A proper mathematical model given by nonlinear Poisson–Nernst–Planck (PNP) equa- tions which describe electrokinetics of charged species is considered. The model is generalized with entropy variables associating the pressure and quasi-Fermi electro- chemical potentials in order to adhere to the law of conservation of mass. Based on a variational principle for suitable free energy, the generalized PNP system is endowed with the structure of a gradient ﬂow. The well-posedness theorems for the mixed formulation (using the entropy variables) of the gradient-ﬂow problem are provided within the Gibbs simplex and supported by a-priori estimates of the solution. Keywords Electrokinetics · Poisson–Nernst–Planck equations · Gibbs simplex · Entropy variables · Gradient ﬂow · Well-posedness analysis · Fermi–Dirac statistics Mathematics Subject Classiﬁcation Primary 35K55; Secondary 37B35 · 80A32 · 82C31 The work is supported by the Austrian Science Fund (FWF) project P26147-N26: “Object identiﬁcation problems: numerical analysis” (PION) and the Austrian Academy of Sciences (OeAW). B Victor A. Kovtunenko victor.kovtunenko@uni-graz.at José Rodrigo González Granada jorodryy@utp.edu.co Department of Mathematics, Universidad Tecnológica de Pereira, Pereira 660003, Colombia Institute for Mathematics and Scientiﬁc Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstr.36, 8010 Graz, Austria Lavrent’ev Institute of Hydrodynamics, Siberian Division of Russian Academy of Sciences, Novosibirsk, Russia 630090 604 J. R. G. Granada, V. A. Kovtunenko 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 gen- eralized 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 uni- formly 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 cou- pled 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 equilib- rium 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 Ω ⊂ R (with natural d ∈ N)be a connected domain with the Lipschitz boundary ∂Ω and the normal vector ν = (ν ,...,ν ) outward to Ω. Here and in what follows the upper symbol stands 1 d for transposition swapping columns and rows. We split ∂Ω into two disjoint parts Γ and Γ corresponding to mixed Dirichlet–Neumann boundary conditions. By this D N consideration we associate Ω to a pore space with a bath boundary Γ , which is D Entropy method for generalized Poisson–Nernst–Planck… 605 Table 1 List of symbols Symbol Unit Name Property k ≈ 1.38e−23 Boltzmann constant Positive u ≈ 1.66e−27 (kg) Uniﬁed atomic mass unit Positive constant θ (K) Absolute temperature Positive constant kg ρ , C Mass concentration of species i = 1,..., n positive i 3 β Volume factors of species i = 1,..., n positive kg kg J Diffusion ﬂuxes of species i = 1,..., n vector m s μ (J) Electro-chemical potentials i = 1,..., n ij m D , D Diffusivity matrices i , j = 1,..., n elliptic Js p (Pa) Pressure z (C) Electric charges of species i = 1,..., n constant φ (V) Electrostatic potential Fkg A Electric permittivity matrix Elliptic Ckg g Electric displacement complement to a solid space (bearing in mind