrm{B}} \theta \nabla \rho _j +\varUpsilon _j(\varvec{\rho }) \nabla \phi \bigr )^\top {\mathsf {D}} =0 \end{aligned}$$

since \(\sum _{j=1}^n (\nabla \rho _j) =\nabla C =0\) and \(\sum _{j=1}^n \varUpsilon _j(\varvec{\rho }) =0\) in (2c) due to the total mass balance in (1f). The assumption (4a) is related to quasi-stochastic matrices. In fact, rearranging the d-by-d matrix entries \({\mathsf {D}}^{ij} =(D^{ij}_{kl})_{k,l=1}^d\) into the n-by-n matrices \({\mathsf {D}}_{kl} =(D^{ij}_{kl})_{i,j=1}^n\), their sum in every column according to (4a) is equal to the same entry of \({\mathsf {D}} =(D_{kl})_{k,l=1}^d\), i.e.

$$\begin{aligned} \sum _{i=1}^n D^{ij}_{kl} =D_{kl}\quad \text{ for } \text{ all }~ j=1,\dots ,n. \end{aligned}$$
(4b)

Such matrices \({\mathsf {D}}_{kl}\) with non-negative entries are called column quasi-stochastic.

The standard assumptions for solvability are the ellipticity and boundedness conditions for diffusivities: there exist \(0<\underline{d}\le \overline{d}\) such that

$$\begin{aligned} \underline{d} \sum _{k=1}^n |\nabla \rho _k|^2 \le \sum _{i,j=1}^n \nabla \rho _j^\top {\mathsf {D}}^{ij} \nabla \rho _i,\quad \sum _{i,j=1}^n \nabla \rho _j^\top {\mathsf {D}}^{ij} \nabla \overline{\rho _i} \le \overline{d} \sum _{k=1}^n |\nabla \rho _k^\top \nabla \overline{\rho }_k|, \end{aligned}$$
(4c)

and for the permittivity: there exist \(0<\underline{a}\le \overline{a}\) such that

$$\begin{aligned} \underline{a} |\nabla \phi |^2 \le \nabla \phi ^\top {\mathsf {A}} \nabla \phi ,\quad \nabla \phi ^\top {\mathsf {A}} \nabla \overline{\phi } \le \overline{a} |\nabla \phi ^\top \nabla \overline{\phi }|. \end{aligned}$$
(4d)

Based on (1)–(4) now we give a weak variational formulation of the generalized PNP problem by excluding the entropy variables \(\varvec{\mu }\) and p with the help of (2c). Find a pair of functions

$$\begin{aligned} \varvec{\rho }\in L^\infty (0,T; L^2(\varOmega ))^n \cap L^2(0,T; H^1(\varOmega ))^n, \quad \phi \in L^\infty (0,T; H^1(\varOmega )) \end{aligned}$$
(5a)

that satisfy the non-negativity and the total mass balance (1f), the Dirichlet condition (3b), and the following variational equations for \(i=1,\ldots ,n\)

$$\begin{aligned}&-\int _0^T \int _\varOmega \rho _i {\textstyle \frac{\partial \overline{\rho }_i}{\partial t}} \,d\mathbf {x} dt +\int _\varOmega \rho ^0_i \overline{\rho }_i(0) \,d\mathbf {x}\nonumber \\&\quad +\int _0^T \int _\varOmega \sum _{j=1}^n \bigl ( k_{\mathrm{B}} \theta \nabla \rho _j +\varUpsilon _j(\varvec{\rho }) \nabla \phi \bigr )^\top {\mathsf {D}}^{ij} \nabla \overline{\rho }_i \,d\mathbf {x} dt =0, \end{aligned}$$
(5b)
$$\begin{aligned}&\int _\varOmega \left( \nabla \phi ^\top {\mathsf {A}} \nabla \overline{\phi } -\sum _{k=1}^n z_k \rho _k \overline{\phi } \right) \,d\mathbf {x} =\int _{\varGamma _{\mathrm{N}}} g \overline{\phi } \,dS_{\mathbf {x}} \end{aligned}$$
(5c)

for all test functions \(\overline{\varvec{\rho }} =(\overline{\rho }_1,\ldots , \overline{\rho }_n)^\top \in H^1(0,T; L^2(\varOmega ))^n \cap L^2(0,T; H^1(\varOmega ))^n\) and \(\overline{\phi }\in H^1(\varOmega )\) such that \(\overline{\varvec{\rho }}(T) =\mathbf {0}\); \(\overline{\varvec{\rho }} =\mathbf {0}\) and \(\overline{\phi } =0\) at \(\varGamma _{\mathrm{D}}\). The well-posedness to (5), (3b), and (1f) was investigated earlier in [18,19,20].

