Xingyu Li

ASYMPTOTIC BEHAVIOR OF NERNST-PLANCK EQUATION

Keywords: Nernst-Planck equation, large time asymptotic, free energy, Fisher information, spectral gap. AMS subject classifications: 35B40, 35P15, 35Q70

published or not. The documents may come

Asymptotic behavior of Nernst-Planck equation

INTRODUCTION

At the end of nineteenth century, Nernst and Planck introduced a system of equations for representing the evolution of charged particles subject to electrostatic forces. The original model is exposed in [START_REF] Nernst | Elektromotorische Wirksamkeit der Jonen[END_REF][START_REF] Planck | Ueber die erregung von electricität und wärme in electrolyten[END_REF]: electrically charged particles diffuse under the action of a drift caused by an electrostatic potential. Nowadays we use this system in various frameworks like, for instance, phenomenological models for electrolytic behaviour in membranes. The original model is the non-confined Nernst-Planck system. If we take into account a mean-field Poisson coupling, in dimension d = 2, the system takes the form

   ∂u ∂t = ∆u + ∇ • (u ∇v) v = G 2 * u u(0, x) = n 0 ≥ 0 x ∈ R 2 , t > 0 , (1)
where G 2 (x) = -1 2 π log |x| denotes the Green function of the Laplacian in R 2 . We shall call this model the Poisson-Nernst-Planck system, which was also considered by Debye and Hückel in [START_REF] Debye | De la theorie des electrolytes. I. abaissement du point de congelation et phenomenes associes[END_REF] and is sometimes called the Debye-Hückel system in the literature. Up to a sign change in the meanfield term, the model is similar to the Keller-Segel model, which is going to be a source of inspiration (see [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF][START_REF] Blanchet | Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model[END_REF][START_REF] Campos | Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane[END_REF] for more details) for the study of the large time behaviour and this is a reason why we consider the twodimensional case of the model. Now let us introduce the notion of confinement. In the whole space, particles repel themselves and a well-known runaway phenomenon occurs: solutions locally vanish while the mass escapes at infinity. This can be prevented using a container (a bounded domain, with convenient boundary conditions) with walls, or a confinement potential. Actually, it is possible to obtain the bounded domain case as a limit of a whole space case with an external potential of confinement taking larger and larger values outside of the domain. Here we shall consider the Poisson-Nernst-Planck system with confinement in R d , where the dimension is d = 2 or d = 2. The density function n solves

   ∂n ∂t = ∆n + ∇ • (n ∇c) + ∇ • (n ∇φ) c = G d * n n(0, x) = n 0 ≥ 0, R d n(0, x) d x = M > 0 x ∈ R d , t > 0 . ( 2
)
The convolution kernel G d is the Green function of the Laplacian in R d , namely

G 2 (x) = - 1 2 π log |x| for any x ∈ R 2 and G 3 (x) = 1 4 π |x| for any x ∈ R 3 .
In other words, we ask that c solves the Poisson equation

-∆c = n x ∈ R d ,
while φ is a given external potential. In the special case of d = 2 and φ(x) = µ 2 |x| 2 for some µ > 0, if we use the change of variables

u(t , x) = R -d n(τ, ξ) , v(t , x) = c(τ, ξ) , ξ = x R , τ = log R , R = R(t ) := 1 + 2 µ t , (3)
then we observe that (n, c) solves (2) if and only if (u, v) solves [START_REF] Addala | Hypocoercivity and large time asymptotics of the linearzied Vlasov-Poisson-Fokker-Planck system[END_REF]. Studying the convergence rates of the solutions of (2) amounts to study the intermediate asymptotics of the solutions of (1) when runaway occurs.

Obviously, the mass of a solution of ( 2) is conserved, and we shall write that R d n(t , x) d x = M for any t ≥ 0. The mass of a solution of (1) is also conserved, but one can prove that, for a solution of (1), the mass contained in any given compact set in R 2 decays to zero.

From here on, we shall assume that M > 0 is fixed. Now let us turn our attention to the conditions on the confinement potential. From now on, we shall assume that φ

∈ W 1,∞ loc (R d ) is such that ∇φ ∈ W 1,∞ (R d ) and lim inf |x|→+∞ φ(x) log |x| > d , ( C1
)
and also that the bounded measure e -φ d x admits a spectral gap (or Poincaré) inequality, i.e., that there exists a positive constant Λ φ such that

R d |∇u| 2 e -φ d x ≥ Λ φ R d |u| 2 e -φ d x ∀ u ∈ H 1 (R d ; e -φ d x) such that R d u e -φ d x = 0 . (C2)
Based on Persson's lemma, a sufficient condition is obtained by requesting that

σ φ := lim r →+∞ infess x∈B c r 1 4 |∇φ| 2 - 1 2 ∆ x φ > 0 and lim r →+∞ infess x∈B c r |∇φ| > 0 .
(C3) Let us refer to [START_REF] Addala | Hypocoercivity and large time asymptotics of the linearzied Vlasov-Poisson-Fokker-Planck system[END_REF] for details and further references. We learn from [START_REF] Arnold | On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations[END_REF][START_REF] Biler | Long time behaviour of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems[END_REF] that the stationary solutions (n ∞ , c ∞ ) of ( 2) are obtained as solutions of the Poisson-Boltzmann equation

-∆c ∞ = n ∞ = M e -c ∞ -φ R d e -c ∞ -φ d x . (4)
Under Assumption (C1) and the additional condition

lim inf |x|→+∞ φ(x) log |x| > 4 + M 2 π if d = 2 , ( C4
)
we know (see [START_REF] Addala | Hypocoercivity and large time asymptotics of the linearzied Vlasov-Poisson-Fokker-Planck system[END_REF]Lemma 5] and earlier references therein) that the unique solution of ( 4) is obtained as a minimizer of the free energy F defined by

F [n] := R d n log n d x + R d n φ d x + 1 2 R d n (-∆) -1 n d x . ( 5
)
Further details are given in Section 2. A simple consequence of the minimization procedure is that

F [n] -F [n ∞ ] ≥ 0 ∀ n ∈ L 1 + (R d )
with the convention that F [n] can take the value +∞ if, for instance n log n is not integrable. For sake of brevity, we shall say that φ is a confinement potential satisfying Assumption (C) if (C1), (C3) and (C4) hold.

Our goal is to study the asymptotic behaviour of a solution of (2) with initial datum n 0 at t = 0 such that F [n 0 ] is finite. It is a standard observation that the free energy F [n(t , •)] of a solution of ( 2) is monotone non-increasing along the flows and obeys to

d d t F [n(t , •)] = -I [n(t , •)] (6)
where the Fisher information I is defined by

I [n] := R d n ∇(log n + c + φ) 2 d x .
Our main result is that, as t → +∞, F [n(t , •)] is bounded by I [n(t , •)] up to a multiplicative constant which shows that n(t , •) converges to n ∞ at an exponential rate. The precise result is not written in terms of the free energy but in terms of a weighted L 2 norm and goes as follows.

Theorem 1.1. Let d = 2 or 3 and consider a potential φ satisfying (C). Assume that n solves (2) with initial datum n(0,

•) = n 0 ∈ L 2 + (n -1 ∞ d x), R d n 0 d x = M , and F [n 0 ] < ∞.
Then there exist two positive constants C and Λ such that

R d n(t , .) -n ∞ 2 n -1 ∞ d x ≤ C e -Λ t ∀ t ≥ 0 .
In section 4, we will characterize Λ as the spectral gap of the linearized operator associated with (2) and observe, as a special case, that Λ = 2 µ if d = 2 and φ = µ 2 |x| 2 , for some µ > 0. Beyond free energy and entropy methods, the study of the large time asymptotics of the Poisson-Nernst-Planck system involves various tools of nonlinear analysis. Proving an exponential rate of convergence is interesting for studies of Poisson-Nernst-Planck systems by methods of scientific computing. Specific methods are needed for the numerical computation of the solutions, see [START_REF] Barcilon | Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study[END_REF][START_REF] Park | Qualitative properties of steadystate Poisson-Nernst-Planck systems: Mathematical study[END_REF]. In [START_REF] Liu | A free energy satisfying finite difference method for Poisson-Nernst-Planck equations[END_REF], Liu and Wang implement at the level of the free energy a finite difference method to compute the numerical solution in a bounded domain. Concerning rates of convergence from a more theoretical point of view, let us mention that the existence of special solutions and self-similar solutions is considered in [START_REF] Biler | Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions[END_REF][START_REF] Biler | The Cauchy problem and self-similar solutions for a nonlinear parabolic equation[END_REF][START_REF] Herczak | Existence and asymptotics of solutions of the Debye-Nernst-Planck system in Rˆ2[END_REF]. We refer to [START_REF] Schönke | Unsteady analytical solutions to the Poisson-Nernst-Planck equations[END_REF] for a discussion of the evolution problem from the point of view of physics.

Variants of the Poisson-Nernst-Planck system with nonlinear diffusions have been considered, for which the sharp rate of convergence is still unknown. Some papers rely on the use of distances related to the L 2 -Wasserstein distance, see [START_REF] Di | Large time behaviour in Wasserstein spaces and relative entropy for bipolar driftdiffusion-Poisson models[END_REF][START_REF] Kinderlehrer | A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations[END_REF][START_REF] Zinsl | Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion[END_REF]. Exponential decay rates should be natural in view of the expected gradient flow structure of the system in this framework. The simpler case of linear diffusions on a bounded domain of R d with d ≥ 3 was studied in [START_REF] Biler | Long time behaviour of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems[END_REF]: the convergence to the stationary solution occurs at an exponential rate. As already mentioned, another related model is the Keller-Segel system in dimension 2. Regularity and asymptotic estimates for this system were discussed in [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF][START_REF] Campos | Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane[END_REF] and are a source of inspiration for the present study, in particular concerning the scalar product and the coercivity estimates. For completeness, let us mention that similar ideas have been recently developed in [START_REF] Li | Flocking: Phase transition and asymptotic behaviour[END_REF] for the study of a McKean-Vlasov model model of flocking, which also involves a non-local coupling.

This paper is organized as follows. In Section 2 we prove that the minimizer of the free energy F is the stationary solution (n ∞ , c ∞ ) and it attracts any solution of (2) as t → +∞. In Section 3, we show that the relative entropy and the relative Fisher information provide us with two quadratic forms which are related by the linearized evolution operator and prove the spectral gap property of this operator. And in Section 4, we give the proof of Theorem 1.1 and give some additional results.

MIMINIZERS OF THE FREE ENERGY AND CONVERGENCE TO THE STATIONARY SOLUTION

The main goal of this section is to prove that the minimizer of the free energy F is the stationary solution (n ∞ , c ∞ ) considered in the introduction and that it attracts any solution of (2) as t → +∞.

X := f ∈ L 1 + (R d ) : R d f (x) d x = M , f log f ∈ L 1 (R d ), f φ ∈ L 1 (R d ) ,
the free energy F is semi-bounded from below.

Proof. According to Assumptionn (C1), we know that e Proof. Standard minimization methods show that a minimizing sequence admits, up to the extraction of a subsequence, a limit which is a minimizer. From the proof above, F is lower bounded and satisfies the coercivity inequality. For a fixed minimizer n ∞ , it should satisfy the Euler-Lagrange equation

-φ ∈ L 1 (R d ). Set ρ(x) := λ e -φ , such that R d ρ(x) d x = M . Since the function x log x is con- vex, we obtain that R d f log f d x ≥ R d f log ρ d x by Jensen's inequality. So F [ f ] ≥ R d f log ρ d x + R d f φ d x + 1 2 R d f (-∆) -1 f d x = M log λ + 1 2 R d f (-∆) -1 f d x . If d = 3, R d f (-∆) -1 f d x ≥ 0 because the Green function G 3 (x) is non- negative. If d = 2,
log n ∞ + φ + c ∞ = λ , c ∞ = (-∆) -1 n ∞ ,
for some Lagrange multiplier λ associated with the mass constraint, which means that (n ∞ , c ∞ ) solve [START_REF] Barcilon | Qualitative properties of steady-state Poisson-Nernst-Planck systems: perturbation and simulation study[END_REF]. By direct computation, with c = (-∆) -1 n, we observe that

F [n] -F [n ∞ ] = R d n log n n ∞ d x + 1 2 R d (n -n ∞ ) (c -c ∞ ) d x . Since R d n d x = R d n ∞ d x = M , we obtain from Jensen's inequality that R d n log n n ∞ d x ≥ 0
and, according to [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF],

R d (n -n ∞ )(c -c ∞ ) d x = R d |∇(c -c ∞ )| 2 d x ≥ 0 . Hence F [n] -F [n ∞ ] ≥ 0 for any n ∈ X , with equality if and only if n = n ∞ .
This means that the minimizer of F is unique.