possibly disconnected set of micro- particles) with the boundary Γ . For time t ∈ R and spatial coordinates x = (x ,..., x ) ∈ R , we look for + 1 d an unknown distribution over the cylinder (0, T ) × Ω (with the ﬁnal time T > 0) of mass concentrations ρ(t , x) = (ρ ,...,ρ ) (natural n ≥ 2) of charged species 1 n (ions) with electric charges z = (z ,..., z ) , electro-chemical potentials μ(t , x) = 1 n (μ ,...,μ ) , the overall electrostatic potential φ(t , x), and the pressure p(t , x) 1 n 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. Our modeling is based on the general law of cross-diffusion ρ = div J , i = 1,..., n, (1a) i i ∂t where the vector-valued diffusion ﬂuxes J (t , x) = ((J ) ,...,(J ) ) are given by i i 1 i d the constitutive law (see [5]) ij J = ρ ∇μ D , i = 1,..., n, (1b) i j j =1 ij d×d with coupling by means of diffusivity matrices D ∈ R , i , j = 1,..., n. Here and in what follows div stands for the divergence, and ∇ for the gradient. Inserting (1b) into (1a) implies a strongly nonlinear equation with respect to ρ and μ. The ﬂuxes have to fulﬁll the mass conservation law: 606 J. R. G. Granada, V. A. Kovtunenko J = 0. (1c) i =1 The electrostatic potential φ is described by the Poisson equation − div(∇φ A) − z ρ = 0, (1d) k k k=1 d×d where the electric permittivity A ∈ R . The Navier–Stokes equation (see e.g. [14]) with zero ﬂow velocity results in the force balance u∇ p =− z ρ ∇φ. (1e) k k i =k The species concentrations should be physically consistent within a Gibbs simplex requiring non-negativity and preserving the total mass C > 0: ρ = C,ρ ≥ 0, i = 1,..., n. (1f) i i i =1 Introducing the Lagrangian function of a free energy (see [20]) E (ρ,φ, p) = k θρ (ln(β ρ ) − 1) + z ρ φ − ∇φ A∇φ B i i i i i Ω i =1 2 + p ρ − C dx + gφ dS (2a) i x C i =1 Γ the governing laws (1) are completed with the thermodynamic equilibrium expressed by functional derivatives δE u μ = = k θ ln(β ρ ) + z φ + p, i = 1,..., n, (2b) i B i i i δρ C 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 ∂ δE ij ρ = div ρ ∇ D , i = 1,..., n. i j ∂t δρ j =1 Since p + K is deﬁned by (1e) up to an additive constant K , all the μ + 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,..., n Entropy method for generalized Poisson–Nernst–Planck… 607 ρ ∇μ = k θ ∇ρ + Υ (ρ)∇φ, Υ (ρ) := ρ z − z ρ , (2c) i i B i i i i i k k k=1 where the functions Υ ,...,Υ are uniformly bounded within the Gibbs simplex 1 n |Υ (ρ)|≤ ρ (|z |+ Z ), Z := |z |, for ρ satisfying (1f). (2d) i i i k k=1 δE Moreover, equating the variation of the function E in (2a) to zero leads to the Gauss δφ law in the form of Poisson equation (1d) and the inhomogeneous Neumann boundary condition below in (3c)for φ. From the optimization viewpoint, the pressure p enters δE (2a) as a Lagrange multiplier to the equality constraint in (1f) implying = 0. δ p The elliptic-parabolic system of nonlinear equations in (1)–(2) is endowed with the standard initial condition ρ = ρ as t =0(3a) and mixed Dirichlet–Neumann boundary conditions 0 0 ρ = ρ ,φ = φ at Γ (3b) ∇φ Aν = g, J ν = 0, i = 1,..., n, at Γ , (3c) ∞ 2 0 ∞ 1 0 for given data functions g ∈ L (0, T ; L (Γ )), φ ∈ L (0, T ; H (Ω)), and ρ = 0 0 1 2 n 1 n (ρ ,...,ρ ) ∈ H (0, T ; L (Ω)) ∩ C ([0, T ]; H (Ω)) such that 1 n 0 0 ρ = C,ρ > 0, i = 1,..., n. (3d) i i i =1 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 ρ, this case was investigated in [18–20]. In order to guarantee the ﬂux balance identity (1c), it sufﬁces to assume ij D = D, j = 1,..., n, (4a) i =1 d×d with an elliptic matrix D ∈ R . Indeed, substituting into (1c) the constitutive equa- tions (1b) together with the expression for ρ ∇μ from (2c) and using the assumption i i (4a), after summation of the ﬂuxes J over i = 1,..., n we have n n n n ij J = ρ ∇μ D = ρ ∇μ D = k θ ∇ρ + Υ (ρ)∇φ D = 0 j j B j j i j j i =1 i , j =1 j =1 j =1 n n since (∇ρ ) =∇C = 0 and Υ (ρ) = 0in (2c) due to the total mass j j j =1 j =1 balance in (1f). The assumption (4a) is related to quasi-stochastic matrices. In fact, 608 J. R. G. Granada, V. A. Kovtunenko ij ij d rearranging the d-by-d matrix entries D = (D ) into the n-by-n matrices kl k,l=1 ij D = (D ) , their sum in every column according to (4a) is equal to the same kl kl i , j =1 entry of D = (D ) , i.e. kl k,l=1 ij D = D for all j = 1,..., n. (4b) kl kl i =1 Such matrices D with non-negative entries are called column quasi-stochastic. kl The standard assumptions for solvability are the ellipticity and boundedness con- ditions for diffusivities: there exist 0 < d ≤ d such that n n n n 2 ij ij d |∇ρ | ≤ ∇ρ D ∇ρ , ∇ρ D ∇ρ ≤ d |∇ρ ∇ρ |, (4c) k i i j j k k=1 i , j =1 i , j =1 k=1 and for the permittivity: there exist 0 < a ≤ a such that a|∇φ| ≤∇φ A∇φ, ∇φ A∇φ ≤ a|∇φ ∇φ|. (4d) Basedon(1)–(4) now we give a weak variational formulation of the generalized PNP problem by excluding the entropy variables μ and p with the help of (2c). Find a pair of functions ∞ 2 n 2 1 n ∞ 1 ρ ∈ L (0, T ; L (Ω)) ∩ L (0, T ; H (Ω)) ,φ ∈ L (0, T ; H (Ω)) (5a) that satisfy the non-negativity and the total mass balance (1f), the Dirichlet condition (3b), and the following variational equations for i = 1,..., n ∂ρ i 0 − ρ dxdt + ρ ρ (0) dx i i ∂t 0 Ω Ω ij + k θ ∇ρ + Υ (ρ)∇φ D ∇ρ dxdt = 0, (5b) B j j 0 Ω j =1 ∇φ A∇φ − z ρ φ dx = gφ dS (5c) k k x Ω Γ k=1 1 2 n 2 1 n for all test functions ρ = (ρ ,..., ρ ) ∈ H (0, T ; L (Ω)) ∩ L (0, T ; H (Ω)) 1 n and φ ∈ H (Ω) such that ρ(T ) = 0; ρ = 0 and φ = 0at Γ . The well-posedness to (5), (3b), and (1f) was investigated earlier in [18–20]. When solving problem (5), (3b), the key issue concerns fulﬁlling explicitly condi- tions (1f). In the following sections we consider the redundant entropy variable μ as independent one, thus allowing to include conditions (1f) implicitly in the problem formulation. Entropy method for generalized Poisson–Nernst–Planck… 609 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 ∞ 1 p ∈ L (0, T ; H (Ω)), u∇ p + z ρ ∇φ ∇ pdx =0(6a) k k k=1 for all test functions p ∈ H (Ω). The next step is to recover μ from (2b). Taking ∇μ, multiplying it with the gradient of