When solving problem (5), (3b), the key issue concerns fulfilling explicitly conditions (1f). In the following sections we consider the redundant entropy variable \(\varvec{\mu }\) as independent one, thus allowing to include conditions (1f) implicitly in the problem formulation.

3 Entropy formulation of the PNP problem

When (5), (3b) is solved, multiplying (1e) with the gradient of a smooth test function and integrating the result over the domain, the pressure can be determined as a solution (defined up to a constant K) to the elliptic equation

$$\begin{aligned} p\in L^\infty (0,T; H^1(\varOmega )),\quad \int _\varOmega \left( \mathrm{u} \nabla p +\sum _{k=1}^n z_k \rho _k \nabla \phi \right) ^\top \nabla \overline{p} \,d\mathbf {x} =0 \end{aligned}$$
(6a)

for all test functions \(\overline{p}\in H^1(\varOmega )\). The next step is to recover \(\varvec{\mu }\) from (2b). Taking \(\nabla \varvec{\mu }\), multiplying it with the gradient of suitable test functions, and integrating over the cylinder, the corresponding electro-chemical potentials

$$\begin{aligned} \mu _i\in L^2(0,T; H^1(\varOmega )),\quad i=1,\ldots ,n, \end{aligned}$$
(6b)

can be looked for as solutions to the mutually independent equations

$$\begin{aligned} \int _0^T \int _\varOmega \left( \nabla \mu _i -k_{\mathrm{B}} \theta {\textstyle \frac{\nabla \rho _i}{\rho _i}} -z_i \nabla \phi -{\textstyle \frac{\mathrm{u}}{{C}}} \nabla p \right) ^\top \nabla \overline{\mu }_i \,d\mathbf {x} dt =0 \end{aligned}$$
(6c)

holding for all test functions \(\overline{\mu }_i\in L^2(0,T; H^1(\varOmega ))\), and satisfying the Dirichlet boundary condition

$$\begin{aligned} \mu _i =k_{\mathrm{B}} \theta \ln (\beta _i \rho ^0_i) +z_i\phi ^0 +{\textstyle \frac{\mathrm{u}}{{C}}}\, p \quad \text {at}~\varGamma _{\mathrm{D}}. \end{aligned}$$
(6d)

In general, this elliptic problem is degenerate, because the operator of (6c) is unbounded due to the presence of factor \({\textstyle \frac{1}{\rho _i}}\). To remedy, we give the following two conditional assertions, which will be justified in Theorem 1 later on.

Proposition 1

(Existence of entropy variable \({\mu }\)) If the solution of the generalized PNP problem (5), (3b), and (1f) satisfies one of the following two conditions:

(i) there exist time \(T_0>0\) (possibly small) and a constant \(\underline{\rho }>0\) such that

$$\begin{aligned} \rho _i(t,\mathbf {x})\ge \underline{\rho }\qquad \text {for all}~ (t,\mathbf {x})\in (0,T_0)\times \varOmega , \end{aligned}$$
(6e)
$$\begin{aligned} \mathrm{(ii)} \,{\text {the inclusion}}~ {\textstyle \frac{\nabla \rho _i}{\rho _i}} \in L^2((0,T_0)\times \varOmega )^n~\text {holds}, \end{aligned}$$
(6f)

then the problem (6b)–(6d) for this index \(i\in \{1,\ldots ,n\}\) is uniquely solvable within this time \((0,T_0)\).

Conversely, any solution to the system (6b)–(6d) satisfies (2b).

Proof

Indeed, if either condition (i) or condition (ii) is satisfied, then existence and uniqueness of the solution to (6b)–(6d) stated in Proposition 1 follows immediately from a general fact on elliptic systems. \(\square \)

Remark 1

Since by the very definition of a solution, \(\rho _i\in L^2(0,T_0;H^1(\varOmega ))\), see (5a), condition (i) in Proposition 1 implies condition (ii).