We may notice that n ∞ is radially symmetric if φ is radially symmetric, as a consequence of the uniqueness result of Lemma 2.2.

We learn from the proof of [9, Lemma 23] that

max |x|→∞ c ∞ + M 2 π log |x| < ∞ if d = 2, max |x|→∞ c ∞ - M 4 π |x| < ∞ if d = 3 ,
and deduce from (4) that, as |x| → ∞,

n ∞ ∼ |x| M 2 π e -φ if d = 2 , n ∞ ∼ e -M 4 π |x| -φ if d = 3 . ( 7
) Proposition 2.1. Let d = 2 or d = 3 and assume that the potential φ satis- fies (C). Then the solutions (n ∞ , c ∞ ) of (4) are such that c ∞ is bounded if d = 3 and ∇c ∞ L q (R 2 ) is bounded for any q ∈ (2, +∞] if d = 2.
Proof. From ( 7) and (C4), we know that n ∞ is bounded outside of a large centered ball of radius R > 0. Let us assume that |x| ≤ R and recall that

c ∞ (x) = κ 3 R 3 e -c ∞ (y)-φ(y) |x -y| d y if d = 3 , with κ 3 = M 4 π R 3 e -c ∞ -φ d x , |∇c ∞ (x)| ≤ κ 2 R 2 e -c ∞ (y)-φ(y) |x -y| d y if d = 2 , with κ 2 = M 2 π R 2 e -c ∞ -φ d x . In dimension d = 3, it is enough to observe that c ∞ = (-∆) -1 n ∞ ≤ 0 and deduce the bound 0 ≤ c ∞ (x) ≤ κ 3 R 3 e -φ(y) |x -y| d y .
In dimension d = 2, we deduce from ( 7) and (C) that for R > 0 large enough, there exists a constant κ > 0 such that

n ∞ (x) ≤ n ∞ (x) 1 |x|<R + κ 1 |x|≥R |x| -4 ,
which allows us to write

c ∞ (x) ≥ - M 2 π log(2 R) - κ 2 π |y|≥R log |x -
d d t R d n(t , x) p d x = -p R d |∇n| 2 n p-2 d x- R d n p+1 d x+ R d n p ∆φ d x ≤ - R d n p+1 d x +C R d n p d x . Using Hölder's inequality R d n d x 1 p R d n p+1 d x p-1 p ≥ R d n p d x, we ob- tain that R d n p+1 d x ≥ M -1 p-1 R d n p d x p p-1 With z(t , •) = R d n(t , •) p d x, the problem reduces to the differential in- equality 1 p -1 z ≤ -M -1 p-1 z p p-1 +C z using R d n 0,k d x ≤ M . It is elementary to prove that z(t ) ≤ (2C ) p-1 M ∀ t ≥ 4C
and conclude that the bound

n(t , •) L p (R d ) ≤ (2C ) p-1 p M 1 p
has a uniform upper bound in the limit as p → +∞. See [START_REF] Blanchet | Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model[END_REF] for further details on a similar estimate.

∈ L 1 + (R d ) such that F [n 0 ] < +∞, there exists a constant C > 0 and a time T > 0 such that ∇c(t , •) L q (R d ) ≤ C ∀ t ≥ T , ∀ q ∈ (2, +∞] . Proof. The method is inspired from [11, Section 3]. If h = (-∆) -1 ρ, then |∇h(x)| ≤ 1 |S d -1 | R d ρ(y) |x -y| d -1 d x
can be estimated by splitting the integral into two parts corresponding to |x -y| ≤ 1 and |x -y| > 1. By applying twice Hölder's inequality, we deduce from 1

|S d -1 | x-y|<1 ρ(y) |x -y| d -1 d y ≤ d d d +1 |S d -1 | -1 d +1 ρ L d +1 (R 3 ) 1 |S d -1 | x-y|≥1 ρ(y) |x -y| d -1 d y ≤ 1 |S d -1 | ρ L 1 (R d ) that ∇((-∆) -1 ρ) L ∞ (R d ) ≤ ρ L 1 (R d ) + d d d +1 |S d -1 | -1 d +1 ρ L d +1 (R d ) (8)
for any ρ ∈ L 1 ∩ L d +1 (R d ). Applying it with ρ = n(t , •) and c = (-∆) -1 n and using Minkowski's inequality ∇c(t ,

•) L q (R d ) ≤ ∇c(t , •)-∇c ∞ L q (R d ) + ∇c ∞ L q (R d ) , the result follows from the estimate ∇c(t , •) -∇c ∞ 2 L 2 (R d ) ≤ 2 F [n 0 ]
together with Proposition 2.1 and Lemma 2.3.

2.3.

Convergence to stationary solutions. The next step is to establish the convergence without rate of the solution of (2) to the stationary solution. For later purpose, let us recall the Aubin-Lions compactness lemma. A simple statement goes as follows (see [START_REF] Gogny | PL Sur les états d'équilibre pour les densités électroniques dans les plasmas[END_REF] for more details). Lemma 2.4. (Aubin-Lions Lemma) Take T > 0, p ∈ (1, ∞), and let ( f k ) k∈N be a bounded sequence of functions in L p (0, T ; H ), where H is a Banach space. If ( f k ) k∈N is bounded in L p (0, T ;V ), where V is compactly imbedded in H and if (∂ f k /∂t ) k∈N is bounded in L p (0, T ;V ) uniformly with respect to k ∈ N, where V is the dual space of V , then ( f k ) k∈N is relatively compact in L p (0, T ; H ). With this result in hand, we are in a position to prove the following result.