suitable test functions, and integrating over the cylinder, the corresponding electro-chemical potentials 2 1 μ ∈ L (0, T ; H (Ω)), i = 1,..., n, (6b) can be looked for as solutions to the mutually independent equations ∇ρ u ∇μ − k θ − z ∇φ − ∇ p ∇μ dxdt = 0 (6c) i B i i ρ C 0 Ω 2 1 holding for all test functions μ ∈ L (0, T ; H (Ω)), and satisfying the Dirichlet boundary condition 0 0 u μ = k θ ln(β ρ ) + z φ + p at Γ . (6d) i B i i D i C In general, this elliptic problem is degenerate, because the operator of (6c)is unbounded due to the presence of factor . To remedy, we give the following two conditional assertions, which will be justiﬁed in Theorem 1 later on. Proposition 1 (Existence of entropy variable μ) 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 (possibly small) and a constant ρ > 0 such that ρ (t , x) ≥ ρ for all (t , x) ∈ (0, T ) × Ω, (6e) i 0 ∇ρ 2 n (ii) the inclusion ∈ L ((0, T ) × Ω) holds, (6f) then the problem (6b)–(6d) for this index i ∈{1,..., n} is uniquely solvable within this time (0, T ). 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. 2 1 Remark 1 Since by the very deﬁnition of a solution, ρ ∈ L (0, T ; H (Ω)),see (5a), i 0 condition (i) in Proposition 1 implies condition (ii). 610 J. R. G. Granada, V. A. Kovtunenko From (2b) we get 1 1 u ρ = exp (μ − z φ − p) , (7a) i i i β k θ C i B and summing up these equations over i = 1,..., n, due to the total mass balance ρ = C we can express the pressure as i =1 k θ 1 1 1 p = C ln exp (μ − z φ) . (7b) i i u C β k θ i B i =1 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 exp (μ −z φ) i i k θ β B ρ = P (μ − zφ) := C , i = 1,..., n. (7c) i i n 1 1 exp (μ −z φ) k k β k θ k B k=1 The normalized probabilities obey the inherent behavior P (μ − zφ) = C , P (μ − zφ) ≥ 0, i = 1,..., n. (7d) i i i =1 Then the non-negativity and the total mass balance (1f) follow straightforwardly from the properties (7d). By this we observe that μ − z φ →−∞ would lead to the i i limit ρ = 0in (7c). This is an admissible behavior, as an example, for the function (μ − z φ)(x) =− ln | ln |x|| as x → 0, which agrees with the H -spatial regularity i i of μ and φ stated in (5a) and (6b). In spite of the fact that μ + K is deﬁned up to an additive constant K , the concentrations ρ = P (μ + K − zφ) in (7c) are deﬁned i i uniquely. Basedon(7), we reformulate the generalized PNP problem as follows. Find a triple of functions ∞ n ∞ 1 2 1 n ρ ∈ L ((0, T ) × Ω) ,φ ∈ L (0, T ; H (Ω)), μ ∈ L (0, T ; H (Ω)) (8a) that satisfy the Fermi–Dirac statistics (7c), the Dirichlet condition (3b), and the fol- lowing variational equations for i = 1,..., n ∂ρ 0 − ρ dxdt + ρ ρ (0) dx i i ∂t 0 Ω Ω ij + ρ ∇μ D ∇ρ dxdt = 0, (8b) j i 0 Ω j =1 ∇φ A∇φ − z ρ φ dx = gφ dS (8c) k k x Ω Γ k=1 Entropy method for generalized Poisson–Nernst–Planck… 611 1 2 n 2 1 n for all test functions ρ = (ρ ,..., ρ ) ∈ H (0, T ; L (Ω)) ∩ L (0, T ; H (Ω)) 1 n and φ ∈ H (Ω) such that ρ(T ) = 0; ρ = 0 and φ = 0at Γ . The nonlinear parabolic equations (8b) imply a degenerate system because its operator