From (2b) we get

$$\begin{aligned} \rho _i ={\textstyle \frac{1}{\beta _i}} \exp \left( {\textstyle \frac{1}{k_{\mathrm{B}} \theta }} (\mu _i -z_i\phi -{\textstyle \frac{\mathrm{u}}{{C}}}\, p) \right) , \end{aligned}$$
(7a)

and summing up these equations over \(i=1,\ldots ,n\), due to the total mass balance \(\sum _{i=1}^n \rho _i=C\) we can express the pressure as

$$\begin{aligned} p ={\textstyle \frac{k_{\mathrm{B}}\theta }{\mathrm{u}}} C \ln \left( {\textstyle \frac{1}{C}} \sum _{i=1}^n {\textstyle \frac{1}{\beta _i}} \exp \bigl ( {\textstyle \frac{1}{k_{\mathrm{B}}\theta }} (\mu _i -z_i\phi ) \bigr ) \right) . \end{aligned}$$
(7b)

Excluding p from (7a) with the help of (7b), we bring the species concentrations in the form of a canonical ensemble of Fermi–Dirac statistics

$$\begin{aligned} \rho _i =P_i(\varvec{\mu } -\mathbf {z} \phi ) :=C {\textstyle \frac{{\textstyle \frac{1}{\beta _i}} \exp \bigl ({\scriptstyle \frac{1}{k_{\mathrm{B}}\theta }} (\mu _i -z_i\phi ) \bigr )}{{\displaystyle \sum \nolimits _{k=1}^n} \frac{1}{\beta _k} \exp \bigl ({\scriptstyle \frac{1}{k_{\mathrm{B}}\theta }} (\mu _k -z_k\phi )\bigr )}},\quad i=1,\ldots ,n. \end{aligned}$$
(7c)

The normalized probabilities obey the inherent behavior

$$\begin{aligned} \sum _{i=1}^n P_i(\varvec{\mu } -\mathbf {z} \phi ) =C,\quad P_i(\varvec{\mu } -\mathbf {z} \phi )\ge 0,\quad i=1,\ldots ,n. \end{aligned}$$
(7d)

Then the non-negativity and the total mass balance (1f) follow straightforwardly from the properties (7d). By this we observe that \(\mu _i-z_i\phi \rightarrow -\infty \) would lead to the limit \(\rho _i =0\) in (7c). This is an admissible behavior, as an example, for the function \((\mu _i-z_i\phi )(\mathbf {x}) =-\ln |\ln |\mathbf {x}||\) as \(\mathbf {x}\rightarrow \mathbf {0}\), which agrees with the \(H^1\)-spatial regularity of \(\varvec{\mu }\) and \(\phi \) stated in (5a) and (6b). In spite of the fact that \(\varvec{\mu } +K\) is defined up to an additive constant K, the concentrations \(\rho _i =P_i(\varvec{\mu } +K -\mathbf {z} \phi )\) in (7c) are defined uniquely.

Based on (7), we reformulate the generalized PNP problem as follows. Find a triple of functions

$$\begin{aligned} \varvec{\rho }\in L^\infty ((0,T)\times \varOmega )^n, \quad \phi \in L^\infty (0,T; H^1(\varOmega )), \quad \varvec{\mu }\in L^2(0,T; H^1(\varOmega ))^n \end{aligned}$$
(8a)

that satisfy the Fermi–Dirac statistics (7c), the Dirichlet condition (3b), and the following variational equations for \(i=1,\ldots ,n\)

$$\begin{aligned}&-\int _0^T \int _\varOmega \rho _i {\textstyle \frac{\partial \overline{\rho }_i}{\partial t}} \,d\mathbf {x} dt +\int _\varOmega \rho ^0_i \overline{\rho }_i(0) \,d\mathbf {x}\nonumber \\&\qquad +\,\int _0^T \int _\varOmega \sum _{j=1}^n \rho _j \nabla \mu _j^\top {\mathsf {D}}^{ij} \nabla \overline{\rho }_i \,d\mathbf {x} dt =0, \end{aligned}$$
(8b)
$$\begin{aligned}&\int _\varOmega \bigl ( \nabla \phi ^\top {\mathsf {A}} \nabla \overline{\phi } -\sum _{k=1}^n z_k \rho _k \overline{\phi } \bigr ) \,d\mathbf {x} =\int _{\varGamma _{\mathrm{N}}} g \overline{\phi } \,dS_{\mathbf {x}} \end{aligned}$$
(8c)

for all test functions \(\overline{\varvec{\rho }} =(\overline{\rho }_1,\ldots , \overline{\rho }_n)^\top \in H^1(0,T; L^2(\varOmega ))^n \cap L^2(0,T; H^1(\varOmega ))^n\) and \(\overline{\phi }\in H^1(\varOmega )\) such that \(\overline{\varvec{\rho }}(T) =\mathbf {0}\); \(\overline{\varvec{\rho }} =\mathbf {0}\) and \(\overline{\phi } =0\) at \(\varGamma _{\mathrm{D}}\).

