Relative entropies for the Vlasov–Poisson system in bounded domains

Abstract Using relative entropies we study irreversibility for the Vlasov–Poisson system with injection conditions on the boundary with or without collisions. If the solution converges for large times, this allows to deduce that the limit has a given trace on the boundary. In the one-dimensional collisionless case and under strong regularity assumptions, this is possible only for the unique stationary solution.


Relative entropies for the Vlasov-Poisson system in bounded domains
Abstract. Using relative entropies we study irreversibility for the Vlasov-Poisson system with injection conditions on the boundary with or without collisions. If the solution converges for large times, this allows to deduce that the limit has a given trace on the boundary. In the one dimensional collisionless case and under strong regularity assumptions, this is possible only for the unique stationary solution. c Académie des Sciences/Elsevier, Paris

Preliminaries and examples
Consider the Vlasov-Poisson system for charged particles transport (with zero or non zero collision term Q(f )) is the measure induced by Lebesgue's measure on Γ. ∇ x φ 0 is an external electrostatic field, whereas ∇ x φ accounts for self-consistent effects. We say that γ has Property (P) if and only if it is defined on (min x∈ω φ 0 (x), +∞), bounded, smooth and strictly decreasing with values in IR + * , and rapidly decreasing at infinity: sup x∈ω +∞ 0 We denote by γ (−1) the inverse function of γ extended by an arbitrary, fixed, strictly decreasing function to IR. For any function g ∈ L 1 (Ω), Q is a collision term preserving the mass: Q(f ) dv = 0. When Q ≡ 0, we define the following stationary solution Note that M is uniquely defined since U [M ] is the unique critical point in H 1 0 (ω) of the strictly convex coercive functional When Q ≡ 0, we still use the same definition for M and require the following property (compatibility of the collisions with the boundary conditions) In this note, we study the irreversibility due to the boundary conditions and compatible collisions,  [2,3] for related results.

Example 3. BGK approximation of the Boltzmann collision operator for fermions
σ is a nonnegative symmetric cross-section and M 0 (v) = (2πθ 0 ) −d/2 exp[− |v| 2 2θ0 ] is a fixed Maxwellian function with a given temperature θ 0 > 0. Here α is a nonnegative parameter (the case α < 0 also makes sense in the context of bosonic particles), γ(u) = (α + exp((u − µ 0 )/θ 0 )) −1 and γ (−1) (f ) = µ 0 − θ 0 log( f 1−αf ). Example 4. Linear elastic collision operator χ is a symmetric positive cross-section, and similarly to Example 1, the choice of γ is left arbitrary. Indeed, for any given function f in L 1 (IR d ), any bounded continuous function ψ and any nondecreasing function H, Consequently, (H1) is satisfied for any function γ having Property (P) and the kernel of Q is spanned by the functions depending only on |v| 2 . Further examples will be provided in a forthcoming paper [4]. The main ingredient we use here is the construction of a relative entropy depending on the boundary data. This denomination generalizes the usual notion of entropy for the Boltzmann equation and tends to be used for other kinetic models or even for parabolic equations (see [5,6]).

Relative entropy and irreversibility
Let us define the relative entropy of two functions g, h of the (x, v) variables by where β γ is the real function defined by β γ (g) = − g 0 γ (−1) (z) dz. Since γ is strictly decreasing, it is readily seen that Σ γ [g|h] is always nonnegative and vanishes if and only if g = h a.e.
Theorem 1 -Let f be a solution of (1) and assume that γ and Q satisfy the properties (P) and (H1) respectively. Then the relative entropy Σ γ [f (t)|M ] where M is defined by (2) satisfies Here Σ + γ is the boundary entropy flux defined by Σ + To prove the theorem, we first recall the following identities satisfied by renormalized solutions of the Vlasov equation [7] (5) Identity (3) is obtained, after some simple calculations, by summing (4) and (5) (with β = β γ ) and taking advantage of (2), and of the identity f = M on Γ − . Since Σ γ [g|h] is always nonnegative, the above theorem provides a uniform in time control on f (t). Like in whole space problems [5,6,8], we obtain a Lyapunov functional for the study of the long time behaviour. The main difference with whole space problems and with previous studies of boundary value problems [2,3] is that the total mass is not conserved.