Proposition 2.2. Suppose that d = 2 or 3. Let n be the solution of (2) and assume that the potential φ satisfies (C). Then for any p ∈ [1, ∞) and any q ∈ [2, ∞), we have

lim t →∞ n(t , •) -n ∞ L p (R d ) = 0 and lim t →∞ ∇c(t , •) -∇c ∞ L q (R d ) = 0 .
Proof. Since F [n(t , .)] is nonnegative and decreasing, by [START_REF] Biler | The Cauchy problem and self-similar solutions for a nonlinear parabolic equation[END_REF] we know that lim

t →∞ ∞ t I [n(s, .)] d s = 0 . ( 9
)
This means that the sequence (n k , c k ) k∈N , defined by

n k (t , •) = n(t + k, •), c k = (-∆) -1 n k , is such that ∇n k +n k ∇c k +n k ∇φ strongly converges to 0 in L 2 (R + × R d )
. By lemma 2.4, this shows that (n k ) k∈N is relatively compact and converges, up to the extraction of a subsequence, to a limit n. Up to the extraction of an additional subsequence, (c k ) k∈N converges to c = (-∆) -1 n so that we may pass to the limit in the quadratic term and know that

∇ n + n ∇ c + n ∇φ = 0 , -∆ c = n .
Since mass is conserved by passing to the limit, we conclude that n = n ∞ and c = c ∞ . The limit is uniquely defined, so it is actually the whole family (n(t , •))

lim t →+∞ n(t , .) -n ∞ L ∞ (R d ) = 0 .
Proof. The main tool is the Duhamel formula: see [START_REF] Campos | Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane[END_REF] for more details. We have

n(t , x) = R 2 K (t , x, y) n 0 (y) d y - t 0 R 2 ∇K (t -s, x, y) • n(s, y) ∇c(s, y) d y d s
where K (t , x, y) is the Green function of the Fokker-Planck equation

∂n ∂t = ∆n + µ ∇(nx)
which is

K (t , x, y) := µ 2 π (1 -e -2t ) e - µ|x-e -t y| 2 2(1-e -2t )
and from the semi-group property we get that

n(t + 1, x) = R 2 K (t , x, y) n(t , y) d y - t +1 t R 2 ∇K (t + 1 -s, x, y) • n(s, y) ∇c(s, y) d y d s . ( 10
)
Notice that the stationary solution n ∞ is a fixed-point of the evolution map, that is,

n ∞ (x) = R 2 K (t , x, y) n ∞ (y) d y - t +1 t R 2 ∇K (t + 1 -s, x, y) • n ∞ (y)∇c ∞ (y) d yd s . ( 11
)
Buy doing the difference between ( 10) and ( 11), we have

n(t + 1, x) -n ∞ (x) = R 2 K (t , x, y) n(t , y) -n ∞ (y) d y - t +1 t R 2 ∇K (t + 1 -s, x, y) n(s, y) ∇c(s, y) d y -n ∞ (y) ∇c ∞ (y) d s .
Hence

n(t + 1, x) -n ∞ (x) L ∞ (R 2 ) ≤ K (t , x, y) L ∞ (R 2 x ;L r (R 2 y )) n(t , x) -n ∞ L 1 (R 2 ) + 1 0 ∇K (s, x, y) L ∞ (R 2 x ;L r (R 2 y )) d s R(t ) where 1 p + 1 q + 1 r = 2 with p ∈ (2, ∞), q ∈ [2, ∞), r ∈ (1, 2), and
R(t ) := sup s∈(t ,t +1) n(s, •) L p (R 2 ) ∇c(s, •) -∇c ∞ L q (R 2 ) + ∇c ∞ L q (R 2 ) n(s, •) -n ∞ L p (R 2 ) . ( 12
)
Notice that

∇K = µ 2 (e -t y -x) 2 π (1 -e -2t ) e - µ |x-e -t y| 2 2 (1-e -2t )
allows us to compute

∇K L r (R 2 y ) = µ 2 2 π (1 -e -2t ) R 2 |x| r e - µ r |x| 2 2 (e 2t -1) d x 1 r
= κ(r ) e 3t e 2t -1 µ

-3 2 + 1 r where κ(r ) = ∞ 0 x r e -1 2 x 2 d x 1 r . So ∇K L r (R 2 y
) is integrable in t ∈ (0, 1) if and only if 1 ≤ r < 2. From Proposition 2.2, R(t ) converges to 0, which completes the proof.

COERCIVITY RESULT OF QUADRATIC FORMS

In this section, we study the quadratic forms associated with the free energy F and the Fisher information I when we Taylor expand these functionals around the stationary solution (n ∞ , c ∞ ) defined by ( 4). Let us consider a smooth perturbation

n = f n ∞ of n ∞ such that R d f n ∞ d x = 0 and suppose that g c ∞ := (-∆) -1 ( f n ∞ ). We define Q 1 [ f ] := lim ε→0 2 ε 2 F [n ∞ (1 + ε f )] = R d f 2 n ∞ d x + R d |∇(g c ∞ )| 2 d x , Q 2 [ f ] := lim ε→0 2 ε 2 I [n ∞ (1 + ε f )] = R d |∇( f + g c ∞ )| 2 n ∞ d x .

A spectral gap inequality.

According to [1, Section 3.2], if the potential φ satisfies (C1), (C2) and (C3), then there exists a positive constant C , such that

R d |∇h| 2 n ∞ d x ≥ C R d h 2 n ∞ d x ∀ f ∈ H 1 (R d n ∞ d x) such that R d h n ∞ d x = 0 . (13)
Here n ∞ is the stationary solution given by (4).

f ∈ H 1 (R d , n ∞ d x) such that R d f n ∞ d x = 0, we have Q 2 [ f ] ≥ C Q 1 [ f ] .
Proof. We apply [START_REF] Cheng-Zhi | A Course of the Ordinary Differential Equation[END_REF]

to h(x) = f (x) + g c ∞ (x) -1 M R d g c ∞ n ∞ d x. Notice that R d h(x) n ∞ d x = 0 from R d f n ∞ d x = 0 and R d n ∞ (x) d x = M . So we obtain that Q 2 [ f ] = R d |∇( f + g c ∞ )| 2 n ∞ d x ≥ C R d ( f + g c ∞ ) 2 n ∞ d x - C M R d g c ∞ n ∞ d x 2 = C R d f ( f + g c ∞ ) n ∞ d x + C R d g c ∞ ( f + g c ∞ ) n ∞ d x - C M R d g c ∞ n ∞ d x 2 = C Q 1 [ f ] + C R d f n ∞ g c ∞ d x + C R d (g c ∞ ) 2 n ∞ d x - C M R d g c ∞ n ∞ d x 2 .
Let us study the term