The nonlinear parabolic equations (8b) imply a degenerate system because its operator loses the ellipticity property compared to (5b). Firstly, because of the cross-diffusion structure induced by matrices \({\mathsf {D}}^{ij}\). Second, due to the presence of the factor \(\rho _j\), which is not uniformly positive in general. These facts do not allow to apply the boundedness-by-entropy method [16].

Nevertheless, the well-posedness results established in [18, 19] guarantee existence of a solution pair \((\varvec{\rho },\phi )\) for the generalized PNP problem in the form of (5), (3b) that satisfies the total mass balance in (1f). From (5b) using (2c) the time derivative \({\textstyle \frac{\partial \varvec{\rho }}{\partial t}}\) can be defined in \(L^2(0,T; H^{-1}(\varOmega ))^n\) as a continuous linear functional

$$\begin{aligned} \int _0^T \left\langle {\textstyle \frac{\partial \rho _i}{\partial t}}, \overline{\rho }_i \right\rangle _\varOmega \,dt =-\int _0^T \int _\varOmega \sum _{j=1}^n \rho _j \nabla \mu _j^\top {\mathsf {D}}^{ij} \nabla \overline{\rho }_i \,d\mathbf {x} dt,\quad i=1,\ldots ,n, \end{aligned}$$
(8b')

where \(H^{-1}(\varOmega )\) stands for the dual space to functions \(\overline{\rho }_i\in H^1(\varOmega )\) such that \(\overline{\rho }_i =0\) on \(\varGamma _{\mathrm{D}}\), with the duality pairing \(\langle \,\cdot \,,\,\cdot \,\rangle _\varOmega \) between them. In fact, \(\varvec{\rho }\) is a continuous function of time, see [3, Remark 1, p. 509]. Since the strongly positive initial distribution \(\varvec{\rho }^0 >\mathbf {0}\) is assumed in (3d), then by the continuity it follows local in time fulfillment of the condition (6e) in Proposition 1. Within uniformly positive \(\varvec{\rho }\ge \underline{\varvec{\rho }}>0\), the electro-chemical potential \(\varvec{\mu }\) in (2b) obeying the regularity (6b) exists, and the triple \((\varvec{\rho },\phi ,\varvec{\mu })\) solves the equivalent problem (8), (7c), and (3b). This leads to the following local existence theorem.

Theorem 1

(Local existence for cross-diffusion entropy system) There exists a time interval \((0,T_0)\) (with \(T_0>0\) maybe small), where the cross-diffusion entropy problem given by (8), (7c), and (3b) has a solution \((\varvec{\rho },\phi ,\varvec{\mu })\). By this, \((\varvec{\rho },\phi )\) solve the problem (5), (3b), and \(\varvec{\mu }\) is from (2b).

For such \(T_0\), the local a-priori estimates hold for all \((t,\mathbf {x})\in (0,T_0)\times \varOmega \)

$$\begin{aligned} 0<\underline{\rho } \le \rho _i <C,\quad \Vert \nabla \mu _i\Vert _{L^2((0,T_0)\times \varOmega )}\le {\textstyle \frac{1}{\underline{\rho }}} K^\mu _1(T_0) +K^\mu _2(T_0), \end{aligned}$$
(8d)

for \(i=1\ldots ,n\), with constant \(K^\mu _1(T_0), K^\mu _2(T_0) \ge 0\). For arbitrary final time \(T>0\), the global a-priori estimates hold with constant \(K_\phi >0\)

$$\begin{aligned} 0\le \rho _i \le C,\quad i=1\ldots ,n,\quad \Vert \nabla \phi \Vert _{L^2(\varOmega )}\le K_\phi . \end{aligned}$$
(8e)

Proof

Here we justify the a-priori estimates. In (8e), the bounds of \(\varvec{\rho }\) follow from (7c) and (7d), and the bound of \(\phi \) is proved in Appendix A, Lemma 1. In (8d), the strict bounds of \(\varvec{\rho }\) are the consequence of continuity due to the strongly positive initial distribution (3d), and below we prove the estimate of \(\varvec{\mu }\).