loses the ellipticity property compared to (5b). Firstly, because of the cross- ij diffusion structure induced by matrices D . Second, due to the presence of the factor ρ , 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 (ρ,φ) for the generalized PNP problem in the form of (5), (3b) that ∂ρ satisﬁes the total mass balance in (1f). From (5b)using (2c) the time derivative can ∂t 2 −1 n be deﬁned in L (0, T ; H (Ω)) as a continuous linear functional T T ∂ρ ij , ρ dt =− ρ ∇μ D ∇ρ dxdt , i = 1,..., n, (8b’) i i ∂t 0 0 Ω j =1 −1 1 where H (Ω) stands for the dual space to functions ρ ∈ H (Ω) such that ρ = 0on i i Γ , with the duality pairing · , · between them. In fact, ρ is a continuous function D Ω of time, see [3, Remark 1, p. 509]. Since the strongly positive initial distribution ρ > 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 ρ ≥ ρ > 0, the electro- chemical potential μ in (2b) obeying the regularity (6b) exists, and the triple (ρ,φ, μ) 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 ) (with T > 0 maybe small), where the cross-diffusion entropy problem 0 0 given by (8), (7c), and (3b) has a solution (ρ,φ, μ).Bythis, (ρ,φ) solve the problem (5), (3b), and μ is from (2b). For such T , the local a-priori estimates hold for all (t , x) ∈ (0, T ) × Ω 0 0 μ μ 0 <ρ ≤ ρ < C , ∇μ ≤ K (T ) + K (T ), (8d) i i 0 0 L ((0,T )×Ω) 1 2 0 ρ μ μ for i = 1 ..., n, with constant K (T ), K (T ) ≥ 0. For arbitrary ﬁnal time T > 0, 0 0 1 2 the global a-priori estimates hold with constant K > 0 0 ≤ ρ ≤ C , i = 1 ..., n, ∇φ ≤ K . (8e) i φ L (Ω) Proof Here we justify the a-priori estimates. In (8e), the bounds of ρ follow from (7c) and (7d), and the bound of φ is proved in Appendix A, Lemma 1.In (8d), the strict bounds of ρ are the consequence of continuity due to the strongly positive initial distribution (3d), and below we prove the estimate of μ. The lower bound of ρ in (8d) allows us to apply Proposition 1. Based on this fact, we substitute p from (6a) and μ = μ into (6c) to obtain i i ∇ρ Υ (ρ) i i ∇μ − k θ − ∇φ ∇μ dxdt = 0, i B i ρ ρ i i 0 Ω 612 J. R. G. Granada, V. A. Kovtunenko where the functions Υ are deﬁned by (2c). With the help of the Cauchy–Schwarz inequality, from the above equality we get k θ ∇μ 2 ≤ ∇ρ 2 + (|z |+ Z )∇φ 2 , i i i L ((0,T )×Ω) L ((0,T )×Ω) L ((0,T )×Ω) 0 ρ 0 0 where the upper bound of Υ (ρ) is from (2d). Applying the estimate of ∇φ from (8e) and the bound of ∇ρ proved in [18–20] for arbitrary T > 0 ρ ρ ∇ρ 2 ≤ K (T ) + K (T )∇φ 2 , i = 1 ..., n, (8f) L ((0,T )×Ω) 1 2 L ((0,T )×Ω) ρ ρ with K (T ), K (T)> 0, we obtain the estimate for ∇μ in (8d) with the bounds 1 1 μ ρ ρ μ K (T ) = k θ K (T )+ K (T )T K and K (T ) = (|z |+ Z )T K , thus ﬁnishing B 0 0 0 φ i 0 φ 1 1 2 2 the proof. For global in time solvability, we require a stronger than (4a) assumption ij D = δ D, i , j = 1,..., n, (9a) ij with the Kronecker δ = 1for i = j, and zero otherwise. The assumption (9a) ij implies (4a) and imposes decoupling in (8b)aswellas(5b). In this case, existence of a solution (ρ,φ) to (5), (3b) satisfying both conditions in (1f) globally in time is proved in [18,19]. Therefore, the existence of the regular entropy variable μ 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 arbi- trary ﬁnal time T > 0. Let (ρ,φ) be a solution of problem (5), (3b) under the decoupling assumption (9a). If the electro-chemical potential μ solving problem (6b)– (6d) exists, then the triple (ρ,φ, μ) 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. 