R d f n ∞ g c ∞ d x. Obviously f n ∞ is in L 2 (R d ) be- cause n ∞ is bounded.
Moreover, for any p ∈ (1, 2), from Hölder's inequality, we infer that ), we also know that R d f n ∞ g c ∞ d x is well defined and learn from [START_REF] Blanchet | Two-dimensional Keller-Segel model: optimal critical mass and qualitative properties of the solutions[END_REF] that the function ∇(g c ∞ ) is bounded in L 2 (R 2 ) using the fact that R d f n ∞ d x = 0. In a word, this means that

R d | f | p n p ∞ d x ≤ R d f 2 d x p 2 R d n 2p 2-p ∞ d x 2-p 2 < ∞ because n ∞ ∈ L 1 ∩ L ∞ (R d ). When d = 3,
R d f n ∞ g c ∞ d x = R d |∇g c ∞ | 2 d x for d = 2 or 3. Next, let us notice that C R d (g c ∞ ) 2 n ∞ d x - C M R d g c ∞ n ∞ d x 2 = C M R d (g c ∞ ) 2 n ∞ d x R d n ∞ d x - C M R d g c ∞ n ∞ d x 2
is nonnegative by Hölder's inequality. Altogether, we conclude that

Q 2 [ f ] ≥ C Q 1 [ f ] + C R d |∇(g c ∞ )| 2 d x ≥ C Q 1 [ f ] .
f ∈ H 1 (R 2 , n ∞ d x) such that R 2 f n ∞ d x = 0, we have Q 2 [ f ] ≥ µQ 1 [ f ] .
Proof. We establish the proof into three steps.

Step 1. Radially symmetric functions and cumulated densities. We first consider the case of a spherically symmetric function f . The probelm is reduced to solving an ordinary differential equation, for which we use a reformulation in terms of cumulated densities. Let

Φ(s) := 1 2 π B (0, s) n ∞ (x) d x , φ(s) := 1 2 π B (0, s) ( f n ∞ )(x) d x and Ψ(s) := 1 2 π B (0, s) c ∞ (x) d x , ψ(s) := 1 2 π B (0, s) (g c ∞ )(x) d x .
Notice that n ∞ and c ∞ are both radial, so they can be regarded as functions of r = |x|. We can easily infer that

n ∞ ( s) = 2 Φ (s) , n ∞ ( s) = 4 s Φ (s) and c ∞ ( s) = 2 Ψ (s) , c ∞ ( s) = 4 s Ψ (s) . The Poisson equation -sc ∞ ( s) = Φ(s) can henceforth be rephrased as -4 s Ψ = Φ ( 14
)
while the equation for the density,

n ∞ ( s) + µ sn ∞ ( s) + n ∞ ( s)c ∞ ( s) = 0 , is now equivalent to Φ + µ 2 Φ + 2 Φ Ψ = 0 . ( 15
)
After eliminating Ψ from ( 14) and ( 15), we can get that Φ satisfies the ordinary differential equation

Φ + µ 2 Φ - 1 2 s Φ Φ = 0 ( 16
)
with initial data Φ(0) = 0 and Φ (0) = a. The solutions of the ODE are parameterized in terms of a > 0.

Let us consider the linearized operator

L f := 1 n ∞ ∇ • f n ∞ ∇(g c ∞ ) .
If f solves -L f = λ f , computations similar to the above ones show that

(n ∞ f )( s) = 2 φ (s) , (n ∞ f )( s) = 4 s φ (s) -2 n ∞ n ∞ φ (s)
which is equivalent to

(g c ∞ )( s) = 2 ψ (s) , (g c ∞ ) ( s) = 4 s ψ (s) . (17)
Using [START_REF] Herczak | Existence and asymptotics of solutions of the Debye-Nernst-Planck system in Rˆ2[END_REF], we find that

-s (g c ∞ ) ( s) = φ(s) , s (n ∞ f )( s) + n ∞ (g c ∞ ) ( s) + λ φ(s) = 0 .
After eliminating Ψ and ψ, we get that Φ and φ satisfy the equation

φ + µ s -Φ 2 s φ + λ -2 Φ 4 s φ = 0 . ( 18
)
Next we check that φ = s Φ (s) is a nonnegative solution of ( 18) with λ = 2 µ. In fact, ( 18) is equivalent to

2 s φ + (µ s -Φ) φ + (µ -Φ ) φ = 0 which is 2 (s φ -φ) + (µ s -Φ) φ = 0 . notice that when φ = s φ , 2(s φ -φ) + (µ s -Φ) φ = s 2 s Φ + (µ s -Φ) φ = 0 .
Hence λ = 2 µ is an eigenvalue of the linearized operator L f .

Step 2. Characterization of the radial ground state. Let us prove that 2 µ is the lowest positive eigenvalue corresponding to a radial eigenfunction. Assume by contradiction that L admits an eigenvalue λ ∈ (0, 2 µ) with eigenfunction f 1 and define the corresponding function φ 1 that satisfy [START_REF] Kinderlehrer | A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations[END_REF]. Let us consider various cases depending on the zeros of φ.