The lower bound of \(\rho _i\) in (8d) allows us to apply Proposition 1. Based on this fact, we substitute p from (6a) and \(\overline{\mu }_i =\mu _i\) into (6c) to obtain

$$\begin{aligned} \int _0^{T_0} \int _\varOmega \bigl ( \nabla \mu _i -k_{\mathrm{B}} \theta {\textstyle \frac{\nabla \rho _i}{\rho _i}} -{\textstyle \frac{\varUpsilon _i(\varvec{\rho })}{\rho _i}} \nabla \phi \bigr )^\top \nabla \mu _i \,d\mathbf {x} dt =0, \end{aligned}$$

where the functions \(\varUpsilon _i\) are defined by (2c). With the help of the Cauchy–Schwarz inequality, from the above equality we get

$$\begin{aligned} \Vert \nabla \mu _i\Vert _{L^2((0,T_0)\times \varOmega )} \le {\textstyle \frac{k_{\mathrm{B}} \theta }{\underline{\rho }}} \Vert \nabla \rho _i\Vert _{L^2((0,T_0)\times \varOmega )} +(|z_i| +Z) \Vert \nabla \phi \Vert _{L^2((0,T_0)\times \varOmega )}, \end{aligned}$$

where the upper bound of \(\varUpsilon _i(\varvec{\rho })\) is from (2d). Applying the estimate of \(\nabla \phi \) from (8e) and the bound of \(\nabla \varvec{\rho }\) proved in [18,19,20] for arbitrary \(T>0\)

$$\begin{aligned} \Vert \nabla \rho _i\Vert _{L^2((0,T)\times \varOmega )}\le K^\rho _1(T) +K^\rho _2(T) \Vert \nabla \phi \Vert _{L^2((0,T)\times \varOmega )},\quad i=1\ldots ,n, \end{aligned}$$
(8f)

with \(K^\rho _1(T), K^\rho _1(T) >0\), we obtain the estimate for \(\nabla \mu _i\) in (8d) with the bounds \(K^\mu _1(T) = k_{\mathrm{B}} \theta \bigl ( K^\rho _1(T_0) + K^\rho _2(T_0) T_0 K_\phi \bigr )\) and \(K^\mu _2(T) = (|z_i| +Z) T_0 K_\phi \), thus finishing the proof. \(\square \)

For global in time solvability, we require a stronger than (4a) assumption

$$\begin{aligned} {\mathsf {D}}^{ij} =\delta _{ij} {\mathsf {D}},\quad i,j=1,\ldots ,n, \end{aligned}$$
(9a)

with the Kronecker \(\delta _{ij} =1\) for \(i=j\), and zero otherwise. The assumption (9a) implies (4a) and imposes decoupling in (8b) as well as (5b). In this case, existence of a solution \((\varvec{\rho },\phi )\) to (5), (3b) satisfying both conditions in (1f) globally in time is proved in [18, 19]. Therefore, the existence of the regular entropy variable \(\varvec{\mu }\) from (2b) is sufficient to state the following theorem (see a relevant work [10]).

Theorem 2

(Conditional global existence for decoupled entropy system) Fix an arbitrary final time \(T>0\). Let \((\varvec{\rho },\phi )\) be a solution of problem (5), (3b) under the decoupling assumption (9a). If the electro-chemical potential \(\varvec{\mu }\) solving problem (6b)–(6d) exists, then the triple \((\varvec{\rho },\phi ,\varvec{\mu })\) solves the decoupled entropy system (8), (7c), and (3b).

The global a-priori estimates (8e) hold. If the Dirichlet data in (3b) are constant, i.e.

$$\begin{aligned} \rho ^0_i ={\textstyle \frac{1}{\beta _i}},\quad i=1\ldots ,n, \quad \text {at}~ \varGamma _{\mathrm{D}}, \end{aligned}$$
(9b)

then the estimate with a constant \(K_\mu (T) >0\) holds (see Appendix A, Lemma 2)

$$\begin{aligned} \Vert \sqrt{\rho _i}\, \nabla \mu _i\Vert _{L^2((0,T)\times \varOmega )}\le K_\mu (T),\quad i=1\ldots ,n. \end{aligned}$$
(9c)

In the next section we investigate an equilibrium state when \(T\nearrow \infty \).