The characterization of the limit Problem
The rigorous treatment of how to pass to the limit in time is beyond the scope of this note. We shall only give the general strategy. We assume that This is satisfied in each of the examples of Section 1. When Q ≡ 0 or Q = Q E , the solution f (t) is uniformly bounded in L ∞ , and we may pass to the limit.
Corollary 1 -Let γ and Q satisfy Property (P) and (H1)-(H2) respectively, and consider a solution f of (1) such that the initial datum is bounded in L 1 ∩ L ∞ (Ω). Consider an unbounded increasing sequence (t n ) n∈I N and (f n , φ n ) defined by When there is no L ∞ uniform bound on f and for γ sufficiently decreasing, in the sense that lim g→+∞ (β γ (g)/g) = +∞, it is still possible to pass to the limit in the equation in some cases (Vlasov-Poisson-Fokker-Planck system or BGK equation, for instance) using renormalized solutions. To identify the limit as t → +∞, one has to prove that Q(f n ) → Q(f ∞ ), which is not straightforward when Q is nonlinear (example 3 with α = 0). This difficulty is not specific to bounded domains.

Solutions with a given trace at the boundary
In this section, we prove that the solution of the limit problem is stationary. If f ∞ ∈ Ker Q depends only on |v| 2 (examples 2, 3 and 4), we may apply the Lemma 1 -Let f be a solution of the Vlasov equation. If f is even (or odd) with respect to the v variable, then it does not depend on t.
For the pure Vlasov-Poisson system (Q ≡ 0) proving that f ∞ is stationary is an interesting open problem. It is true when d = 1 if the potential is analytic.
Theorem 2 -Assume that γ satisfies Property (P) and consider a solution (f, φ) of the Vlasov-Poisson system on the interval ω = (0, 1) such that f (x, v, t) = γ( 1 2 |v| 2 + φ 0 (x)) for any t ∈ IR + , (x, v) ∈ Γ and assume that φ is analytic in x with C ∞ (in time) coefficients. If −∆φ 0 ≥ 0 on ω, Analyticity results are available only for whole space or periodic evolution problems [9] (note that analyticity is the standard framework for the study of the Landau damping [1,10,11]). We are not aware of any analyticity result for boundary value problems. In the above theorem, the assumption on φ 0 is made only to avoid closed characteristics [12]. We shall refer to [4] for a complete proof and will only provide a sketch of the computations. Let us write φ(x, t) = +∞ n=0 a n (t)x n with a n ∈ C ∞ . The first term a 0 (t) vanishes identically because of the boundary conditions and inf t>0 a 1 (t) > 0 by Hopf's lemma.
Next, we shall consider particles injected in the domain with a small velocity at x = 0: their exit time is small too (φ + φ 0 is strictly concave). As long as v does not change sign, we may parametrize the characteristics with x by writing dv dt = ∂φ ∂x (x, t) and dv 2 dx = −2 ∂φ ∂x (x, t) and using dt = dx/v. Letting e ± (x) = |v| 2 2 (x) for particles going forward (+) and backward (−) respectively, we have dy e ± (y) ) , e ± (x 0 ) = 0 , where x 0 is the point where the velocity changes sign (at time t = t 0 ). The monotonicity of γ and the control of f on the whole boundary Γ implies that e + (0) = e − (0) for any x 0 > 0, small, and an expansion in ǫ = √ x 0 shows that ∂ ∂t ∂ n φ ∂x n (0, t) = 0 for any t ∈ IR + . ⊔ ⊓ As a concluding remark, we notice that if the stationary Vlasov-Poisson problem has a unique solution (this is true for example in dimension d = 1, when −∆φ 0 ≥ 0), this solution is the long time limit of f (t). The results contained in this note can be extended to relativistic or periodic in momentum models simply by replacing |v| 2 /2 by an energy ε(k) depending on the wave vector (or momentum) k, v by ∇ k ε(k) and dv by dk, provided k → ε(k) is even, smooth and non degenerate.