• Assume that φ 1 is always strictly positive or strictly negative in (0, ∞). Suppose without losing generality that φ 1 (s) > 0 in (0, ∞). On the one hand, if we multiply [START_REF] Kinderlehrer | A Wasserstein gradient flow approach to Poisson-Nernst-Planck equations[END_REF] written for the eigenvalue 2 µ and for the eigenvalue λ respectively by φ 1 and φ, we obtain that

φ 1 φ - Φ Φ φ 1 φ + 2 µ -2 Φ 4 s φ 1 φ = 0 , φ φ 1 - Φ Φ φ φ 1 + λ -2 Φ 4 s φ φ 1 = 0 .
By subtracting the second identity from the first one, we have

φ 1 φ(s) -φ 1 φ (s) Φ (s) ∞ 0 = (0,∞) 2 µ -λ 4 s φ φ 1 d s > 0 . ( 19
)
On the other hand, define

h(s) := 1 2 π B (0, s) f 2 1 n ∞ (r )d r .
From the cumulated mass formulation of Step 1, we find that

h (s) = 1 2 f 2 1 n ∞ ( s) = 2 Φ (s) 2 n ∞ ( s)
is in L 1 (0, ∞). So, for some constant κ > 0, we have

φ 1 (s) 2 = (s,∞) φ 1 (s) d s 2 ≤ (s,∞) φ 1 (s) 2 n ∞ ( s) d s (s,∞) n ∞ ( s) d s ≤ κ (s,∞) s -α 2 e -s 2 d s ≤ κe -µ s 4
when s is large enough. As a consequence, we known that

lim s→∞ φ 1 (s) = lim s→∞ φ(s) = lim s→∞ s φ(s) = 0 .
We also claim that lim

s→∞ s φ 1 (s) = 0 . (20)
In fact, for any large enough x 1 , x 2 , by integrating on (x 1 , x 2 ), we have

φ 1 (x 2 ) -φ (x 1 ) + µ 2 φ 1 (x 2 ) -φ 1 (x 1 ) - Φ 2 s φ 1 (x 2 ) -φ 1 (x 1 ) - x 2 x 1 φ 1 s φ -Φ 2s 2 d s + (x 1 .x 2 ) λ -Φ 4 s φ 1 d s = 0 .
Using again that φ 1 (s) ≤ κe -µ s 4 , we get that there exists a constant c 2 which is independent of x 1 and x 2 , such that |φ 1 (x 2 ) -φ 1 (x 1 )| ≤ c 2 . So φ 1 (s) is bounded. As a result, φ 1 (s) is also bounded, with a bound c 3 . If [START_REF] Lions | Equations differentielles operationnelles: et problémes aux limites[END_REF] is not true, then there exists a constant c 1 and a strictly increasing, diverging sequence (s k ) k∈N such that s k φ 1 (s k ) ≥ c 1 . For any interval (s k , ∞), we have that

c 1 s k ≤ C e -µ s k 4
which is impossible as k → ∞. So from [START_REF] Lions | Equations differentielles operationnelles: et problémes aux limites[END_REF], we obtain that

lim s→∞ φ 1 φ(s) -φ 1 φ (s) Φ (s) = lim s→∞ s φ 1 -φ 1 1 + s φ Φ = lim s→∞ s φ 1 -φ 1 1 - µ s -Φ 2 = 0 .
(21) From ( 19), [START_REF] Liu | A free energy satisfying finite difference method for Poisson-Nernst-Planck equations[END_REF], we have

0 = φ 1 φ(s) -φ 1 φ (s) Φ (s) ∞ 0 = (0,∞) 2 µ -λ 4 s φ φ 1 d s > 0 a contradiction.
• Assume that φ 1 has a zero in (0, ∞). By Sturm comparison theorem (see [START_REF] Cheng-Zhi | A Course of the Ordinary Differential Equation[END_REF]), we get that φ(s) = s Φ (s) has a zero in (0, ∞). It means that

n ∞ ( s) = 2 Φ (s)
has a zero between (0, ∞). But according to the definition of n ∞ , it is impossible. Hence we have shown that 2 µ is the best constant.

Step 3. Spherical harmonics decomposition.

We now deal with the non-radial modes of L . Notice that n ∞ and c ∞ are radial functions: we can use a spherical harmonics decomposition as in [START_REF] Campos | Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane[END_REF]. In dimension d = 2, we use radial coordinates and a Fourier decomposition for the angular variables. On the k t h mode we can write the operator L corresponding to the radial functions f and g as

-f - f r + k 2 f r 2 + µ r + c ∞ f + (g c ∞ ) -n ∞ f = λ f , -(g c ∞ ) - (g c ∞ ) r + k 2 g c ∞ r 2 = n ∞ f , for any integer k ≥ 1,
It is obvious that in non-radial functions, k = 1 realizes the infimum of the spectrum of L . We now check that when k = 1, λ = µ and f = -n ∞ /n ∞ is an eigenstate. In fact, we can choose g c ∞ = -c ∞ , so that f = µ r + c ∞ , and notice that

-c ∞ - c ∞ r = n ∞ , f + f r = 2 µ + c ∞ + c ∞ r = 2 µ -n ∞ ,
for the first equation, and

-f - f r + k 2 f r 2 + µ r + c ∞ f + (g c ∞ ) + n ∞ f = -f + f r -n ∞ + µ µ r + c ∞ = µ f
for the second equation, while

-(g c ∞ ) - (g c ∞ ) r + k 2 g c ∞ r 2 = -c ∞ + c ∞ r = -n ∞ = n ∞ f .
It is easy to prove that f is nonnegative and that f 1 (r ) := r f (r ) solves -L f 1 = (λ + µ) f 1 among the radial functions: we are back to the Step 2and find that λ = µ. Let us summarize: the spectral gap λ associated with the operator L is achieved either among radial functions and λ = 2 µ in this sense, or it is achieved among the functions in one of the non-radial components (in the sense of harmonics decomposition), which has to be the k = 1 component, and in that case we have found that λ + µ = 2 µ, that is λ = µ. Obviously λ = µ is optimal, which completes the proof of Lemma 3.1.

LINEARIZED EQUATION AND THE LARGE TIME BEHAVIOUR

This section is primarily devoted to the proof Theorem 1.1 but also collects some additional results.