4 Equilibrium state

Let \(\lim _{T\rightarrow \infty } g(T,\mathbf {x}) =:g^\infty (\mathbf {x})\in L^2(\varGamma _{\mathrm{N}})\) in (3c), and the limit in (3b)

$$\begin{aligned} \lim _{T\rightarrow \infty } (\varvec{\rho }^0,\phi ^0)(T,\mathbf {x}) =(\varvec{\rho }^{(0,\infty )}, \phi ^{(0,\infty )}) \end{aligned}$$
(10a)

be constant independent of \(\mathbf {x}\) and satisfying according to (3d) the properties

$$\begin{aligned} \sum _{i=1}^n \rho ^{(0,\infty )}_i=C,\quad \rho ^{(0,\infty )}_i \ge 0,\quad i=1,\ldots ,n. \end{aligned}$$
(10b)

We consider a stationary counterpart of the entropy system (8), (7c), and (3b) under the decoupling assumption (9a). Find a triple of functions

$$\begin{aligned} \varvec{\rho }^\infty (\mathbf {x})\in L^\infty (\varOmega )^n, \quad \phi ^\infty (\mathbf {x})\in H^1(\varOmega ), \quad \varvec{\mu }^\infty (\mathbf {x})\in H^1(\varOmega )^n \end{aligned}$$
(11a)

that satisfy the Fermi–Dirac statistics

$$\begin{aligned} \rho ^\infty _i =P_i(\varvec{\mu }^\infty -\mathbf {z} \phi ^\infty ),\quad i=1,\ldots ,n, \end{aligned}$$
(11b)

the Dirichlet condition

$$\begin{aligned} \varvec{\rho }^\infty =\varvec{\rho }^{(0,\infty )},\quad \phi ^\infty =\phi ^{(0,\infty )} \quad \text {at}~ \varGamma _{\mathrm{D}}, \end{aligned}$$
(11c)

and the following variational equations for \(i=1,\ldots ,n\)

$$\begin{aligned}&\int _\varOmega \rho ^\infty _i (\nabla \mu ^\infty _i)^\top {\mathsf {D}} \nabla \overline{\mu }_i \,d\mathbf {x} =0, \end{aligned}$$
(11d)
$$\begin{aligned}&\int _\varOmega \Bigg ( (\nabla \phi ^\infty )^\top {\mathsf {A}} \nabla \overline{\phi } -\sum _{k=1}^n z_k \rho ^\infty _k \overline{\phi } \Bigg ) d\mathbf {x} =\int _{\varGamma _{\mathrm{N}}} g^\infty \overline{\phi } \,dS_{\mathbf {x}} \end{aligned}$$
(11e)

for all \(\overline{\varvec{\mu }} \in H^1(\varOmega )^n\) and \(\overline{\phi }\in H^1(\varOmega )\) such that \(\overline{\varvec{\mu }} =\mathbf {0}\) and \(\overline{\phi } =0\) at \(\varGamma _{\mathrm{D}}\).

Theorem 3

(Existence of equilibrium) A solution of problem (11) exists, which is given by the following relations

$$\begin{aligned} \varvec{\mu }^\infty =\mathbf {0},\quad \rho ^\infty _i =P_i(-\mathbf {z} \phi ^\infty ) =C {\textstyle \frac{{\textstyle \frac{1}{\beta _i}} \exp \bigl (-{\scriptstyle \frac{z_i}{k_{\mathrm{B}}\theta }} \phi ^\infty \bigr )}{{\displaystyle \sum \nolimits _{k=1}^n} \frac{1}{\beta _k} \exp \bigl ( -{\scriptstyle \frac{z_k}{k_{\mathrm{B}}\theta }} \phi ^\infty \bigr )}},\quad i=1,\ldots ,n, \end{aligned}$$
(12a)

with the unique solution \(\phi ^\infty (\mathbf {x})\in H^1(\varOmega )\) satisfying the Dirichlet condition

$$\begin{aligned} \phi ^\infty =\phi ^{(0,\infty )} \quad \text {at}~\varGamma _{\mathrm{D}} \end{aligned}$$
(12b)

and the quasilinear variational equation for the electrostatic potential

$$\begin{aligned} \int _\varOmega \bigl ( (\nabla \phi ^\infty )^\top {\mathsf {A}} \nabla \overline{\phi } -\sum _{k=1}^n z_k P_k(-\mathbf {z} \phi ^\infty ) \overline{\phi } \bigr ) \,d\mathbf {x} =\int _{\varGamma _{\mathrm{N}}} g^\infty \overline{\phi } \,dS_{\mathbf {x}} \end{aligned}$$
(12c)

for all test functions \(\overline{\phi }\in H^1(\varOmega )\) such that \(\overline{\phi } =0\) at \(\varGamma _{\mathrm{D}}\).

