Long time behavior of solutions to Nernst-Planck and Debye-Hückel drift-diffusion systems

We study the convergence rates of solutions to drift-diiusion systems (arising from plasma, semiconductors and electrolytes theories) to their self-similar or steady states. This analysis involves entropy-type Lyapunov functionals and logarithmic Sobolev inequalities.


Introduction
We consider the long time asymptotics of solutions to drift-diffusion systems u t = ∇ · (∇u + u∇φ) , where E d is the fundamental solution of the Laplacian in IR d , the system (1.1)-(1.3) was also studied by P. Debye and E. Hückel in the 1920's. (1.6) signifies a conducting boundary of the container, while in the case of a bounded domain the "free" boundary condition (1.7) corresponds to a container immersed in a medium with the same dielectric constant as the solute. These equations, together with their generalizations including e.g. an exterior potential, known as drift-diffusion Poisson systems, also appear in plasma physics and (supplemented with some mixed linear boundary conditions instead of (1.4)-(1.5)) in semiconductor device modelling.
To determine completely the evolution, the initial conditions ≥ 0, as well as the total charges Here M u , M v are not necessarily the same, i.e. the electroneutrality condition is not, in general, required. Condition (1.10) must be satisfied in the case of the homogeneous Neumann boundary conditions ∂φ ∂ν = 0 (i.e. an isolated wall of the container) leading together with (1.4)-(1.5) to Our results (Theorem 1.2 below) are valid in that case, with even a simpler proof.
The asymptotic properties of solutions to (1.1)-(1.3), (1.7) have been studied recently in [1]. The authors proved that (for d ≥ 3, M u = M v = 1 and u 0 and v 0 regular enough) u, v tend to their self-similar asymptotic states at an algebraic rate. We improve these results by relaxing assumptions on the initial data and showing a stronger (still algebraic) decay rate, which we expect to be optimal (see Theorem 1.1 below).
In the case of a bounded domain, the convergence (with no specific speed) in the L 1 -norm of u and v solving (1.1)-(1.5) to their corresponding steady states has been proved in [5] (as well as the L ∞ -convergence for more regular u 0 , v 0 ). Here we prove the exponential convergence towards the steady states with a decay rate depending on Ω ⊂ IR d , d ≥ 2, and the initial value of the entropy functional only (see Theorem 1.2 below).
Notation. The L p -norm in IR d or Ω ⊂⊂ IR d is denoted by | · | p , and inessential constants (which may vary from line to line) are denoted generically by C.
Define the asymptotic states in IR d by where the charges of the solution u,v of (1.1)-(1.2) are given by (1.9), and the entropy functional by and where In the case of a bounded domain, define the entropy functional for the solution u,v,φ of (1.1)-(1.5), (1.6) or (1.7), (1.8) and the unique steady state U,V,Φ of the Debye-Hückel system with Note that for the condition (1.6) the fifth and the sixth terms in W (t) take the form 1

18)
and We begin with a rescaling of the system (1.1)-(1.3) which will lead to a system with a quadratic confinement potential, and therefore (eliminating the dispersion) to the expected exponential convergence to the steady states. This idea was applied in [8] and [7], as well as in [1], to a variety of problems ranging from kinetic equations to porous media equations. Letx ∈ IR d , τ > 0, be the new variables defined bȳ and consider the rescaled functionsū,v,φ such that This whole section will deal with the rescaled system, so omitting the bars over x, u, v, φ will not lead to confusions with the original system, which now takes, after rescaling, the form The scaling (2.2) preserves the L 1 -norms, so the rescaled initial data u 0 , v 0 still satisfy Of course, going back to the original variables x, t, u ∞ ,v ∞ corresponds to the asymptotic state u as ,v as defined by (1.11)-(1.12). Writing φ = βψ with β = β(τ ) = e −τ (d−2) → 0 as τ → +∞, we introduce the relative entropy corresponding to the original entropy functional L in (1.13). The evolution of W is given by 10) with U, V denoting the local Maxwellians so that ∇U /U = −(x + ∇φ), ∇V /V = −(x − ∇φ). Using the notation (2.10) can be rewritten as The quantity J in (2.13) can be estimated from below using the Gross logarithmic Sobolev inequality valid for each a > 0, see e.g. [11] or a thorough discussion of different versions of logarithmic Sobolev inequalities in [2]. (2.15) becomes an equality if and only if f (x) = C exp(−|x| 2 /(2a)) (up to a translation). Taking a = 1 in (2.15), the relation (2.14) leads to (d−2)/2 Σ d/2 , because by the Hardy-Littlewood-Sobolev inequality and an interpolation Clearly, (2.16) implies and, after one integration, we obtain Since from the Csiszár-Kullback inequality (cf. (1.9) in [2], App. D in [7], [6] or [10]) W (τ ) controls the L 1 -norm of u − u ∞ and v − v ∞ , we get the same decay rates as in (2.17)-(2.19) for (2.20) Returning to the original variables x, t, this implies, of course, the estimates (1.14)-(1.15) of Theorem 1.1 in the general case.
In the electroneutrality case (1.10): and L(0) only, and is independent of e.g. |u 0 | r , |v 0 | r with some r > d/2 -as it was in fact in [1]. Conditions like |u 0 | r + |v 0 | r < ∞ are sufficient for (local in time) existence of solutions to the considered systems (cf. Theorem 2 in [5]), but they can be relaxed -as it was done for a related parabolic-elliptic system describing the gravitational interaction of particles in [4]. Thus, compared to [1], Theorem 1.1 gives not only an improvement of the exponents but also gets rid of the unnecessary dependence on quantities other than L(0), M u , M v . We do not know if the exponents in Theorem 1.1 are optimal, but such a conjecture is supported by the calculations in the proof of the following and hence for each solution u,v to the Nernst-Planck system.