4.1. The scalar product and the linearized operator. We adapt the strategy of [START_REF] Campos | Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane[END_REF]. Notice that

f 1 , f 2 := R d f 1 f 2 n ∞ d x + R d n ∞ f 1 G d * ( f 2 n ∞ ) d x ( 22
)
is a scalar product on the admissible set

A := f ∈ L 2 (R d , n ∞ d x) : R d f n ∞ d x = 0 because Q 1 [ f ] = f , f
. Now come back to the Poisson-Nernst-Planck system with confinement (2). For any x ∈ R d and t ≥ 0, let us set

n(t , x) = n ∞ (x) 1 + f (t , x) , c(t , x) = c ∞ 1 + g (t , x)
and rewrite the evolution problem in terms of f and g as

n ∞ ∂ f ∂t = ∆(n ∞ f )+∇• n ∞ f ∇φ +∇• n ∞ ∇(c ∞ g ) + n ∞ f ∇c ∞ + n ∞ f ∇(c ∞ g ) .
After observing that

∆(n ∞ f ) + ∇ • (n ∞ f ∇φ) + ∇ • (n ∞ f ∇c ∞ ) = ∇ • (n ∞ ∇ f ) , it turns out that n ∞ ∂ f ∂t = ∇ • (n ∞ ∇ f ) + ∇ • n ∞ ∇(c ∞ g ) + ∇ • n ∞ f ∇(c ∞ g ) . Hence ( f , g ) solves    ∂ f ∂t -L f = 1 n ∞ ∇ • f n ∞ ∇(g c ∞ ) -∆(g c ∞ ) = f n ∞ x ∈ R d , t > 0 ( 23
)
for any x ∈ R d , t ≥ 0, where the linear operator L is defined by

L f := 1 n ∞ ∇ n ∞ ∇ f + g c ∞ .
Lemma 4.1. The linearized operator L is self-adjoint on A with the scalar product defined in [START_REF] Nernst | Elektromotorische Wirksamkeit der Jonen[END_REF], which means that f

1 , L f 2 = L f 1 , f 2 for any f 1 , f 2 ∈ A , and moreover, -f , L f = Q 2 [ f ] for any f ∈ A . Proof. Set g 1 c ∞ = (-∆) -1 ( f 1 n ∞ ), g 2 c ∞ = (-∆) -1 ( f 2 n ∞ ). By direct com- putation, we obtain that L f 1 , f 2 = R d f 2 ∇ • n ∞ ∇( f 1 + g 1 c ∞ ) d x + R d g 2 c ∞ ∇ • n ∞ ∇( f 1 + g 1 c ∞ ) d x = - R d n ∞ ∇( f 1 + g 1 c ∞ ) • ∇( f 2 + g 2 c ∞ ) d x ,
which proves the lemma.

Proof of Theorem 1.1.

Proof. For the equations ( 23), we find that

d d t Q 1 [ f ] = -2Q 2 [ f ]-2 λ(t ) with λ(t ) := R d ∇( f + g c ∞ ) • f n ∞ ∇(g c ∞ ) d x .
According to the Cauchy-Schwarz inequality, we have that

(λ(t )) 2 ≤ Q 2 [ f ] R d f 2 n ∞ d x ∇(g c ∞ ) 2 L ∞ (R d ) ≤ Q 2 [ f ]Q 1 [ f ] ∇(g c ∞ ) 2 L ∞ (R d ) .
So we obtain

d d t Q 1 [ f ] ≤ -2 1 - ∇(g c ∞ ) L ∞ (R d ) C * Q 2 [ f ] ≤ -2 C * 1 - ∇(g c ∞ ) L ∞ (R d ) C * Q 1 [ f ] .
We know from Proposition 2.2 that lim t →+∞ ∇(g c ∞ ) L ∞ (R d ) = 0, which proves that lim sup

t →∞ e 2 (C * -ε) Q 1 [ f (t , •)] < ∞
for any ε ∈ (0, C * ). It remains to prove that we can also obtain this estimate with ε = 0. Suppose that C * is the optimal constant without losing generality. Let us give a more accurate estimate of λ(t ). If d = 2, according to [START_REF] Biler | The Debye system: existence and large time behaviour of solutions[END_REF]

applied to ρ = f n ∞ , we have ∇(g c ∞ ) L ∞ ≤ C ( f n ∞ L 1 + f n ∞ L 3 )
where

f n ∞ L 1 ≤ M f n ∞ L 2 , f n ∞ L 3 ≤ f n ∞ 2 3 L 2 f n ∞ 1 3 L ∞ n ∞ 1 3 L ∞ . Notice that from f n ∞ 2 L 2 ≤ Q 1 [ f ], we deduce that ∇(g c ∞ ) L ∞ = O Q 1 [ f (t , •)] 1 3 (24)
which leads to

λ(t ) ≤ O Q 1 [ f (t , •)] 4 3
as t → +∞ .

As a result, we read from

d d t Q 1 [ f ] ≤ -2 C Q 1 [ f ] + O (Q 1 [ f ]) 4 3 that lim sup t →∞ e 2 C t Q 1 [ f (t , •)] < ∞ .
When d = 3, we have the estimate

∇(g c ∞ ) L ∞ ≤ C f n ∞ L 1 + f n ∞ L 4
and similarly obtain that [START_REF] Rudin | Principles of mathematical analysis[END_REF] which allows us to write that

f n ∞ L 1 ≤ M f n ∞ L 2 , f n ∞ L 4 ≤ f n ∞ 1 2 L 2 f n ∞ 1 2 L ∞ n ∞ 1 4 L ∞ . Using again f n ∞ 2 L 2 ≤ Q 1 [ f ], we have ∇(g c ∞ ) L ∞ = O Q 1 [ f (t , •)] 1 4
λ(t ) ≤ O Q 1 [ f (t , •)] 5 4
as t → +∞ . We conclude as above, which completes the proof of Theorem 1.1. Proof. From the Cauchy-Schwarz inequality, we read that

n(t , •) -n ∞ L 1 (R d ) ≤ n ∞ L 1 (R d ) R d |n(t , •) -n ∞ | 2 n ∞ d x 1 2 ≤ C M e -t
for some C > 0 if t is taken large enough, and we also know also that

n(t , •) -n ∞ L p (R d ) = O e -t p ( 26
)
for any p ∈ [1, ∞). By definition of Q 1 [ f ], we have

∇c(t , •) -∇c ∞ L 2 (R d ) ≤ C e -t
for some C > 0 if t is taken large enough, according to Lemma 3.1. Moreover, according to [START_REF] Planck | Ueber die erregung von electricität und wärme in electrolyten[END_REF], [START_REF] Rudin | Principles of mathematical analysis[END_REF] and Theorem 1.1, we obtain that

∇c(t , •) -∇c ∞ L ∞ (R d ) = O e -t d +1 . ( 27
)
This proves that

∇c(t , •) -∇c ∞ L q (R d ) = O e - t (q+2d ) (d +1)q
for any q ∈ [2, ∞) by interpolating between [START_REF] Schönke | Unsteady analytical solutions to the Poisson-Nernst-Planck equations[END_REF] and (27).