Proof

We transform (11b) to the form (2b)

$$\begin{aligned} \mu ^\infty _i =k_{\mathrm{B}} \theta \ln (\beta _i \rho ^\infty _i) +z_i\phi ^\infty +{\textstyle \frac{\mathrm{u}}{{C}}}\, p^\infty . \end{aligned}$$

Consider the function

$$\begin{aligned} \mu ^{(0,\infty )}_i := k_{\mathrm{B}} \theta \ln (\beta _i \rho ^{(0,\infty )}_i) +z_i\phi ^{(0,\infty )} +{\textstyle \frac{\mathrm{u}}{{C}}}\, p^\infty \end{aligned}$$

defined in \(\varOmega \). Then it holds for \(i=1,\ldots ,n\)

$$\begin{aligned} \mu ^\infty _i -\mu ^{(0,\infty )}_i = k_{\mathrm{B}} \theta \ln \Bigl ( {\textstyle \frac{\rho ^\infty _i}{\rho ^{(0,\infty )}_i}} \Bigr ) +z_i (\phi ^\infty -\phi ^{(0,\infty )}), \end{aligned}$$
(13a)

and due to the assumption of constant \((\varvec{\rho }^{(0,\infty )}, \phi ^{(0,\infty )})\) in (10a)

$$\begin{aligned} \nabla \mu ^{(0,\infty )}_i ={\textstyle \frac{\mathrm{u}}{{C}}}\nabla p^\infty ,\quad i=1,\ldots ,n. \end{aligned}$$
(13b)

By virtue of the Dirichlet conditions (11c) we have \(\mu ^\infty _i -\mu ^{(0,\infty )}_i =0\) at \(\varGamma _{\mathrm{D}}\) and hence can substitute (13a) for the test function \(\overline{\mu }_i\) into (11d). Summing over \(i=1,\ldots ,n\) and taking into account that \(\underline{d} |\nabla \mu ^\infty _i|^2\le (\nabla \mu ^\infty _i)^\top {\mathsf {D}} \nabla \mu ^\infty _i\) with some \(\underline{d}>0\) by ellipticity of \({\mathsf {D}}\) in (9a), we obtain

$$\begin{aligned}&0\le \underline{d} \int _\varOmega \sum _{i=1}^n \rho ^\infty _i |\nabla \mu ^\infty _i|^2 \,d\mathbf {x} \le \int _\varOmega \sum _{i=1}^n \rho ^\infty _i (\nabla \mu ^\infty _i)^\top {\mathsf {D}} \nabla \mu ^\infty _i \,d\mathbf {x}\\&\quad =\int _\varOmega \sum _{i=1}^n \rho ^\infty _i (\nabla \mu ^\infty _i)^\top {\mathsf {D}} \nabla \mu ^{(0,\infty )}_i \,d\mathbf {x} ={\textstyle \frac{\mathrm{u}}{{C}}} \int _\varOmega \sum _{i=1}^n \rho ^\infty _i (\nabla \mu ^\infty _i)^\top {\mathsf {D}} \nabla p^\infty \,d\mathbf {x} =0. \end{aligned}$$

Here we used the equality (13b) and the flux conservation (1c) implying \(\sum _{i=1}^n \rho ^\infty _i (\nabla \mu ^\infty _i)^\top {\mathsf {D}} =0\). This identity is sufficed by \(\varvec{\mu }^\infty =\mathbf {0}\), which is also necessary when \(\varvec{\rho }^\infty >\mathbf {0}\). Henceforth, \(\rho ^\infty _i =P_i(-\mathbf {z} \phi ^\infty )\) according to (11b), thus implying (12a), and (11e) turns into (12c).

The solvability of problem (12b), (12c) is proved in Appendix A, Lemma 1. \(\square \)