Remark 2.3
The interest of this proposition is that the constants controlling the convergence of W (t), L(t), and hence |u − u as | 1 , |v − v as | 1 in (1.15), depend on the initial values of W (0), L(0) only (and not on |u| 1 = M u , |v| 1 = M v , which are quantities not comparable with, say, ulog u dx, v log v dx in the whole IR d space case). However, the exponent λ -which is evaluated explicitly -is not as good as the one in Theorem 1.1.
Proof of Proposition 2.2. Using (2.9), (2.13), (2.14), we may write for any positive λ Observe that if we define Define now By the Cauchy-Schwarz inequality we have and thus

Using (2.22) we get
with X = E/F ≥ 0. For either d ≥ 4 and λ ≤ 2, or d = 3 and λ ≤ 1, we have µ ≤ 2. The right hand side of (2.23) (positive for X ≥ µ/2) equals (for X ≤ µ/2 ≤ 1) guarantees dW dτ + λW ≤ 0, which implies (2.21). The condition (2.24) is equivalent to λ ≤ λ(d). In particular, λ(d) is an increasing function of d, In the case of one species of particles, i.e. v ≡ 0 as was in [3] and [4], the result of Proposition 2.2 still holds. Finally, we remark that there is, in general, no hope to have λ > 2 in nontrivial cases. This can be inferred from the formula (2.22), where for each χ > 1, J − χ ulog u u∞ dx + v log v v∞ dx could be negative and dominate the other terms (for instance, in the limit M u , M v → 0 + ).

Proof of Theorem 1.2
First, we recall that steady states U,V,Φ of (1.1)-(1.3) satisfy the relations Together with (1.3) this leads to the Poisson-Boltzmann equation This equation, supplemented with the Dirichlet boundary condition (1.6) or the free condition (1.7), for every M u ,M v ≥ 0, has a unique (weak) solution Φ, see [9] or Proposition 2 in [5] (and this solution is classical whenever ∂Ω is of class C 1+ǫ for some ǫ > 0). The evolution of the Lyapunov functional defined by (1.16) in the case of the Dirichlet boundary condition (1.6) or in the case (1.7) is given by cf. (35) in [5], where the above relation is obtained for weak solutions to the Debye-Hückel system. Concerning the global in time existence of solutions to the Debye-Hückel system with nonlinear boundary conditions (1.4)-(1.5), we note that this was proved for d = 2 only in Theorem 3 of [5]. Thus, in higher dimensions d ≥ 3, we assume that u(t),v(t) exists for all t ≥ 0. If equations (1.1)-(1.3) are supplemented with linear type boundary conditions (as it is the case in semiconductor modelling), the assumption u 0 , v 0 ∈ L r (Ω) with an exponent r > d/2 (cf. Theorem 2 (ii) in [5] and [1] for the case of the whole space IR d ) guarantees the existence of u(t),v(t) for all t ≥ 0.
First, we represent the entropy production terms in (3.3) as where W = ulogu dx + v logv dx + 1 2 is as in (1.16), and is a strictly convex functional reaching its minimum at Φ. Now it is clear from (3.3), (3.5)-(3.6) and (3.7) that for some λ = λ(Ω) > 0 dW dt + λW ≤ 0, i.e. W (t) decays exponentially in t W (t) ≤ W (0) e −λt . (3.8) By the Csiszár-Kullback inequality (as was in Section 2), W (t) controls the L 1 -convergence to the unique steady state, so the conclusion (1.19) of Theorem 1.2 follows from (3.8). This improves (34) in Theorem 6 of [5] in two ways. First, there is an exponential decay rate. Second, (34) is proved under the assumption W (0) < ∞, which is much weaker than the assumption on the L 2 -boundedness in time of the solution u,v in Theorem 6 of [5]. Evidently, this result is also valid for one species case (M u or M v equal to 0), so Theorem 2 in [3] is also improved. 2