0 1 ρ = , i = 1 ..., n, at Γ , (9b) i β then the estimate with a constant K (T)> 0 holds (see Appendix A, Lemma 2) ρ ∇μ 2 ≤ K (T ), i = 1 ..., n. (9c) i i μ L ((0,T )×Ω) In the next section we investigate an equilibrium state when T ∞. 4 Equilibrium state ∞ 2 Let lim g(T , x) =: g (x) ∈ L (Γ ) in (3c), and the limit in (3b) T →∞ N 0 0 (0,∞) (0,∞) lim (ρ ,φ )(T , x) = (ρ ,φ ) (10a) T →∞ Entropy method for generalized Poisson–Nernst–Planck… 613 be constant independent of x and satisfying according to (3d) the properties (0,∞) (0,∞) ρ = C,ρ ≥ 0, i = 1,..., n. (10b) i i i =1 We consider a stationary counterpart of the entropy system (8), (7c), and (3b) under the decoupling assumption (9a). Find a triple of functions ∞ ∞ n ∞ 1 ∞ 1 n ρ (x) ∈ L (Ω) ,φ (x) ∈ H (Ω), μ (x) ∈ H (Ω) (11a) that satisfy the Fermi–Dirac statistics ∞ ∞ ∞ ρ = P (μ − zφ ), i = 1,..., n, (11b) the Dirichlet condition ∞ (0,∞) ∞ (0,∞) ρ = ρ ,φ = φ at Γ , (11c) and the following variational equations for i = 1,..., n ∞ ∞ ρ (∇μ ) D∇μ dx = 0, (11d) i i ∞ ∞ ∞ (∇φ ) A∇φ − z ρ φ dx = g φ dS (11e) k x Ω Γ k=1 1 n 1 for all μ ∈ H (Ω) and φ ∈ H (Ω) such that μ = 0 and φ = 0at Γ . Theorem 3 (Existence of equilibrium) A solution of problem (11) exists, which is given by the following relations i ∞ exp − φ k θ ∞ ∞ ∞ B μ = 0,ρ = P (−zφ ) = C , i = 1,..., n, (12a) k ∞ exp − φ β k θ k B k=1 ∞ 1 with the unique solution φ (x) ∈ H (Ω) satisfying the Dirichlet condition ∞ (0,∞) φ = φ at Γ (12b) and the quasilinear variational equation for the electrostatic potential ∞ ∞ ∞ (∇φ ) A∇φ − z P (−zφ )φ dx = g φ dS (12c) k k x Ω Γ k=1 for all test functions φ ∈ H (Ω) such that φ = 0 at Γ . D 614 J. R. G. Granada, V. A. Kovtunenko Proof We transform (11b) to the form (2b) ∞ ∞ ∞ u ∞ μ = k θ ln(β ρ ) + z φ + p . B i i i i C Consider the function (0,∞) (0,∞) (0,∞) u ∞ μ := k θ ln(β ρ ) + z φ + p B i i i i C deﬁned in Ω. Then it holds for i = 1,..., n (0,∞) ∞ i ∞ (0,∞) μ − μ = k θ ln + z (φ − φ ), (13a) B i (0,∞) i i (0,∞) (0,∞) and due to the assumption of constant (ρ ,φ ) in (10a) (0,∞) u ∞ ∇μ = ∇ p , i = 1,..., n. (13b) (0,∞) By virtue of the Dirichlet conditions (11c)wehave μ − μ = 0at Γ and hence i i can substitute (13a) for the test function μ into (11d). Summing over i = 1,..., n and ∞ 2 ∞ ∞ taking into account that d|∇μ | ≤ (∇μ ) D∇μ with some d > 0 by ellipticity i i i of D in (9a), we obtain n n ∞ ∞ 2 ∞ ∞ ∞ 0 ≤ d ρ |∇μ | dx ≤ ρ (∇μ ) D∇μ dx i i i i i Ω Ω i =1 i =1 n n (0,∞) ∞ ∞ u ∞ ∞ ∞ = ρ (∇μ ) D∇μ dx = ρ (∇μ ) D∇ p dx = 0. i i i C i i Ω Ω i =1 i =1 Here we used the equality (13b) and the ﬂux conservation (1c) implying ρ i =1 i ∞ ∞ (∇μ ) D = 0. This identity is sufﬁced by μ = 0, which is also necessary when ∞ ∞ ∞ ρ > 0. Henceforth, ρ = P (−zφ ) according to (11b), thus implying (12a), and (11e) turns into (12c). The solvability of problem (12b), (12c) is proved in Appendix A, Lemma 1. Acknowledgements Open access funding provided by Austrian Science Fund (FWF). V.A.K. thanks the Colombian Institute for Education and Technical Studies Abroad (ICETEX) and Universidad Tecnológica de Pereira (UTP) for supporting his visit of UTP and the Conference ICAMI 2017 at San Andrés Island, Colombia; he is also thankful to O.O. Vasilieva. Compliance with ethical standards Conﬂict of interest The authors declare that they have no conﬂict of interest. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Entropy method for generalized Poisson–Nernst–Planck… 615 A Appendix Here we present auxiliary lemmas proving global a-priori estimates. The proof utilizes the standard boundary trace theorem and Poincare inequality as follows 2 2 2 2 2 φ ≤ K φ , φ =φ +∇φ , (14a) 2 tr 1 1 2 2 L (Γ ) H (Ω) H (Ω) L (Ω) L (Ω) 2 2 φ ≤ K ∇φ for φ = 0at Γ . (14b) 2 P 2 D L (Ω) L (Ω) In Theorem 1,(8e) and in Theorem 3 there was used the ﬁrst lemma. Lemma 1 (Global well-posedness of electrostatic potential φ) For given measur- ∞ n ∞ 2 0 able function μ ∈ L ((0, T ); M(Ω)) , data g ∈ L (0, T ; L (Γ )) and φ ∈ ∞ 1 ∞ 1 L (0, T ; H (Ω)), a solution φ ∈ L (0, T ; H (Ω)) satisfying the Dirichlet condi- tion φ = φ on Γ in (3b) and the quasilinear parabolic equation in the variational form (inserting (7c) into (8c)) ∇φ A∇φ − z P (μ − zφ)φ dx = gφ dS (15a) k k x Ω Γ k=1 for all φ ∈ H (Ω) such that φ = 0at Γ , exists and satisﬁes the a-priori estimate ∇φ 2 ≤ K , (15b) L (Ω) with the positive constant 2 4K 1 2 4(1+K ) 1 2 2a 0 2 P P K = + |Ω|(CZ ) + K + g + 2 φ , 2 2 tr 2 2 2 1 a a L (Γ ) a H (Ω) a a N (15c) where the bounds are from (4d), (7d), (14), and recalling Z = |z |. k=1 Proof A solution to the quasilinear elliptic problem (15a) exists by the Schauder– Tikhonov ﬁxed point theorem. Indeed, the principal term in (15a) is linear, coercive and bounded due to (4d). The nonlinear term is uniformly bounded according to (7d) ∂ P (μ) C and Lipschitz continuous, because ≤ for i , k = 1,..., n, from which it ∂μ k θ i B C n follows that |∇ P (μ)|≤ for μ ∈ R and μ k k θ C n 1 2 1 2 z P (μ − zφ ) − P (μ − zφ ) ≤ |z(φ − φ )|Z . (15d) k k k k θ k=1 616 J. R. G. Granada, V. A. Kovtunenko Testing (15a) with φ = φ − φ provided by zero on Γ , due to (4d) we evaluate a∇φ ≤ ∇φ A∇φ dx L (Ω) 0 0 = ∇φ A∇φ + z P (μ − zφ)(φ − φ ) dx k k k=1 + g(φ − φ ) dS . Applying Young’s inequality to the following terms 0 2 a 0 2 ∇φ A∇φ dx ≤ ∇φ + ∇φ , 2 2 4 a L (Ω) L (Ω) z P (μ − zφ)(φ − φ ) dx k k k=1 a K a 2 a 0 2 2 ≤ φ + φ + + |Ω|(CZ ) , 2 2 4K a a P L (Ω) L (Ω) 4a 0 2 a 0 2 g(φ − φ ) dS ≤ φ + φ x 2 2 4K (1+K ) aK tr P L (Γ ) tr L (Γ ) N N aK K (1+K ) tr 2 tr P + + g , 2 2 a L (Γ ) 4a N where (7d) and Z = |z | were used, after summation of these inequalities, with k=1 the help of (14) we arrive at the estimate (15b) and the bound (15c). In Theorem 2, the estimate (9b) follows from the next lemma. Lemma 2 (Global a-priori estimate for electro-chemical potential μ) For the quadru- ∞ 1 ∞ 1 ple (ρ,φ, μ, p),let φ ∈ L (0, T ; H (Ω)),p ∈ L (0, T ; H (Ω)) solve (6a), ∂ρ ∞ n 2 −1 n ρ ∈ L ((0, T ) × Ω) within the Gibbs simplex (1f), and ∈ L (0, T ; H (Ω))) . ∂t 2 1 n If μ ∈ L (0, T ; H (Ω)) satisﬁes the thermodynamic relations (2b), the initial con- dition (3a), the boundary conditions in (3b) with the constant Dirichlet data 0 1 ρ = , i = 1 ..., n, at Γ (16a) i β subject to conditions (3d), and the nonlinear parabolic equations (8b’) in the decou- pled by (9a) form T T ∂ρ , ρ dt + ρ ∇μ D∇ρ dxdt = 0, i = 1,..., n, (16b) i i i ∂t 0 0 Ω 2 1 n for all ρ ∈ L (0, T ; H (Ω)) such that ρ = 0 at Γ , then the a-priori estimate holds 2 d 2 2 2 2 ρ ∇μ ≤ C 2 |z| + nZ ∇φ + θ S . 2 n 2 L ((0,T )×Ω) d L ((0,T )×Ω) d t =0 (16c) Entropy method for generalized Poisson–Nernst–Planck… 617 Here the bounds are from (2d), (4c),Z = |z |, and the function of entropy k=1 ∂E R → R, S := − =− k ρ ln(β ρ ) − 1 dx (16d) + B i i i ∂θ i =1 is uniformly bounded as follows k |Ω|C min 0, 1 − ln(β C ) ≤ S ≤ k |Ω|C . (16e) B i B i =1 Proof We start with deﬁning the entropy function in (16d) according to (2a). Since the inequality −x (ln x +1) ≤ 1 holds for all x ≥ 0, we have −ρ (ln(β ρ ) −1) ≤ .The i i i upper bound for S in (16e) follows by summing the latter inequality over i = 1,..., n, multiplying the result by k , integrating over Ω, and using = C due to k=1 β (16a) and the total mass balance (3d). While the lower bound of S in (16e) holds for ρ ∈ (0, C ). The time derivative of entropy can be calculated with the help of thermodynamic relations (2b) in the form n n ∂ S ∂ρ ∂ρ u i i θ =− , k θ ln(β ρ ) =− ,μ − z φ − p . (16f) B i i i i ∂t ∂t ∂t C Ω Ω i =1 i =1 Next we remark that the function μ −z φ − p = k θ ln(β ρ ) is zero on Γ because i i B i i D of the assumption (16a) and the Dirichlet condition (3b). Thus it can be inserted for a test function into (16b) and we get the following relations d ρ ∇μ ≤ ρ ∇μ D∇μ dxdt i i 2 i i L ((0,T )×Ω) 0 Ω T T u ∂ρ u = ρ ∇μ D z ∇φ + ∇ p dxdt − ,μ − z φ − p dt i i i i C ∂t C 0 Ω 0 (16g) thanks to the lower bound in (4c) and (9a). From (2b) and (2c) we easily get u Υ (ρ) z ∇φ + ∇ p =∇(μ − k θ ln(β ρ )) = ∇φ. (16h) i i B i i C ρ Combining this equality with the upper bound in (4c) and (9a), and taking into account Υ (ρ) the inequality ρ ≤ C and estimate (2d)for , we can estimate the former integral in the right-hand side of (16g) from above using Young’s inequality as follows: √ √ Υ (ρ) ( ρ ∇μ ) D ρ ∇φ dxdt i i i 0 Ω 2 d 2 2 ≤ ρ ∇μ + C (|z |+ Z ) ∇φ . (16i) i i i 2 n 2 2 2d L ((0,T )×Ω) L ((0,T )×Ω) 618 J. R. G. Granada, V. A. Kovtunenko After summation over i = 1,..., n, the latter integral in the right-hand side of (16g) ∂ S agrees with θ dt due to (16f). 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Published: Nov 2, 2018
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