The proof of the case d = 2 is inspired by [START_REF] Campos | Asymptotic estimates for the parabolic-elliptic Keller-Segel model in the plane[END_REF]Remark 5]. We reconsider R(t ) defined in [START_REF] Debye | De la theorie des electrolytes. I. abaissement du point de congelation et phenomenes associes[END_REF] in Section 2 with p = 7r 5r -4 , q = 7r 2r -3 . We obtain from Corollary 4.1 that

n(t , •) -n ∞ L ∞ (R 2 ) = O e -5r -4 7r t .
This is the first step of a proof by induction. If

n(t , •) -n ∞ L ∞ (R 2 )
= O e -at , then one has

n(t , •) -n ∞ L ∞ (R 2 )
= O e -5r -4+(2r +4)a 7r t .

By iterating this estimate infinitely many times, we finally have

n(t , •) -n ∞ L ∞ (R 2 ) = O e -λt
for any λ < 1. The proof of the corollary is complete.

u ∞ (x, t ) = 1 1 + 2t n ∞ 1 2 log(1 + 2t ), x 1 + 2t , ( 28
) v ∞ (x, t ) = c ∞ 1 2 log(1 + 2t ), x 1 + 2t , ( 29
)
where (n ∞ , c ∞ ) are the stationary solutions of (2) given by ( 4) with the harmonic potential φ(x) =

u(t , •) -u ∞ L 1 (R 2 ) = O (1 + 2t ) -1 2 , u(t , •) -u ∞ L p (R 2 ) = O (1 + 2t ) -λ 2 - d (p-1) 2p
, (ii) for any q ∈ (2, ∞) and any λ < 1,

∇v(t , •) -∇v ∞ L 2 (R 2 ) = O (1 + 2t ) -1+ d 4 , ∇v(t , •) -∇v ∞ L q (R 2 ) = O (1 + 2t ) -λ+1 2 + d 2q .

2. 1 .Lemma 2 . 1 .

 121 Minimizers of the free energy and stationary solutions. Let d = 2 or d = 3 and assume that the potential φ satisfies (C). On the set

Lemma 2 . 2 .

 22 the result has been established in [? , Corollary 1.2] as a consequence of Assumption (C4). Let d = 2 or d = 3 and assume that the potential φ satisfies (C). There exists a unique minimizer n ∞ of F in X .

Corollary 2 . 1 .

 21 Let d = 2 or d = 3 and assume that the potential φ satisfies (C). For any solution n of (2) with initial datum n 0

Proposition 3 . 1 .

 31 Let d = 2 or d = 3 and assume that the potential φ satisfies (C). Then for any

  we directly obtain from the Hardy-Littlewood-Sobolev inequality thatR d f n ∞ g c ∞ d x is well defined and equal to R d |∇g c ∞ | 2 d x. When d = 2, by log-Hölder interpolation, | f n ∞ | log | f n ∞ | is integrable.From the logarithmic Hardy-Littlewood-Sobolev inequality (see[? ]

3. 2 .Lemma 3 . 1 .

 231 Optimal spectral gap in a special case. As a conclusion, let us give the optimal coercivity constant in the special case that the dimension d = 2 and the harmonic function φ = µ 2 |x| 2 , µ > 0, Suppose that d = 2, φ = µ 2 |x| 2 , where µ > 0. Then for any

4. 3 .Corollary 4 . 1 .

 341 Uniform rate of convergence. Let us give additional results on the convergence in various norms of the solution of (2) to the stationary solution. Under the assumptions of Theorem 1.1, if φ(x) = 1 2 |x| 2 , the solution n of (2) is such that n(t , •)n ∞ L p = O e -t p and ∇c(t , •) -∇c ∞ L q = O e -t (q+2 d ) (d +1) q as t → +∞, for any p ∈ (1, ∞) and any q ∈ (2, ∞). Additionally, if d = 2, then n(t , •)n ∞ L ∞ = O e -λtas t → +∞, for any λ < 1.

4. 4 .

 4 Intermediate asymptotics of the Nernst-Planck equation with Poisson term. Let us come back to the equation (1). The self-similar solution of (1) has the expression

1 2 |x| 2 .Theorem 4 . 1 .

 241 Using Theorem 1.1 and Corollary 4.1, we achieve a result on the intermediate asymptotics for the solutions of the Nernst-Planck equation with Poisson term in absence of any external potential of confinement. Assume that u solves (1) with initial datum u(0, •) = n 0 ∈ L 2 + (n -1 ∞ d x), R d n 0 d x = M , and F [n 0 ] < ∞.Let us consider the self-similar solution defined by[START_REF] Zinsl | Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion[END_REF] and (29) of mass M . Then, as t → +∞, we have (i) for any p ∈ (1, ∞) and any λ < 1,

Uniform convergence in L ∞ norm in the harmonic potential case.

  The issue of the convergence of n(t , •) to n ∞ and of ∇c(t , •) to ∇c ∞ in L ∞ (R d ) was left open in Section 2.3. As in the case of the Keller-Segel model, see[START_REF] Blanchet | Asymptotic behaviour for small mass in the two-dimensional parabolic-elliptic Keller-Segel model[END_REF], better results can be achieved in the case of the harmonic potential.

t >0 which converges as t → +∞ to n ∞ and lim t →+∞ F [n(t , •)] = F [n ∞ ], then proving by the Csiszár-Kullback inequality that lim t →∞ n(t , •)-

n ∞ L 1 (R d ) = 0 (see [? ]) and lim t →∞ ∇c(t , •) -∇c ∞ L 2 (R d ) = 0.

The result for any p ∈ [1, ∞) and any q ∈ [2, ∞) follows by Hölder interpolation. 2.4. Proposition 2.3. Set d = 2, φ = µ 2 |x| 2 , for some µ > 0. Then for any solution n of (2) is such that

© 2019 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes.

Email address: li@ceremade.dauphine.fr

Acknowledgments

This work has been supported by the Project EFI ANR-17-CE40-0030 of the French National Research Agency.